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Non-commutative differential geometry

Non-commutative differential geometry by ALAIN Introduction This is the introduction to a series of papers in which we shall extend the calculus of differential forms and the de Rham homology of currents beyond their customary framework of manifolds, in order to deal with spaces of a more elaborate nature, such as, a) the space of leaves of a foliation, b) the dual space of a finitely generated non-abelian discrete group (or Lie group), c) the orbit space of the action of a discrete group (or Lie group) on a manifold. What such spaces have in common is to be, in general, badly behaved as point sets, so that the usual tools of measure theory, topology and differential geometry lose their pertinence. These spaces are much better understood by means of a canonically associated algebra which is the group convolution algebra in case b). When the space V is an ordinary manifold, the associated algebra is commutative. It is an algebra of complex-valued functions on V, endowed with the pointwise operations of sum and product. A smooth manifold V can be considered from different points of view such as o~) Measure theory (i.e. V appears as a measure space with a fixed measure class), 8) Topology (i.e. V appears as a locally compact space), y) Differential geometry (i.e. V appears as a smooth manifold). Each of these structures on V is fully specified by the corresponding algebra of functions, namely: ~) The commutative von Neumann algebra L~ of classes of essentially bounded measurable functions on V, 8) The &-algebra G0(V ) of continuous functions on V which vanish at infinity, y) The algebra G~~ of smooth functions with compact support. CO~N-~q'ES ALAIN CONNES It has long been known to operator algebraists that measure theory and topology extend far beyond their usual framework to A) The theory of weights and yon aVeumann algebras, B) C*-algebras, K-theory and index theory. Let us briefly discuss these two fields, A) The theory of weights and yon Neumann algebras To an ordinary measure space (X,$) correspond the von Neumann algebra L~~ ~) and the weight ~: r =fxfd VfeL~ X, ?t) +" Any pair (M, ~) of a commutative von Neumann algebra M and weight q~ is obtained in this way from a measure space (X, ~). Thus the place of ordinary measure theory in the theory of weights on von Neumann algebras is similar to that of commutative algebras among arbitrary ones. This is why A) is often called non.commutative measure theory. Non-commutative measure theory has many features which are trivial in the commutative case. For instance to each weight q~ on a yon Neumann algebra M corresponds canonically a one-parameter group a T e Aut M of automorphisms of M, its modular automorphism group. When M is commutative, one has aT(x ) = x, V t e R, V x e M, and for any weight ~ on M. We refer to [i 7] for a survey of non-commutative measure theory. B) C*-algebras, K.theory and index theory Gel'land's theorem implies that the category of commutative C*-algebras and .-homomorphisms is dual to the category of locally compact spaces and proper conti- nuous maps. Non-commutative C*-algebras have first been used as a tool to construct von Neumann algebras and weights, exactly as in ordinary measure theory, where the Riesz representation theorem [6o], Theorem 2.14, enables to construct a measure from a positive linear form on continuous functions. In this use of C*-algebras the main tool is positivity. The fine topological features of the " space " under consideration do not show up. These fine features came into play thanks to Atiyah's topological K-theory [2]. First the proof of the periodicity theorem ofR. Bott shows that its natural set up is non-commutative Banach algebras (cf. [7I]). Two functors K0, K 1 (with values in the category of abelian groups) are defined and any short exact sequence of Banach algebras gives rise to an hexagonal exact sequence of K-groups. For A = C0(X), the commutative C~ associated to a locally compact space X, Kj(A) is (in a natural manner) isomorphic to KJ(X), the K-theory with compact supports of X. 258 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Since (cf. [65] ) for a commutative Banach algebra B, Kj(B) depends only on the Gel'fand spectrum of B, it is really the C*-algebra case which is most relevant. Secondly, Brown, Douglas and Fillmore have classified (cf. [rx]) short exact sequences of (]*-algebras of the form o ---> Yf ~ A -+ C(X) ~o where ~ is the C*-algebra of compact operators in Hilbert space, and X is a compact space. They have shown how to construct a group from such extensions. When X is a finite dimensional compact metric space, this group is naturally isomorphic to KI(X), the Steenrod K-homology of X, cf. [24] [38]. Since the original classification problem of extensions did arise as an internal question in operator and C*-algebra theory, the work of Brown, Douglas and Fillmore made it clear that K-theory is an indispensable tool even for studying C*-algebras per se. This fact was further emphasized by the role of K-theory in the classification of C*-algebras which are inductive limits of finite dimensional ones (cf. [IO] [26] [27] ) and in the work of Cuntz and Krieger on C*-algebras associated to topological Markov chains ([22]). Finally the work of the Russian school, of Mi~enko and Kasparov in particular, ([5 ~ [42] [43] [44]), on the Novikov conjecture, has shown that the K-theory of non- commutative C*-algebras plays a crucial role in the solution of classical problems in the theory of non-simply-connected manifolds. For such a space X, a basic homotopy invariant is the F-equivariant signature cr of its universal covering X, where F = r~l(X ) is the fundamental group of X. This invariant a lies in the K-group, K0(C*(F)) , of the group C* algebra C*(F). The K-theory of C*-algebras, the extension theory of Brown, Douglas and Fillmore and the Ell theory of Atiyah ([3]) are all special cases of Kasparov's bivariant functor KK(A, B). Given two Z/2 graded C*-algebras A and B, KK(A, B) is an abelian group whose elements are homotopy classes of Kasparov A-B bimodules (cf. [42] [43]). For the convenience of the reader we have gathered in appendix 2 of part I the defi- nitions of [42] which are relevant for our discussion. After this quick overview of measure theory and topology in the non-commutative framework, let us be more specific about the algebras associated to the " spaces " occurring in a), b), c) above. a) Let V be a smooth manifold, F a smooth foliation of V. The measure theory of the leaf space " V/F " is described by the von Neumann algebra W*(V, F) of the foliation (cf. [I4] [I5] [I6]). The topology of the leaf space is described by the C*-algebra C*(V, F) of the foliation (cf. [i4] [I5] [66]). b) Let F be a discrete group. The measure theory of the (reduced) dual space is described by the von Neumann algebra ~,(F) of operators in the Hilbert space t*(F) which are invariant under right translations. This von Neumann algebra is the weak closure of the group ring CF acting in t2(F) by left translations. The topology of the 259 44 ALAIN CONNES (reduced) dual space F is described by the C*-algebra C,*(F), the norm closure of CF in the algebra of bounded operators in t~(F). b') For a Lie group G the discussion is the same, with C~(G) instead of CF. r Let F be a discrete group acting on a manifold W. The measure theory of the " orbit space " W/F is described by the von Neumann algebra crossed product L~(W) >q F (cf. [5I]). Its topology is described by the C*-algebra crossed product c0(v) r (of. [51]). The situation is summarized in the following table: Space V V/F F G W/F Measure theory L~~ W*(V, F) ;~(F) ),(G) Lc~ )~ F Topology Co(V ) C*(V, F) C;(F) C;(G) Co(W ) >~ F It is a general principle (cf. [5] [18] [7]) that for families of elliptic operators (Du)ve Y parametrized by a " space " Y such as those occurring above, the index of the family is an element of K0(A), the K-group of the C*-algebra associated to Y. For instance the F-equivariant signature of the universal covering X of a compact oriented manifold is the F-equivariant index of the elliptic signature operator on X. We are in case b) and , e K0(C;(F)). The obvious problem then is to compute K.(A) for the C*-algebras of the above spaces, and then the index of families of elliptic operators. After the breakthrough of Pimsner and Voiculescu ([54]) in the computation of K-groups of crossed products, and under the influence of the Kasparov bivariant theory, the general program of computation of the K-groups of the above spaces (i.e. of the associated C*-algebras) has undergone rapid progress in the last years ([16] [66] [52] [53] [68] [69] ). So far, each new result confirms the validity of the general conjecture formulated in [7]. In order to state it briefly, we shall deal only with case c) above (x). By a fami- liar construction of algebraic topology a space such as W/F, the orbit space of a discrete group action, can be modeled as a simplicial complex, up to homotopy. One lets I" act freely and properly on a contractible space EF and forms the homotopy quotient W � EP which is a meaningful space even when the quotient topological space W/I' is patho- logical. In case b) (F acting on W = {pt}) this yields the classifying space BF. In case a), see [I6] for the analogous construction. In [7] (using [16] and [18]) a map ix is defined from the twisted K-homology K.,,(W � EF) to the K group of the C'-algebra C0(W) >~ F: ~z : K,,,(W � EF) ~ K,(Co(W ) N V). The conjecture is that this map ~ is always an isomorphism. At this point it would be tempting to advocate that the space W � EF gives a sufficiently good description of the topology of W/F and that we can dispense with (x) And we assume that I" is discrete and torsion free, cf. [7] for the general case. 260 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY C*-algebras. However, it is already clear in the simplest examples that the C*-algebra A = G0(W ) :~ F is a finer description of the " topological space " of orbits. For instance, with W = S z and F = Z, the actions given by two irrational rotations R0, , R0, yield isomorphic C*-algebras if and only if 01 = 4- 02 ([54] [55]), and Morita equivalent C*-algebras if and only if 0 z and 02 belong to the same orbit of the action of PSL(2, Z) on PI(R) [58]. On the contrary, the homotopy quotient is independent of 0 (and is homotopic to the 2-torus). Moreover, as we already mentioned, an important role of a " space " such as Y = W/F is to parametrize a family of elliptic operators, (Dy)yey. Such a family has both a topological index Indt(D), which belongs to the twisted K-homology group K,.~(WXrEI`), and an analytic index Ind,(D) = tz(Ind~(D)), which belongs to K,(C0(W ) � F) (cf. [7] [20]). But it is a priori only through Inda(D ) that the analytic properties of the family (Dr)re Y are reflected. For instance, if each Dy is the Dirac operator on a Spin Riemannian manifold M u of strictly positive scalar curvature, one has Inda(D) ---= o (cf. [59] [2o]), but the equality Indt(D ) = o follows only if one knows that the map Vt is injective (cf. [7] [59] [2o]). The problem of injectivity of is an important reason for developing the analogue of de Rham homology for the above " spaces ". Any closed de Rham current C on a manifold V yields a map q~c from K*(V) to C q~c(e) = (G, che) VeeK*(V) where ch:K*(V) -+ H*(V, R) is the usual Ghern character. Now, any " closed de Rham current " G on the orbit space W/l` should yield a map q~o from K,(G0(W ) xl l') to G. The rational injectivity of tz would then follow from the existence, for each co e H*(W � EF), of a " closed current " G(co) making the following diagram commutative, K,,,(W � Er) ~--+ K,((Co(W) ~ r) ob, ---~C H,(W � El', R) Here we assume that W is F-equivariantiy oriented so that the dual Chern character ch, : K,,, ~H, is well defined (see [2o]). Also, we view co eH*(W � EF, C) as a linear map from H,(W � El-', R) to C. This leads us to the subject of this series of papers which is i. The construction of de Rham homology for the above spaces; 2. Its applications to K-theory and index theory. The construction of the theory of currents, closed currents, and of the maps q% for the above " spaces " requires two quite different steps. 261 46 ALAIN CONNES The first is purely algebraic: One starts with an algebra M over C, which plays the role of C~ and one develops the analogue of de Rham homology, the pairing with the algebraic K-groups K0(d), Kl(d), and algebraic tools to perform the computations. This step yields a eontravariant functor H~ from non commutative algebras to graded modules over the polynomial ring C(~) with a generator a of degree 2. In the definition of this functor the finite cyclic groups play a crucial role, and this is why H~ is called cyclic cohomology. Note that it is a contravariant functor for algebras and hence a covariant one for " spaces ". It is the subject of part II under the title, De Rham homology and non-commutative algebra The second step involves analysis: The non-commutative algebra d is now a dense subalgebra of a C*-algebra A and the problem is, given a closed current C on d as above satisfying a suitable conti- nuity condition relative to A, to extend q0 c : K0(M ) ~ C to a map from K0(A ) to C. In the simplest situation, which will be the only one treated in parts I and II, the algebra d C A is stable under holomorphic functional calculus (el. Appendix 3 of part I) and the above problem is trivial to handle since the inclusion M C A induces an isomorphism K0(d ) m K0(A ). However, even to treat the fundamental class of W/P, where I" is a discrete group acting by orientation preserving diffeomorphisms on W, a more elaborate method is required and will be discussed in part V (cf. [2o]). In the context of actions of discrete groups we shall construct C(co) and ~c(,~} for any cohomology class co e H*(W � EF, C) in the subring R generated by the following classes: a) Chern classes of F-equivariant (non unitary) bundles on W, b) F-invariant differential forms on W, c) Gel'fand Fuchs classes. As applications of our construction we get (in the above context): ~) If x e K,,~(W � EF) and (ch, x, co) # o forsome o~ in the above ring R then ~z(x) # o. In fact we shall further improve this result by varying W; it will then apply also to the case W = {pt), i.e. to the usual Novikov conjecture. All this will be discussed in part V, but see [2o] for a preview. ~) For any o~ e R and any family (Du)ue ~ of elliptic operators parametrized b2 Y = W/F, one has the index theorem: ~c(Ind=(D)) = (ch. Ind,(D), co). When Y is an ordinary manifold, this is the cohomological form of the Atiyah-Singer index theorem for families ([5]). 262 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY It is important to note that, in all cases, the right hand side is computable by a standard recipe of algebraic topology from the symbol of D. The left hand side carries the analytic information such as vanishing, homotopy invariance... All these results will be extended to the case of foliations (i.e. when Y is the leaf space of a foliation) in part VI. As a third application of our analogue of de Rham homology for the above " spaces " we shall obtain index formulae for transversally elliptic operators, that is, elliptic operators on those " spaces " Y. In part IV we shall work out the pseudo- differential calculus for crossed products of a C'-algebra by a Lie group (cf. [19]), thus yielding many non-trivial examples of elliptic operators on spaces of the above type. Let A be the C* algebra associated to Y, any such elliptic operator on Y yields a finitely summable Fredholm module over the dense subalgebra ~ of smooth elements of A. In part I we show how to construct canonically from such a Fredholm module a closed current on the dense subalgebra .~/. The title of part I, the Chern character in K-homology is motivated by the specialization of the above construction to the case when Y is an ordinary manifold. Then the K homology K.(V) is entirely described by elliptic operators on V ([9] [18]) and the association of a closed current provides us with a map, K,(V) --> H,(V, C) which is exactly the dual Chern character ch,. The explicit computation of this map oh, will be treated in part III as an intro- duction to the asymptotic methods of computations of cyclic cocycles which will be used again in part IV. As a corollary we shall, in part IV, give completely explicit formulae for indices of finite difference, differential operators on the real line. IfD is an elliptic operator on a " space" Y and C is the closed current C = oh, D (constructed in part I), the map ~c : K,(A) ---> s makes sense and one has 9c(E) ---- (E, [D] ) ---- Index D E V E e K,(A) where the right hand side means the index of D with coefficients in E, or equivalently the value of the pairing between K-homology and K-cohomology. The integrality of this value, Index D E e Z, is a basic result which will be already used in a very efficient way in part I, to control K,(A). The aim of part I is to show that the construction of the Chern character ch. in K homology dictates the basic definitions and operations---such as the suspension map S--in cyclic cohomology. It is motivated by the previous work of Helton and Howe [3o], Carey and Pincus [12] and Douglas and Voiculescu [25]. There is another, equally important, natural route to cyclic cohomology. It was taken by Loday and Q uillen ([46]) and by Tsigan ([67]). Since the latter's work is independent from ours, cyclic cohomology was discovered from two quite different points of view. 263 48 ALAIN CONNES There is also a strong relation with the work of I. Segal [6I] [62] on quantized differential forms, which will be discussed in part IV and with the work of M. Karoubi on secondary characteristic classes [39], which is discussed in part II, Theorem 33. Our results and in particular the spectral sequence of part II were announced in the conference on operator algebras held in Oberwolfach in September i98i ([2I]). This general introduction, required by the referee, is essentially identical to the survey lecture given in Bonn for the 25th anniversary of the Arbeitstagung. This set of papers will contain, I~ The Ghern character in K-homology. II. De Rham homology and non commutative algebra. III. Smooth manifolds, Alexander-Spanier cohomology and index theory. IV. Pseudodifferential calculus for C* dynamical systems, index theorem for crossed products and the pseudo torus. V* Discrete groups and actions on smooth manifolds. Foliations and transversally elliptic operators. VI. VII. Lie groups. Parts I and II follow immediately the present introduction. 264 I. -- THE CHERN CHARACTER IN K-HOMOLOGY The basic theme of this first part is to " quantize " the usual calculus of differential forms. Letting ~ be an algebra over (3 we introduce the following operator theoretic definitions for a) the differential df of any f ~ ~, b) the graded algebra ~ = @ ~q of differential forms, c) the integration co --~ f co ~ (3 of forms co E ~n, df = i[F,f] = i(Ff -- fF) V f ~ M', nq = { xf o df 1 ... dfq, fJ }, ~". f~ = Trace(,o)) V co The data required for these definitions to have a meaning is an n-summable Fredholm module (H, F) over d. Definition 1. -- Let d be a (not necessarily commutative) Z/2 graded algebra over G. An n-summable Fredholm module over s~ is a pair (H, F), where, z) H = H+ ~ H - is a Z[2 graded Hilbert space with grading operator ~, ~---- (-- z) a'gr for all ~ eH +, 2) H is a Z/2 graded left d-module, i.e. one has a graded homomorphism rc of d in the algebra .9~(H) of bounded operators in H, 3) F~.s F 2= I, F~=--~F andforany a~d one has Fa -- (-- I) d~ a aF e .%f"(H) where ,Lf"(H) is the Schatten ideal (cf. Appendix 1). When d is the algebra C~176 of smooth functions on a manifold V the basic examples of Fredholm module over d come from elliptic operators on V (cf. [3]). These modules are p-summable for any p > dim V. We shall explain in section 6, theorem 5 how the usual calculus of differential forms, suitably modified by the use of the Pontrjagin classes, appears as the classical limit of the above quantized calculus based on the Dirac operator on V. The above idea is directly in the line of the earlier works of Helton and Howe [3o], Carey and Pincus [z2], and Douglas and Voiculescu [25]. The notion of n-summable Fredholm module is a refinement of the notion of Fredholm module. The latter is due to Atiyah [3] in the even case and to Brown, Douglas and Fillmore [i i] and Kas- 265 5o ALAIN CONNES parov [4 2] in the odd case. The point of our construction is that n-summable Fredholm modules exist in many situations where the basic algebra d is no longer commutative, cf. sections 8 and 9. Moreover, even when d is commutative it improves on the previous works by determining all the lower dimensional homology classes of an extension and not only the top dimensional "fundamental trace form ". This point is explained in section 7. Let then d be a not necessarily commutative algebra over C and (H, F) an n-summable Fredholm module over ~. We assume for simplicity that d is trivially Z/2 graded. For any a e~r one has da = i[F, a] es For each q e N, let f~q be the linear span in .L~/g(H) of the operators (a ~ da ida 2 ... da q, a ted, X ~C. Since .~/ql � .W,/q~ C .W "/Iqs + q~) (el. Appendix I) one checks that the composition of operators, f~q~ � flq~ ~ f~q~ +q~ endows ~ = + f~J with a structure of a graded j=0 algebra. The differential d, do~ = i[F, to] is such that d 2 = o, d(o l = (do 0 + (-- I) d~ do2 V c0z e Thus (f~, d) is a graded differential algebra, with d 2 = o. Moreover the linear func- tional f: f~" -+ C, defined by = Trace(coo) V to e f~" has the same properties as the integration of the trace of ordinary matrix valued diffe- rential forms on an oriented manifold, namely, for any o~je~V, j= 1,2, ql+q2=n. Thus our construction associates to any n-summable Fredholm module (H, F) over d an n-dimensional cycle over d in the following sense. Definition 2. -- a) a cycle of dimension n is a triple (f~, d, f) where ~ = + ~i is j=0 graded algebra over C, d is a graded derivation of degree i such that d ~ = o, and f : ~" ~ C is a closed graded trace on ~. and b) Let d be an algebra over C,. Then a cycle over ~r is given by a cycle (f~, d, f) a homomorphism p : d --~ ~o. As we shall see in part II (cf. theorem 32) a cycle of dimension n over d is essentially determined by its character, the (n + i)-linear function % 9 (a ~ ..., a") = fp(a ~ d(p(al)) d(p(a~)) ... d(p(a")) V a t e~r 266 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 5I Moreover (cf. part II, proposition i), an n + I linear function -r on ~r is the character of a cycle of dimension n over ~qr if and only if it satisfies the following two simple conditions, ~) "r(a ~, a ~, ..., a n, a ~ = (-- I)" "r(a ~ ..., a") V a ~ e ,.4, ~) ~ (-- I) ~'t'(a 0, ..., a ~a j+l, ..., a n+l ) + (-- I) n+lz(a n+la 0, a 1, ..., a n ) = o. There is a trivial manner to construct functionals -: satisfying conditions ~) and ~). Indeed let C~(~) be the space of (p + x)-linear functionals on ~ such that, 9(a 1,...,a v,a ~ = (-- I) ~(a ~ ~) Va ~e~r Then the equality, bq~(a~ "'" ap+l) ---- ~0 (- I)j q)(a0' "" "' aj aJ+X' "" "' ap+l) + (-- I) '+l~(a '+ta ~ ...,a ~) defines a linear map b from C~(d) to C~+I(J~) (cs part 11, corollary 4). Obviously conditions 0:) and ~) mean that v ~ C~ and bx = o. As b ~" = o, any b9, q~ ~ C~-1(.~), satisfies a) and ~). The relevant group is then the cyclic cohomology group H~,(,~) ={': e C~,(,~), b'r = o)/{&p, ~O e C~-1(.~)}. The above construction yields a map ch*: {n summable Fredholm modules over ~r I-I~(ad). Since ~r is trivially Z[2 graded, the character -r e C~(~r of any n summable Fredholm module over d turns out to be equal to o for n odd. Let us now restrict to even n's. The inclusion .~eP(I-I) C .LPq(H) for p g q (ef. Appendix i) shows that an n summable Fredholm module (H, F) is also n + 2k summable for any k = I, 2, ... We shall prove (cf. section 4) that the (n + 2k)-dimensional character % + ~k of (H, F) is deter- mined uniquely as an element of H~+2k(d) by the n-dimensional character % of (H, F). More precisely, there exists a linear map S:H~(d) ~ I-I~+~(sr such that "r,+~ = Sk'rn in H~+~(~). The operation S : H~(M) -+ H~+*(d) is easy to describe at the level of cycles. Let Y~ be the 2-dimensional cycle over the algebra B = C with character cr, a(I, I, I) = 2ir~. Then given a cycle over ~ with character % S'r is the character of the tensor product of the original cycle by Z. The reason for the normalization constant 2ir~ appears clearly from the computation of an example (cf. section 2). It corresponds to the following normalization for f co, co e ~", n = 2m, fo) = m!(2i~)" Trace(r162 267 ALAIN CONNES 5 2 Let now H~(d) = (fl~ H~(d). The operation S turns H~(.qr into a module n=0 over the polynomial ring C(a), S being the multiplication by a. Let, n*(0~r ~-- Lim(H~(~r S) = H~(~r | where (3(a) acts on (3 by P(a) z = P(I) z for z ~ (3. The above results yield a map ch*: {finitely summable Fredholm modules over a~r H*(ad). We shall show (section 5) that two finitely summable Fredholm modules over ad which are homotopic (among such modules) yield the same element of H'(ad). When off = C~(V), where V is a smooth compact manifold, one has H~ont(ad) ---- H.(V, (3) where H~ont means that the (n -]- I)=linear functionals ? e C~(0ff) are assumed to be continuous, and H.(V, (3) is the ordinary homology of V with complex coefficients. We can now explain what our construction has to do with the Chern character in K-homology. The latter is (cf. [9]) a natural map, ch, : K,(V) ---> H.(V, (3) where the left side is the K-homology of V ([9])- By [24] the left side is isomorphic to the Kasparov group KK(C(V), (3) of homotopy classes of *Fred.holm modules over the C*-algebra C(V) (1). The link between our construction and the ordinary dual Chern character ch. is contained in the commutativity of the following diagram: homotopy classes of finitely summable / .~ H~o.t(C~(V)) *Fredholm modules over C~ ) KK(C(V), C) oh. > H.(V, (3) For an arbitrary algebra 0d over (3, let Ko(d ) be the algebraic K-theory of ad (cf. [40]). One has (cf. part II, proposition I4) a natural pairing < , > between K0(ad ) and the even part of H*(od). Moreover the following simple index formula holds for any finitely summable Fredholm module (H, F) over d: <[e], ch*(H, F)) -~ Index F + V e ~Proj Mk(d ). Here e is an arbitrary idempotent in the algebra of k � k matrices over d, [e] is the corresponding element of K0(ad), and F + is the Fredholm operator from e(H+| C k) to e(H- | (3 k) given by e(F | I) e. This formula is a direct generalisation of [2o], [34]. It follows that any element .~ of H*(0d) which is the Chern character of a finitely sum- mable Fredholm module has the following integrality property, < Ko(d), -: > C Z. (1) A Fredholm module over a *algebra 0d is a *Fredholm module if and only if < a~, ~] > = < ~, a* B > for aEd, ~,~ ~ H. 268 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY To illustrate the power of this result we shall use it to reprove a remarkable result of M. Pimsner and D. Voiculescu: the reduced C*-algebra of the free group on 2 gene- rators does not contain any non trivial idempotent. Letting -= be the canonical trace on C~(F), and e e Proj(C~(F)), one knows that v(e) 9 [o, i]. Using a suitable Fredholm module (cf. [56], [23], [37]) with character x we shall get -r(e) 9 Z and hence "r(e)~{o, I}, i.e. e=o or e= I. Part I is organized as follows: CONTENTS I. The character of a i-summable Fredholm module ........................................... 53 2. Higher characters for a p-summable Fredholm module ....................................... 56 3- Computation of the index map from any of the characters "r n ................................ 6o 4- The operation S and the relation between higher characters .................................. 6i 5- Homotopy invariance of ch*(H, F) .......................................................... 63 6. Fredholm modules and unbounded operators ................................................ 66 7. The odd dimensional case ................................................................. 69 8. Transversally elliptic operators for foliations ................................................. 77 9- Fredholm modules over the convolution algebra of a Lie group ............................... 8o APPENDICES L Schatten classes ........................................................................... 86 2. Fredholm modules ........................................................................ 88 3- Stability under holomorphic functional calculus .............................................. 92 I. The character of a x-snrnmable Fredholm module Let d be an algebra over G, with the trivial Z/2 grading. Let (H, F) be a i-summable Fredholm module over ~. Lemma 1. -- a) The equality ~(a) = i Trace(,F[F, a]), V a 9 ~, defines a trace on ~. b) The index map, K0(d ) -+ Z, is given by the trace .~: Index F, + = ('~ | Trace) (e) V e 9 Proj Ms(d ). ca=de for all a 9 Proof. -- a) Since d is trivially Z/2 graded, one has As zF = -- Fe one has eF[F, a] ---- eF ~ a -- eFaF = ,F 2 a + FasF = ea + FaeF since F 2 = i. Thus, eF[F, a] = [F, a] cF. Then "r(ab) = ~ Trace(sF[F, ab]) = ~ Trace(eF[F, a] b + cFa[F, hi) 2 2 = - Trace([F, a] ~Fb + [F, b] cFa), Thus (ab) = (ba) for a, b d. which is symmetric in a and b. 269 ALAIN CONNES b) Replacing d by Mq(d), and (H, F) by (H| q, F| I) we may assume that q=I. Let F=[; oW"[ so that PQ=IH_ , QP=I m. With Hx=eH +, H2 = ell- we let P' (resp. Q') be the operator from I-I t to t-I s (resp. H2 to Hx) which is the restriction of eP (resp. eQ.) to t-I x (resp. t-Is). Since [F, e] 9 .o.WX(H) one has P' O' -- In, 9 .ogal(H2) , O~' 1 )' -- IH1 e ,,oCfl(Hx). Thus (proposition 6 of Appendix I) one has Index P' -- Trace(Iri 1 -- O~ 1 )') -- Trace(in, -- P' Q') = Tracem(e -- eOePe ) -- Tracea_(e -- ePeQe) = Trace(~(e -- eFeFe)). But Trace(r -- eFeFe)) = Trace(~(e -- FeFe) e) = Trace(eF(Fe -- eF) e) = I Trace(r e] + ,F[F, e] e) = I Trace(r e]) = -r(e). [] 2 2 Then its character Definition 2. -- Let (H, F) be a I-summable Fredholm module over d. is the trace ~ on d given by lemma I a). Then Corollary 3. -- Let "r be the character of a I-summable Fredholm module over d. < Ko(ag), ~) C Z. Now let A be a C*-algebra with unit and "~ a trace on A such that 1:(x'x) ~o for xeA, I) %" is positive, i.e. 2) v is faithful, i.e. x 4= o :,- x(x* x) > o (cf. [55]). Corollary 4. -- Let A be a C*-algebra with unit and 9 a faithful positive trace on A such that ":(I) = I. Let (H, F) be a Fredholm module over A (cf. Appendix 2) such that a) d ---= {a 9 A, [17, a] 9 .s } /s dense in A, b) -~/d is the character of (H, F). Then A contains no non trivial idempotent. Proof. -- By proposition 3, Appendix 3, the subalgebra d of A is stable under holomorphic functional calculus. Hence (Appendix 3) the injection d ~ A yields an isomorphism, K0(d)~ K0(A ). Thus the image of K0(A) by -r is equal to the image of K0(d ) by the restriction of-r to d so that, by corollary 3, it is contained in Z. If e is a selfadjoint idempotent one has ~(e) 9 [o, I] n Z = {o, i } and hence, since v is faithful, one has e = o or e = I. It follows that A contains no non trivial idem- potent f, f~ =f. [] Before we give an application of this corollary, let us point out that its proof is exactly in the spirit of differential topology. The result is purely topological; it is a state- ment on a C*-algebra, which, for A commutative, means that the spectrum of A is connected. But to prove it one uses an auxiliary " smooth structure " given here by the subalgebra d = {a 9 A, [F, a] 9 s176 270 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY As an application we shall give a new proof of the beautiful result of M. Pimsner and D. Voiculescu that the reduced &-algebra of the free group on two generators does not contain any non trivial idempotent [56]. This solved a long standing conjecture of R. V. Kadison. We shall use a specific Fredholm module (H, F) over the reduced C*-algebra of the free group which already appears in [56] and in the work ofJ. Cuntz [23], and whose geometric meaning in terms of trees was clarified by P. Julg and A. Valette in [37]. Definition 5. -- Let r be a discrete group. Then the reduced C*-algebra A -~ c~(r) of F is the norm closure of the group ring CF in the algebra .Cf(t2(F)) of operators in the left regular representation of F (el. [5i]). Now let I" be an arbitrary free group, and T a tree on which F acts freely and transitively. By definition T is a i-dimensional simplicial complex which is connected and simply connected. For j -~ o, I let T i be the set ofj-simplices in T. Let p e T O and 0 : T~ ~ TI be the bijection which associates to any q ~ T ~ q 4:p, the only i-simplex containing q and belonging to the interval [p, q]. One readily checks that the bijection 0 is almost equivariant in the following sense: for all g ~ F one has o(gq) = go(q) except for finitely many q's (cf. [23], [37]). Next, let H + = 12(T~ H- = I2(T 1) @ C. The action of F on T O and T 1 yields a C~(F)-module structure on /Z(Ti), j ---- o, i, and hence on H + if we put a(~, ),) = (a~, o) V ~ eta(T1), ;~ e C, a e c;(r). Let P be the unitary operator P:H + -> I-I- given by P~v: (o, I), Pgq----r Vq4:p (where for any set X, (r is the natural basis off'(X)). The almost equivariance of 0 shows that Lemma 6. -- The pair (H, F), where H = H + @ H-, F = is a Fredholm [; o module over A and ~/ = { a, IF, a] ~ .5f ~ (H)} is a dense subalgebra of A. Proof. ~ For any g ~ F the operator gP -- Pg is of finite rank, hence the group ring CI" is contained in d = {a ~ C;(F), [F, a] E ~~ }. As CF is dense in C~(F) the conclusion follows. [] Let us compute the character of (H, F). Let a~, then a--P-aaP~~ and _I Trace(~F[F, a]) = Trace(a -- p-1 aP). Let 9 be the unique positive trace on A such that "r(Yag g) = at, where x e F is the unit, for any element a = ag g of CF. Then for any a ~ A = C;(F), a- "~(a) I belongs to the norm closure of the linear span of the elements g ~ F, g 4: I. 27'1 56 ALAIN CONNES Since the action of F on T~ is free, it follows that the diagonal entries in the matrix of a--v(a) I int2(T j) are all equal to o. This shows that for any ae~r one has, Trace(a -- p-1 aP) = x(a) Trace(i -- p-1 IP) = v(a). Thus the character of (H, F) is the restriction of v to d and since -r is faithful and positive (cf. [5I]), corollary 4 shows that Corollary 7. -- (Cf. [56]). Let F be the free group on 2 generators. Then the reduced C*-algebra C*(F) contains no non trivial idempotent. 2. Higher characters for a p-snmmable Fredholm module Let d be a trivially Z[2 graded algebra over C. Let (H, F) be a p-summable Fredholm module over ~. As explained in the introduction we shall associate to (H, F) an n-dimensional cycle over d, where n is an arbitrary even integer such that n >= p. In fact we shall improve this construction so that we only have to assume that n => p -- i, i.e. that (H, F) is (n + I)-summable. Let ffbe obtained from d by adjoining a unit which acts by the identity operator in H. For any T e.Sf(H) let dT = i[F, T] where the commutator is a graded commutator. For each j e N we let ~J be the linear span in .~f(H) of the operators of the form a ~ 1...da t, a ke Lemma 1. -- a) d ~T=o VTe~(H). b) d(T x T2) = (dT1) T z + (-- i)eT, Ta dT, V Tx, T, e .~(H). c) d~ t C ~J +1. d) h j x ~k C ~J + k; in particular each fit is a two-sided ~-module. e) ~k C ~~ Proof. -- a) If T is homogeneous, then F(FT -- (-- I) ~ TF) -- (-- I)eT+I(FT -- (-- I) ~T TF) F = F 2 T -- TF ~ = o. b) The map T ~ [F, T] is a graded derivation of .Lf(H). c) Follows from a), b). d) It is enough to show that for a ~ j, ae~ one has (a ~ da 1. . . da j) a e ~j. This follows from the equality (da j) a = d(a t a) -- d da, by induction. e) Since (H,F) is n+ I summable one has dae.~+l(H) for all aed and I I I e) follows from the inclusion .Lr p � -oq ~q C ~" for r = p + q (cf. Appendix I). [] 272 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Lemma i shows that the direct sum ~ = G ~J of the vector spaces ~J is naturally j=0 endowed with a structure of graded differential algebra, with d2= o. Lemma 2. -- For any T 9 .Lf(H) suck that [F, T] 9 ~I(H) let Tr,(T) = I- Trace(r T])). a) If T is homogeneous with odd degree, then Trs(T ) = o. b) If T 9 s then Trs(T ) = Trace(~T). c) One has IF, ~"] C s and the restriction of Trs to ~n defines a closed graded trace on the differential algebra ~. Proof. -- a) Since F[F, T] is homogeneous with odd degree one has ~F[F, T] = -- F[F, T] and Trace(eF[F, T]) = Trace(F[F, T] s) = -- Trace(eF[F, T]) thus Trace(r T]) = o. b) i_ Trace(eF[F, T]) = i Trace r -- FTF) for all T with 0T = o (mod 2). 2 2 If T 9 .~al(H) then Trace(r = -- Trace(FcTF) = -- Trace(eT), so that - Trace(r T]) = Trace(r c) One has [F,~n]Cf~+tC.LPl(H) by lemma x. Since d z=o one has Tr,(dco) = o V co 9 f/n-~. It remains to show that for coa 9 f~", co~ 9 ~', nl + n~ = n one has Tr,(coa co2) = (-- I)"'"' Tr,(co, %), or equivalently, that Trace(r d(co 1 co,)) = (-- x)" Trace(r d(coz col)). Since cF commutes with dcoj and dco,, one has Trace(r d(cox co2)) = Trace(r co2) + (-- I) n' Trace(r dco2) = Trace(r 2 dcox) + (-- i)"' Trace((r dco,) col) = (-- I) ~' Trace(r d(m~ cot) ). [] We can now associate an n-dimensional cycle over ~r to any n + I summable Fredholm module (H, F). Definition 3. -- Let n = 2m be an even integer, and (H, F) an (n + i )-summable Fredholm module over d. Then the associated cycle over ~' is given by the graded differential algebra (fl, d), integral the f co = (2iz:)" m ! Trs(co) V co 9 ~2" and the homomorphism r: : ~r -+ G 0 C .W(H) of definition I. 8 ALAIN CONNES The normalization constant (2irc)"mI is introduced to conform with the usual integration of differential forms on a smooth manifold. To be more precise let us treat the following simple example. We let F C 13 be a lattice, and V = 13[F. Then V is a smooth manifold and the 0 operator yields a natural Fredholm module over C~~ We consider 0 as a bounded operator from the Sobolev space H + = { ~ ~ L~(V), 0~ e L2(V) } to H- = Lz(V). The algebra C~ acts in H + by multiplication operators, and the operator F is given by , where ~ ~ 13, i, r F  the orthogonal of ~+~ the lattice P (to ensure that 0 + e is invertible). We let (eg-)ger" be the natural orthonormal basis of Lz(V) = H-, s~-(z) = 113/r -" exp i(g, z> for z e 13[F, and (%+) be the corresponding basis of H +, Cg+ = (0 + e)-x ~-. Thus ,+(z) = (ig + ,)-~ r for z e 13]F, and we may as well assume that the r form an orthonormal basis of H +. For each gsP let UgeC*(V) be given by Uo(z ) =expi(g,z), then Uoa+o = = UglUg= for gl, gz e F" and the algebra G| is naturally isomor- phic to the convolution algebra 5e(F') of sequences of rapid decay on F', C~ ={ZagU0, a e S'~ One has Ugr = %~ and _ i(g + k) + r %++~, U~ ,~ + = U~(ik + r e~-= (ik + ~)-x so+~ -- ik + for any g,k~F t . We are now ready to prove Lemma d. ~ With the above notations, (H, F) is a 3.summable G~ and I .I 0 1 Try(f~ i[F'fl] i[F'f2]) = ~i~ f df ^ df 2 V f~ 2 e C~ where V is oriented by its complex structure. Proof. -- For g ~ P" one has (!(g + k) + (FU 0-UgF)~k + =F~ ik+s _ o+k--%+k _ /g _ _ {i(g+k)_+e I %+k -- ik + r ~g+k -- ig ~++~. Since 5~(F  C tl(P it follows that and similarly (FUg -- U s F) ~- -- ih + e (H, F) is p-summable for any p > 2. To prove the equality of the lemma we may assume that f~ = Ug i with go, gi, g~ e F I. From the above computation we get N, uj [r, Uj IF, UgJ I + g~ + k -- i g~ + k -- i ~go+gl+g~+k 274 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY and IF, U.o] [F, [F, Uj +g2+k" = -- - Trace(r Ug0] [F, Ugx] [F, Ug,]) is equal Thus Trs(Ug ~ i[F, Ugl] i[F, Uj) 2 to o if go + gl + g*, # o and otherwise to: ker  1-k g2 -b k -- i ~ --k k -- i This sum can be computed as an Eisenstein series ([70]). More precisely let u, v be generators of r -L with Ira(v/u) > o and El(z ) the function N M El(z ) = lim ~ (Lira ~ (z-4-k)-l) where k=~u-bvv. Then the above expression coincides with g1(E1(-- is) -- E1(g ~ -- is)) -- g~(E1(g~ -- i~) -- E~(gl + g~ -- is)) = =i=(n, m, - nl where g~=niu+m~v (cf. [7o], p. I7). Let (~,[~) be the basis of C over R dual to (u,v). Then P=27r(Za A-Z~), Ug(xa-]-y[~) =e~"*ei"Y for all x,y~R, g=nu +mveF  For go+gl+g, 4~ one has (v Ug0 dUg, dUo, = o and otherwise fV Ug0 dUg1 dUg2 = f2rc f~ ( (inx) (im2) -- (in~) (iml) ) dx dy = (2i.) 2 > (n 1 m 2 -- n 2 ml). [] A similar computation yields the factor (2in)"m! for n = 2m. Proposition 8. -- Let n = 2m, (H, F) be an (n A- i)-summable Fredholm module over ~r and 9 be the character of the cycle associated to (H, F), -r(a ~ ..., a ~) = (2i7r)" m! Tr~(a ~ da ~ ... da"). Then a) x(a 1,...,a",a ~ ='~(a ~ for a jed; b) ~ (-- I) j'c(a ~ ..., a j a j+l, ..., a "+1) + (-- I) "+lx(a "+1 a ~ ..., a") = o. Proof. -- Follows from proposition i of part II. [] With the notation of part II, corollary 4, one has -~ e C[(~1) by a) and bv = o by b), i.e. v ~ Z[(zr Remark 6. -- All the results of this section extend to the general case, when],~' is not trivially Z/2 graded. The following important points should be stressed, ~) Since a ~ ~ .~r can have non zero degree rood =, it is not true in general that f to=o for c0ef~", n odd. 276 60 ALAIN CONNES ~) Since the symbol d has degree i, the n-dimensional character -r, of an (n q- i)- summable Fredholm module (H, F) over ~r is now given by the equality, x,,(a ~ ..., a") = c,(-- x) e~+e~"+ "'" + e~*+~+ ... Tr,(a 0 da ~ ... da"). n+2 Here c, is a normalization constant such that c,+~ = 2ir~--c,,, we take Y) In general, the conditions a), b) of proposition 5 become n Oaj) t 0 a') "r(a', . . ., an, a ~ = (-- i)n( - I) ~176 "tea,a' , . . ., a") V aJ e.~ r (-- I) j'l~(a 0, ..., a j a t§ ' ..., a n+l ) b') .i=o + (_ 0.+,(_ "r(a TM a ~ a 1, . .., a") = o. The general rule (cf. [49]) is that, when two objects of Z/2 degrees ~ and ~ are permuted, the sign (-- i) ~ is introduced. 3" Computation of the index map from any of the characters ~. Let d be an algebra over C, with trivial Z[~ grading. Let n = 2m be an even integer, (H, F) an (n + I)-summable Fredholm module over d, and -r, the n- dimensiona character of (H, F). Let (x,) be the class of x, in H~(~) = Z~(d)/bC~-X(d). By part II, propo- sition I4, the following defines a bilinear pairing ( , ) between Ko(.~t ) and H~(~): (e, 9) = (2i~)-m(m!)-'(9 # Wr) (e, ..., e) for any idempotent e e Mk(d ) and any 9 ~Z~(d)- Here 9 # Tr ~Z~(Mk(~r is defined by (9 # Tr) (a ~174 ~ ..., a"| m") = 9(a ~ ..., a") Trace(m ~ ... m") for any a jed, m jeMk(C ). When the algebra d is not unital, one first extends 9 e Z~(d) to ~ e Z'~(ff), where ~r is obtained from ~r by adjoining a unit, ~(a~ + X~ I, .. ., a" + X" I) =9(a ~ Va~e~ r hieC. Then one applies the above formula, for e e Mk( ~. Theorem 1. -- (Compare with [25] and [34]). Let n = 2m and (H, F) an (n + I)- summable Fredholm module over zi. Then the index map K0(~ ) ---> Z /s given by the pairing of K0(d ) with the class in H~(d) of the n-dimensional character v, of (H, F): Index F + = ([e], (v,)) for e e Proj Mq(d). 276 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 6i Proof. -- As in the proof of lemma I. I b) we may assume that k = i, that ~1 is unital and that its unit acts in H as the identity. Let F=[; oQ], sothat PQ=Ia_ , Q P= In+. Let H l=eH +, H,=eH-, and P' (resp. O~) be the operator fromH 1 to Hz (resp. H~ to Hx) which is the restriction of eP (resp. eO) to H x (resp. H~). Thus ~HI- Q' P' (resp. Ia,- P' O~) is the restriction to H i (resp. H~) of e--eFeFe. As e--eFeFe=--e[F,e]*e, and [F,e] ~"+I(H), we get (Appendix I, propo- sition 6) Index P' = Trace e(e -- eFeFe) "+l. One has (e, %,) -- (- r)" Trace(r e]~"+~). As IF, e] ---- elF, e] + [F, e] e, one has Trace(r e]) 2"+t) = Trace(eFe[F, e] [F, e] ~"~) + Trace(r e] e[F, e]2"). cF=--Fr F[F,e] m+l=-[F,e] m+lF, so that Now Trace(r e] ~ + 1) = _ Trace(F,e[F, e] ~ +l) = -- Trace(,e[F, e] ~+a F) = Trace(,eF[F, e]m+l). As elF, e] ~= IF, e] ze we get Trace(eF[F, e] z'+l) = 2 Trace(r e] ,[F, e] ~'') = 2(-- I) "~ Trace r -- eFeFe) m+l. [] 4" The operation S and the relation between higher characters In part II, theorem 9, we show that the operation of tensor product of cycles yields a homomorphism (% +) ~ q> # + of Z~(d) � Z~(&) to Z~+m(d| for any algebras d, ~ over C. Taking ~=C and aeZ~(C), ~(X0, Xl, X,) =2inX 0xlkS yields the map S, Sq~ = 9 # ~ from Z~(d) to Z~+2(~1| C) = Z~+2(~1). By part II, corollary Io, one has SB~(~r B~+2(~r Now let n = 2m be even, (H, F) be an (n + i)-summable Fredholm module over d. As .L~+t(H) C .oq~+S(H), the Fredholm module (H, F) is (n + 3)-summable, and hence has characters %, -~,+z of dimensions n and n + 2. Theorem 1. -- One has %+2 = S% in H~+~(~r Proof. -- By construction, -r. is the character of the cycle (~, d, f) associated to (H, F) by definition 3. Thus (part II, corollary Io) S% is given by n+l I S-~,(a ~ ..., a "+z) = 2i Y, (a ~ da 1 ... da j-l) a j aJ+i(da j+9" ... da "+z) 0 9 n+l = (2/~) "+1 mt Z Tr,((a ~ dat.., da j-i) a t aJ+l(da ~+~ ... da"+2)). 277 6~ ALAIN CONNES By definition, % +2 is given by %+2(a ~ ..., a "+2) = (2iTr)"+t(m + I)l Tr,(a ~ da t ... da"+Z). We just have to find % e C~+1(~qr such that b?o = S~r, -- %+2. We shall construct ? E C~+t(,~r such that n+l b~(aO, ..., an+2 ) = -2 y~ TL((aO da t ... daJ_t) aj aj+t(daJ+2 ... da,+2) ) n+l We take ? -- ~o (- I)j qfl' where q~i(aO ' .-., a . +1) = Trace(eFa ~ da j + t ... daJ-1). One has a~ da j+l.., da j-t ~f~"+t C S~ so that the trace makes sense; moreover by construction one has ? ~ C~+t(.~r To end the proof we shall show that (-- i) j-t b?J(a ~ ..., a n+2) Tr,(a ~ da 1 ... da" + z) + - (-- I) j Tr,((a ~ dal.., da j-t) a j aJ+t(daJ +2 ... dan+9~)). Using the equality d(ab) = (da) b+adb, with a,b~d, we get b~(a ~ ..., a" + 2) ---_ Trace(eF(a i+t da j+2 ... da "+z) a~ 1 ... daJ)) + (-- i)i -1 Trace(eFaJ+l(dai+Z... da~ da i-1) a j) + Trace(eFaJ(da ~+t ... da "+z) a~ t ... dai-t)). ... da "+2) a~ t ... da j-t) E f~". Using the equality Let [3 = (da ~+2 Trace(~ d[3) = TL(~ d[3) = TL(i[F, 0c] [3) V 0c ~ SO(H), ~0~ = -- 0re, we get (-- i)J -t Trace(r j+2 ... da ~ ... da j-t) a j) = TL(i[F , a j Fa j+t] [3). Thus, bgJ(a ~ ..., a ~+z) = Trace(da j cFa j+l [3) + TL(i[F, a j Fa j+l] [3) + Trace(r jda j+l [3) = Tr,((Fd(a j a ~+1) + i[F, a i FaJ+l]) [3). One has Fd(a ja j+l) +i[F,a iFa j+t] :--i(da jda j+t-2a ~a ~+t) and the above equality follows easily. [] This theorem leads one to introduce the group HeY(d) which is the inductive limit of the groups I-~fl(d) with the maps, 278 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY With the notation of part II, corollary io, one has, n~ : H[~(a/) | C where C(a) acts on C by P(~) -+ P(t) (cf. part II, definition t6). Definition 2. -- Let (H, F) be a finitely summabte Fredholm module over ~. We let ch*(H, F) be the element 0f H*(d) given by any of the characters "~2,,, m large enough. By part II, corollary t 7, one has a canonical pairing ( , ) between Hev(~/) and Ko(~) and theorem 3. t implies the following corollary. Corollary 3. -- Let (H, F) be a finitely summable Fredholm module over d. Then the index map K0(~r ) -+ Z is given by Index F + -~ (ch.(e), ch*(H, F)) V e ~ Proj Mk( ~. For such a formula to be interesting one needs to solve two problems: t) compute H*(~); ~) compute ch*(H, F). In part II we shall develop general tools to handle problem t. 5" Homotopy invariance of ch'(H, F) Let d be an algebra over C. In this section we shall show that the character ch*(H, F) ~ Hov(M) of a finitely summable Fredholm module only depends upon the homotopy class of (H, F). Let H 0 be a Hi]bert space and H the Z/2 graded Hi]bert spacewith H+=Ho, H-=H o. Let F~L~~ F=[~ Io], Lemma 1. -- Let p = 2m be an even integer. For each t ~ [o, I] let ~t be a graded homomorphism of d in 5e(H) such that I) t -+ IF, nt(a)] is a continuous map from [o, i] to .~q~P(H)for any a ~ d, 2) t -+ rq(a) ~ is a C 1 map from [o, I] to H for any a ~ ~, ~ ~ H. Let (Ht, F) be the corresponding p-summable Fredholm modules over d. Then the class in H~ + 2( ~t) of the (p + 2).dimensional character of (Ht, F) is independent of t ~ [o, i]. Proof. ~ Replacing d by ffwe can assume that ~ is unital and that nt(i) = i, V t ~ [o, i]. By the Banach Steinhaus theorem, the derivative 8,(a) of the map t-+ nt(a) is a strongly continuous map from [o, I] to Lf(H). Moreover, ~t(ab) = ~t(a) 8,(6) + ~,(a) ~t(b) for a, b ~ d, t ~ [o, I]. For t ~ [o, 1] let ~, be the (p + 2)-linear functional on d given by p+l ~t(a ~ ..., a p+t) = Y~ (-- t) k-lTrace(~t(a ~ [F, nt(al)] ... k=l [r, ~,(ak-1)] ~,(a k) [F, 7r,(ak+l)] ... [F, 7r,(a'+l)]). 279 64 ALAIN CONNES Using the equality 8t(ab) = rq(a) ~t(b) + ~t(a) ~t(b), V a, b e ~r one checks that q~t is a Hochschild cocycle, i.e. bgt----o, where p+l b~pt(a ~ . . ., a 7'+~) ---- Z (-- r)q q~t(a ~ .... , aq a q+l, . . ., a p+~) q=0 ..~ (__ i)P+2 q)t(alo+2 a O, ~1, "'', aV+l), V a j e ~r Let 9 be the (p + 2)-linear functional on ~r given by q~(a ~ ..., a p+I) = %(a ~ a p+I) dt. fo "., (Since II~,(a)II and IIn,(~)II are bounded, the integral makes sense.) One has One has b~=o and s ~ v+l) =o if a J= I for some j4=o. dt Y, (-- i)kTrace(~[F, ~t(a~ ... q~(l,a ~ t , ...,a v) =f~ k=0 [r, ~,(~- 1)] 8,(~) [F, ':t(~ + ~)] --. [r, ~,(~')]). Let .r,(a ~ ..., a p) = Trace(r ~ [F, ~,(al)] ... [F, ~xt(aV)]). One has /- (.rt+~(a ~ ..., a v) -- -rt(a ~ ..., aV)) ---- Trace (r ~(nt+ s(a ~ -- nt(a~ ~,+ s(at)]... IF, 7~,+ s(al~)]) + Trace (r ~ IF, rq(at)]... [F, ~(rq+,(a v) -- rq(aV))]). When s --+ o one has, using t) and 2), Trace (~7:t(a ~ [F, rq(at)]... IF, rq(a k- 1)] iF, ~(rq +s(a ~) -- rq(a~))]... [F, rq +8(aV)]) = (-- I) kTrace (e[F, ~t(a~ ... [r, ~,(a~-l)] I-s (~'+8(ak) - ~'(a~)) [r, ~,(~+t)] ... [r, ~,+s(~')]) -+ (-- ~)~ Trace(r rq(a~ ... [F, rc,(ak-1)] ~t(a k) [F, r:,+~(a~+~)] ... [F, ~,+,(aV)]). Thus ~(I, a 0, ..., a p) = ~'~ dt -= "q(a ~ ..., a v) - "ro(a ~ ..., a p) and the result fol- lows from Part II, lemma 34, since bq~=o and B oq~='r 1-%. [] 280 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 65 Theorem 2. -- Let d be an algebra over 13, H a Z/2 graded Hilbert space. Let (Ht, Ft) be a family of Fredholm modules over d with the same underlying Z/2 graded Hilbert space H. [o Let O~ be the corresponding homomorphisms of ~ in ~,~ and F t = . Assume that for some p < oo and any a e d, P~ I) t ~ p+(a) -- Qt p~-(a) Pt is a continuous map from [o, I] t0 ~'(H), 2) t ~ p+(a) and t ~ Qt p~(a) Pt are piecewise strongly C 1. Then ch*(Ht, Ft) e H~'(~ r is independent of t ~ [o, I]. [:o] [: :] Proof.- Let T t = , then TtFtT~-I = and Q, T,p,(a) TT,=[p+~a) o ]. Q, O-[(a) P, Then the result follows from lemma i and the invariance of the trace under similarity. [] Corollary 3. -- Let (H, Ft) be a family of p.summable Fredholm modules over d with the same underlying d-module H and such that t ~ F t is norm continuous. Then ch*(H, Ft) is independent of t e [o, r]. Proof. ~ Since the set of invertible operators in ..q'(H +, H-) is open, one can replace the homotopy F t by one such that t ~ Pt is piecewise linear and hence pieeewise norm differentiable. [] Let now A be a C*-algebra and .~r C A a dense *subalgebra which is stable under holomorphie functional calculus (cf. Appendix 3). By theorem 2, the value ofch*(H, F) only depends upon the homotopy class of (H, F). We thus get the following commu- tative diagram, Homotopy classes offinitelysummable} oh. HeV(d ) *Fredholm modules over d KK(A, 13) , Hom(Ko(A), Z) C I-Iom(K,(A), 13) where a) the left vertical arrow is given by proposition 4 of Appendix 3, b) the right vertical arrow is given by the pairing of Ko(~ ) with HeY(d) of part II, corollary 17 together with the isomorphism Ko(d ) ~ K0(A ) (Appendix 3, pro- position 2), c) the lower horizontal arrow is given by the pairing between KK(A, 13) and KK(13, A) = K0(A ). 9 66 ALAIN CONNES 6. Fredholm modules and unbounded operators Let d be an algebra over C. In this section we shall show how to construct p-summable Fredholm modules over ~ from unbounded operators D between d-modules. We shall then apply the construction to the Dirac operator on a manifold. We let H be a Z/2 graded Hilbert space which is an ~-module and D a densely defined closed operator in H such that x) cD= --De, 2) D is invertible with D -1 9 ~r 3) for any a e%t the closure of a--D-laD belongs to .LPV(H) (where p e [I, oo[ is fixed). [o ~9.[. Let H1 be the Z[~ graded ~t.module Proposition 1. -- a) Write D = D1 given by H + = H +, H~ = The Hilbert space H + with a~ = D~ -1 a Dl~, for ~ 9 DomD 1 a 9 Let Fi=[~ Io]. Then (H1, F~) isap"summableFredholmmodule over ~'. b) The following equality defines an element .c e Z~(~), n = 2m, n > p -- I z(a ~ ..., a n) : (2~i)"*m! Trace(D-l[D, a ~ ... D-'[D, an]), V a s ~'. c) Let (H~, F,) be constructed as in a) from H- and D,. Then .~ : x i-n: s where v~ is the character of (H~, F j). Proof. ~ a) For a 9 ~, let ~(a) be the operator in H + defined as the closure of D~ -lad 1. Since a--D -laD 9 pp, we see that ~(a)--a is bounded and belongs to ACP(H+). Since D1D~ -1 : I one has n(ab) = n(a) n(b) for a, b 9 Thus the module H~- is well defined and one has IF1, a] ~ .~~ , V a 9 d. b) Follows from c). [a-- D~-laD1 o ] c) One has D-I[D, a] ---- so that, for any a s 9 ~r o a -- D~-I a D2 x(a ~ ..., a") = TraceK+((a0 -- D~ -1 ao D1) ... (a, -- Di -1 a, D1) ) -- TraceH-((ao -- D~ -1 a0 D2) ... (a, -- D~ -1 a, D2)). Now the character v i of (Hi, F1) is given by = (2~i)" m! TraceH§ ~ -- Di -1 a ~ D1) ... (an -- Di -1 a ~ D1)). Similarly one has %(a ~ ..., a") = (~xi)"m! TraceE-((a 0 -- D~ -1 a 0 D2) ... (a, -- D~ -1 a~ D~)) [] 282 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Let us now assume that off is a *algebra and H a *module (i.e. (a*~, n) = (~, an>, V~,~H, aeoff). For any ~0~C~(off), let ?* be defined by o*(a ~ . . ., a n ) : 9(a~,...,a;) Va~eoff. One checks that ~*~ C~ and that (bq~)* : (-- r)" bq~*. Corollary 2. -- If D is selfadjoint, then .r = ((2ni)-m(mt) -1 -rl) + ((2ni)-"(m!) -a ~:1)*. Proof. -- One has D 2 = D~, thus TraceH-((a o -- D~ -t ao D~) ... (a, -- D~ -~ a, D2)) = TracerI-((a0 -- D~ -1 a0 D~) ... (a, -- D~ -i a, DE) ) = (Tracea-((a~ -- D1 a~, D~ -1) ... (4 -- I)1 a~ D~-~))) - = (Tracea,((Di -1 a~ D1 -- a~) ... (a~ -- D~ -1 a; D1)))- = -- ((2rci)-"(ml) -~ ~)* (a0, ..., a,). [] We shall define the character of a pair (H, D) satisfying i) 2) 3) as -r(a ~ ..., a") -- (2hi)" m! I Trace(r D-lID, a ~ ... D-~[D, a']). When D = F with F2= I we get the same formula as in section 2. Since = ~ ('rl -- ~:z) where vj is the character of a Fredholm module determined by (H, D), = S k the results of section 4 still hold for the character % i.e. "r,+z~ v, in H~+~(d) for any k -- i, 2 ... We let ch*(H, D) be the element of HeV(off) determined by any of the -r,. Corollary 3. -- Let d be a 9 algebra, H a Z/2 graded Hilbert space which is a 9 module over off, and D a (possibly unbounded) selfadjoint operator in H such that, ~) ~ D = -- De, ~) the domain of D is invariant by any a ~ off and [D, a] is bounded, 2') D -~ e .oq 'p. Then D satisfies conditions I) 2) 3) above and for any selfadjoint idempotent e ~ Mk(off), the operator D, = e(D | i) e /s selfadjoint in e(H | Ck). Its kernel is finite dimensional and invariant under ~, with Signature ,/Ker D, : ( [el, ch*(H, D) ). Proof. -- Since D -1 ~ Lap, one has D-lID, a] ~ .WP for all a s off, so that D satisfies I) 2) 3). For the rest of the proof we may assume that k = I. By ~), D, is densely defined in ell. It is selfadjoint by [57], since D -- (e De + (I -- e) D(I -- e)) is a bounded operator. Let f be the closure of D-le D, then f is a bounded operator with f-- e ~ Ae~ and f2 = f Let us show that the kernel ofj~ in eI-I is the same as the kernel of D,. Clearly ~ ~ Ker D, implies ~ s Kerfe. Conversely, let ~ ~ Kerfe. Let us show that e~ e Dora e De. Let ~, ~ Dora D, ~, ~ e~. Let ~, =f~, = D -x e D~,. One has fe~=o, hence ~,~o. Thus ~,--~,~DomD, ~,--~%~e~ and eD(~,~--~q,) ----eD~,--eD~,=o. This shows that e~DomeD and that eDe~=o. 283 68 ALAIN CONNES Now, as f--e e Lot, fe defines a Fredholm operator from eH to fH, and its kernel is finite dimensional. The operator e commutes with fe and one has, Signature (~/Ker D,) = dim Ker(fe), m -- dim Ker(fe),H-. Let us show that the codimension of the range offe infH + is equal to dim Ker(fe),a-. In fact both are equal to the codimension of the range of e De D -1 H- in ell-. Thus, Signature (~/Ker D,) = Index J? :eH + ->fH +. With the notation of proposition I the right side of the above equality is the index of (F~+), so that, by theorem 3. i, it is equal to .-., The conclusion follows from corollary 5 combined with the equality e = e*. [] In corollary 3 the condition " D is invertible " is still unnatural, we shall now show how to replace it by 7') (I + D~) -1 e S ~ Let H be a Z[2 graded Hilbert space which is a module over the algebra .~r Let D be a (possibly unbounded) selfadjoint operator in H verifying 0t) and [~) of Corollary 3- To make D invertible we shall form its cup product (cf. [6]) with the following simple Fredholm module (He, F0) over the algebra C. We let H e be the Z/2 graded Hilbert space H~=G, we let G act on the left in Heby X-+[~ :]eoL~a(H0), andwe let Proposition 4. -- a) Let H ---- H ~3 H c be the graded tensor product of H by H c viewed as an d | E = sr left module. For any m 4= o, m e R, the operator D,~ = D ~ i + mi ~ F c is an invertible selfadjoint operator in H which satisfies ~) ~) y) if D satisfies o~) ~) 7'). b) Corollary 3 still holds under this weaker hypothesis. c) ch*(H, Din)= [zm] e HOV(d) is independent of m (where % is the character of (H, D~,)). Proof. -- a) One has D~ = (D ~ + m 2) ~ i, so that D,~ is invertible. Moreover [D~tl = (D2+ m2)-m~ i e.Sf p. Since conditions ~) ~) are obviously satisfied by D,, we get a). b) Let ec=[: :]e~oqa(HG). Forany e=eZ=e*e~r onehas (e (~ %) D,~(e @ ec) = e De Q ec, thus Signature (e/Ker e De) = Signature (r | ec/Ker(e (~ ee) D,,(e G e~)) = <[e], ch*(H, D,,)> by corollary 3. c) Follows from Corollary 9 and Lemma 5. I. [] 284 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY The above construction of the operator D,, from the operator D associates to the Dirac operator in R 3 the Dirac Hamiltonian with mass m. Let now V be a compact even dimensional Spin manifold. Let g be a Riemannian metric on V, S the bundle of complex spinors and D the Dirac operator in L*(V, S) = H. By construction H is a Z/2 graded Hilbert space, with H  ---- L2(V, S +) and is a module over ~r C~176 One has: ~) ~D------D~; 9) the domain of D is invariant under any f e d and [D,f] is bounded; y') (I 4-D2) -les for any p>dimV. Thus proposition 4 applies and combined with proposition I b) it yields for each m e R, m # o an element %, of z~mv(c~(v)), the character of (H, Din). Theorem 5. -- a) With the above notation, vm(f ~ ...,f") is convergent, when m -+ 0% for any fo, . . .,f" e C~r b) The limit "c(f ~ ...,f") is given by .r ( fO, . . .,f") = f fO dfl ^ ... ^ df" 4- (S 2~1) (fo, ...,f") 4- (S co~) (fo, ...,f") 4- ... 4- S,/2 cos(,/4)(f o, ...,J") where S is the canonical operation Z~-+ Z~ +2 (el. Part 11I), o~j is the differential form Aj(px, ..., P~) describing the component of degree 4J of the .~ genus of V in terms of the curvature matrix of the metric g, and is considered as an element of Z~-4J(d) by the formula ,~(fo, fl, ...,f"-4J) = fvfO dfl A ... h df "-4j ^ co s . Here the manifold V is oriented by its Spin structure. This theorem will be proven in part III using the technique introduced by E. Getzler in [28]. 7- The odd dimensional case For nuclear C*-algebras A, there are two equivalent descriptions of the K-homology KI(A). The first, due to Brown, Douglas and Fillmore ([ii]) classifies extensions of A by the algebra o~f of compact operators, i.e. exact sequences, of C*-algebras and homomorphisms o -+ ~f -+ 8 -+ A -+ o. The second, due to Kasparov classifies Fredholm modules over the Z/2 graded C*-algebra A | C1 where C 1 is the following Z/2 graded Clifford algebra over C, C + ={7,1,),eC}, i the unit of C 1 C~ =(X~,XeC}, ~=I. 285 ALAIN CONNES 7o In the work of Helton and Howe on operators with trace class commutators and in the further work [23] [I2] [20], differential geometric invariants on V are assigned to an exact sequence of the form, o ---> -oq~ ---> a -+ C~(V) ~ o. In this section we shall clarify the link of these invariants with our Chern character. We show that, given a trivially Z]2 graded algebra d over C, :) ap-summable Fredholm module (H, F) over d | C1 yields an exact sequence, o ~.oq ~p/2 -+8 -+d ~o; 2) the cohomology class I-r] eH~"-l(d), m GN, m>p/2 of the character of the above Fredholm module only depends upon the associated exact sequence, and can be defined directly (without (H, F)); 3) when .~ = C=(V), the fundamental trace form ? of Helton and Howe ([3:]) is obtained from the character -r by complete antisymmetrisation: ? = Zr -r ~ Hence, using the results of part II (lemma 45 a) and theorem 46) we see that ? is the image of-r under the canonical map :: H~-I(C~(V)) -. H'-I(C~(V), C~(V)*). Since the kernel of I is the direct sum of the de Rham homology groups H2,,_a(V , C) @ H,,,_5(V, C) | @ Hi(V, C), we see that some information is lost in the process when the latter group is not trivial. This fits with the results of [3:] and [25] where the fundamental trace form is used either in low dimensions or for spheres. Our formalism thus gives an explicit formula for the lower homology classes of Helton and Howe ([3:]). Let us begin with :). We letH tbe theZ/2 gradedHilbert space H + =C, H~" =C. We let C 1 act inHlby, Lemma 1. -- Let d be a trivially Z/~ graded algebra, (K, P) a pair, where K is a Hilbert space in which d acts (by bounded operators), while P e .o~(K) satisfies the conditions a) [P,b] e.L~'P(K), Vbed, b) P~= i. (H, F) is a p-summable Fredholm module over the Z]2 graded algebra d | C 1. Proof. -- By construction H is a Z/2 graded d | C1 module. The operator F satisfies eF=--Fe, F 2= :. Finally for any x=aQI -q-b| x the graded commutator [F, x] is given by [--[P,b] --[P,a]] iEF, x] = [P, a] [ P, b] e .~P(H). [] 286 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 7I Lemma 2. -- Let % be the n-dimensional character of (H, F) for n >_ p -- I. Then, a) if n is even one has %=o; b) if n is odd, one has % = ~ | Y, where Y is the graded trace on Gt, Y(Z + ~t~) = ~t V X + ~t~ e Gx, and where n--1 , 0 %(a,...,a") = (--I) z c, Trace(P[P,a ~ [P,a x] ... [P,a"]), Va ie~r c) one has x~ e Z~.(..~r Proof. -- One has by definition (cf. remark i .6) %(x ~ ..., x") = (-- I) ~ c, Tr,(x ~ dx t ... dx"), dx j = i[F, x~], for x ~ ...,x"e~r174 x ~ homogeneous, q=~deg(x~+l). Replacing .~r by ~qr we may assume that ~1 is unital and that its unit acts as the identity in K. We shorten the notation and replace x| by 0tin ~r174 It acts in H by the matrix i- -i o~= [: :1. One has x~=ax for xe~r174 t and (r162 this shows i. J that when n is even, any to ~ f~" satisfies IXto ~ tool,, As e~ = -- ~r this shows that for n even, n~p-- I, one has Tr,(to) =o Vt0e~". By remark I .6, %(x ~ . .., x") = o for x i ~ d| Gx, x i homogeneous, Let n be odd. Z Ox ~ = o (mod ~). for a i ~ ,~, ei e { o, I }, Since Foc=--~F, one has d~=o, and hence, '~r = I (mod 2), %(a ~ 0r ..., a" o~',) = %(0ca ~ a a, ..., a"). Now ,or oo. = al = t ."l [ o thus, with --I O ~ to = o~a ~ da 1 ... da n, _x F(Feto -- toFr = _I iF~e da ~ ... da" 2 2 2 (- I) ~ (PIP, a ~ ... [P, a"]) | This shows that .. = ~,(a, ..., a") V(~'0 ~" ... ~). Tn(a 0 0~o, ., an o~n) , o It follows that Finally, the character % satisfies conditions a') b') of remark i. 6 y). < ~ z~(d). [] 287 ALAIN CONNES 7~ To any pair (K, P) verifying the conditions a) b) of lemma r, we have thus associated, for any odd n>p--i, the (n + 1)-linear functional -r~ e Z~(~r We shall now show that the class of z~ in H~(~r only depends on the extension of ~1 by .oq a~/z associated to (K, P) as follows, Iq-P Proposition 3. -- Let d, K, P be as above andput E 0 -- -- 9 .~e(K), E = Range of E0, p(b) =E 0bE 0 9 for any b 9 a) One has p(ab) -- p(a) o(b) 9 .LaP/2(E) for all a, b 9 d. b) Let g = p(zt)+ LaP/Z(E) C-qa(E), and d~" be the quotient of ~t by the ideal d" = {a 9 ~1, p(a) 9 .~~ then one has a natural exact sequence, o + sePn(E) -+ ~" -+ d' -+ o. Proof. -- a) Since p2= I, one has F_~=E 0. Hence E o abE o -- E o aE 0 bE o = -- E0[Eo, a] [Eo, b] 9 Na'/2(K). b) By a), g is a subalgebra of .la(E). One has s C ~' and p yields an isomorphism p' of ,~r with g'/.lap~. [] Let J=.i~ 'p/zC g. Then for any integer m>p/2 we have jmc.9,1, so that the trace defines a linear functional -~ on J" such that x(ab) = x(ba) for a 9 b 9 k -k- q_> m. Moreover p : ~1 -+ g is muhiplicative modulo J. We shall show in this generality how to get an element qh,,-1 of Z~'-x(~r and relate it to -r" in the above situation. Proposition 4. -- Let Y~ be an algebra, J c Z a two-sided ideal, m e N, and .r a linear functional on J'~ such that "r(ab) = v(ba) for a e J k, b 9 kq-q=m. Let p : d ~ Y, be a linear map which is multiplicative modulo J. a) Let ~ be the 2m-linear functional on d given by ~(a ~ ..., a ~"-1) = ~(~o ~ ... ~m-~) - ~(~ ~3 . . . ~-0, where ,j = p(a ~ a s+l) -- p(a ~) p(a~+l), j ---- o, I, ..., ~m -- I. Then ~ 9 Z~'~-l(d). b) Let p' : d --+ Z satisfy the same conditions as p, with p(a) -- p'(a) 9 for a 9 ~r then, with obvious notation, one has ~" -- r 9 B~m-~(~r c) Let (K, P) satisfy conditions a) b) of [emma I and "~" be given by lemma 2, for n=2m--I, m_>p. Let 2~---- g, J=LfP/2(E), and p be as in proposition 3. Then the corresponding ~ 9 z~m-l(d) satisfies v = - (2 -("+~) cZ ~) ~'. 288 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Proof. -- a) One has, by construction, ~(2,...,a 2m-l,a ~ =--@(a ~ Va jEd. With the notations of a), let ~+ = X(r ... r One has O(aJaJ-blaJ+2) __ ~(aJa j§ ~(a j+2) -- (o(aJa j+l a j+2) _ p(a ~) o(aJ+laJ+2)) = o(a j) p(a j+l a j+2) _ o(a j a j+l) o(a j+~) _-- o(a j) ej+l -- r p(aJ+2). Using this equality, we get b?+(a ~ a TM ) ?+(a "+1 a ~ 2, 9 9 *, -- o 9 a t~) = ~(p(a ~ sl ~... ~. - So... s._l p(.+l)). Similarly, with @- -- ?+ -- ~, we have b?-(a ~ . . ., a "+l) - ?-(a ~ a 1, a s, . . ., a "+1) = - ~(p(a 1) ~ ~... s.+~ - s~ ~.... s. pCa~ Thus bg(a ~ ..., a "+1) = cp+(a "+l a ~ ..., a") -- "r(~ 0 ... s,_ l p(a"+l)) 9 .., a n+l) ...... -- (?-(a ~ 2, + "r(p(2) ~ s.+l) = x(A~ s._l) where A : p(a "+1 a ~ a 1) -- p(a "+1 a ~ p(a 1) -- p(a "+1) r (P( a"+l a~ - p(a "+l) p(a ~ a~)) + s.+, p(a ~) = o. Therefore ~p e Z~(.~). b) Let L=p'--O; then Lis alinear map from a~ to J. With Pt----o+tL it is enough to show that the cocycle cpt associated to Pt, satisfies ~ ~p~ = bd h for a continuous family +t r C~-1(~) 9 Clearly it is enough to do it for t----o. Letting ---- ?t we have (J,) t=0' ,'(a ~ ..., a") =-r(A-- B), where A=g~...s,_l+SoV~S 4...s._l+ ... +~o~... r i t t B -- sl ~s 9 9 9 s, + s i s s ... s, + ... + S I S8 . 9 . ~n and cj = L(a ~ a j +1) _ 0(a j) L(a j +1) _ L(a ~) 9(a j +1). Let +o(a ~ "-1) =x(L(a ~ l~3... ~,-2) and let %(a ~ ..., a n-1 ) = d/o(aJ , a j +l, ..., aJ-'). Using the same equality as in a) we obtain b+~(a ~ ..., a") = "r((p(a ~ s 1 sa-.. L(a2u+l) ." s,-1) -- (~0 ... ~z~-2 P(a~) L(a2~+I) ..- s,-1) + (So... ~-~ L(a2~a2~+a) -.. r -- (r a~-2L(a~) P(a2~+1) "" s.-1) + (So ... ~.~_, L(a ~) ~+~ ... s._~ ~(a")) -- (p(a" a ~ 2) -- p(a" a ~ 0(2)) ~... s,~_2 L(a ~) ~a~+1 ..- s._,.)). 10 74 ALAIN CONNES The last two terms cancel the first two in bhbzk_l(a ~ ..., a") = "r((p(a" a ~ a 1) -- p(a") p(a ~ at)) ~... ~k-2 L(a~) -.. *,,-z) - (pCa ~) ~... L(a ~) ... ~,) - (~,... ~_~ ~_~ 9 .. ~,) -- (r L(a2k-1) ... r P(a~ 9 Thus we get, for k= 1, 2, . . ., m -- I, b(q,k-~ + +,k) ( a~ ..., a") = x((o(a ~ ,1 ... ~k-t L(a2*+t) ... r -- (O(a~ r ~k-3L(a*~-l) .-. *,-~) + (~o... *~k-24k''' e,-t) - (~ ... ~_~ ~,_~ ... ~,)). As bd~o(a ~ ..., a") = "r((o(a ~ L(a 1) ~... *,-1) -- (P(a~ ~1 ... L(a")) + (~ ~... ~.-1) - (,, ... ~.-~ ~')), one obtains n--1 y. %=r $=0 c) Let o(a)= EoaE o ~.oq'(E), where Eo---. One has o(a ~ aa) -- o(a ~ p(a 1) = -- Eo[Eo, a ~ [Eo, a 1] = -- I Eo[p ' aO ] [p, at]. Therefore, since E o commutes with [P, a ~ [P, al], m--1 rl (p(a~ a ~k+l) - p(a ~) p(a~+l)) = (_ 4)-'~ ~ [1', ~q. k=o S=o Thus we obtain q~( a~ -.-, a~-l) = Trace(eo e2 ... ~m--2) -- Trace(c1% .. 9 ~m-t) = (-- 4) -m Trace(Eo( h [P, a s] -- fi [P, a s + t])). j=o j=o Similarly, if we let Eo= t--Eo, E'=Range of Eo, p'(a) :EoaEo~.o~a(E ') for a e ~r we have, with obvious notation, 9'(a ~ ..., a ~-1) = (-- 4) -m Trace(E~( fi [P, a j] -- II [P, aS+l])). S =o S=o One has P=2E o-I =E o-Eo, thus ,P - ,P' = (-4)-" (- ~)"-~ (G.)-~ "~..' Since F_~+E 6= I, one has (,p + ,p') (a~ a-) = (- 4)-" Trace( n [1', aq -- I] [P,,~s+~]) = o. S=o S=o Thus ~=--2- ~-tc~-1%. [] 290 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY The construction of the character of an extension of d by -~P/~ can be summarized as follows: Theorem 5. -- a) Let E be a Hilbert space, p a linear map of ~t in .o~~ which is multi- plicative modulo .oq~ then the following functional %, n = 2m- I, re>p[2 belongs to %(a ~ ..., a") : -- 2 "+2 c, Trace((~ 0 ~... ~,-1) -- (s1r "" r where ~j = p(a ~ a j+t) -- p(a j) p(aJ+l). b) The class of % in H~(~r depends only on the quotient homomorphism c) The class of % in H~(~r is unaffected by a homotopy Ot such that i) [1 p,(ab) -- p,(a) p,(b)[[p~ is bounded on [o, I] for any a, b ~ ~; 2) for ae~r ~ eE, the map t-+p~(a) ~ is C 1. d) The index map Kl(d ) ---> Z is given by Index~(u) = ([u], %) V u e OL(d). e) One has SIr,] = [%+~] in H[+~(d). Proof. -- a) and b) follow from proposition 4. c) Follows from the proof of proposition 4 b). d) Follows from the equality Index ~(u) = Trace(i -- ~'(u- 1) ~(u))" -- Trace(I -- ~(u) ~'(u- 1)),, (of. Appendix I) and the definition of the pairing between Kl(d ) and I-IX(d ) (Part II, proposition I5). e) Follows from the following algebraic lemma, whose proof is left as an exercise to the reader. Lemma 6. -- With the notation of proposition 4 one has (2z-~ S)(~,.-1)= 4 (m + ~) ~2,.+1 We shall thus define the Chern character of the given extension as the element of H~d(~) = lim(H~"-l(d), S) given by any of the characters %, n odd. Let us now clarify the relation between v, and the fundamental trace form of Helton and Howe ([3I]). We assume now that ~r is commutative. The fundamental trace form is defined, under the hypothesis of theorem 5, by the equality, T(a ~ ..., a") : Trace(Zr p(a ~176 ... p(a~ where ~ runs through the group ~,+1 of all permutations of {o, i, ..., n} and c(e) denotes its signature. 291 76 ALAIN CONNES Proposition 7. -- Let d be a commutative algebra, p and E be as in theorem 5, p: ~r -+ ~e(~.)/~e,~(v.). a) FOr p = I trig fundamental trace form T(a", a x) is equal to ~-~ q(a ~ al). b) For p> I, one has (-- I) m (n -q- I) ]~((~) Tn(a0 aO(1 ) .... , aO(,))" T(a ~ ..., a") = 9 "+sc,, Proof. -- a) One has, by definition, "rl(a~ , a l) = (Trace(p(a ~ a 1) -- p(a ~ p(al)) -- Trace(p(d a ~ -- p(a 1) p(a~ As a la ~ ~ x one gets the result. b) For any n + i linear functional d/ on d, let Oqb be given by 0+(a ~ ..., a") = Z ~(~) +(a n(~ ..., an(")). ~n+l Since v, satisfies ~,(a 1, ..., a", a ~ = -- v,(a ~ ..., a"), one has Z r "r,(a ~ a "(1), ..., a a(")) _ ~ ~ ~(~) ~,(r ..., a-C-)) _ ~ K. n -~- I n~@n+X n + I Let us write T n = T: -- "7n, where, with the notation of theorem 5 a), + 0 2n+2 x, (a, ..., a") = -- c, Trace(% e~... r One has x[(a ~ .. .,a") = x, + (a, 1 ..., a~ and hence 0r, = 0~ + -- 0r; = 20~ +. As in the proof of a) one has p(a 2k a 2~+I) _ p(a ~) p(a 2k+I) _ (p(a 2k+I a ~) -- o(a 9"k+I) p(a~)) = [v(a~+~), v(a~)]. Let ~k be the transposition between 2k and 2k + I ; then m--1 + ).~ ( 1-I (I -- 0~2,k) ) T n = (-- I (-- 2n+2Cn) k=0 � Trace([o(a~ o(a~)] ... [p(a"-~), p(a")]). Since 0(I -- ~-a~) = 20, we get 0r + = ~-~ 0q~, where +(a ~ .... , a") = (-- I) ~ (-- e+~c,) � Trace([p(a~ o(al)] ... [p(a"-1), p(a")]). The result now follows easily. [] 292 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 8. Transversally elliptic operators for foHations Let (V, F) be a compact manifold with a smooth foliation F, given as an integrable subbundle F of TV. We shall show that any differential or pseudodifferential ope- rator D on V, which is transversally elliptic with respect to F yields a finitely summable Fredholm module over the convolution algebra d = C~(Graph(V, F)) ([i5] , [i6]). We deal here with the obvious notion of transversally elliptic operator; a more general notion will be handled in Part VI. Let E q- be complex vector bundles on V which are equivariant for the action of Graph(V, F) on V. This means that for any 7 ~ Graph(V, F), s(7) = x, r(7 ) =y, one is given a linear map 4 -+ 74 of F_~ to E~ with the obvious smoothness and compa- tibility conditions. Definition 1. -- Let D be a pseudodifferential operator of order n from E + to E-. Then D is transversalty elliptic with respect to F if and only if its principal symbol is a) invariant under holonomy, and b) invertible for 4  F, 4 9 o. More explicitly, a) means that for any 7 e Graph(V, F), 7:x~y, one has a((dT) t 4) = Yg(4) 7 -1, V 4 e F~, where d7 is the differential of the holonomy, a linear map from TflF, to Ty[F u. Let E = E + 9 E-, and let us show that each of the usual Sobolev spaces W'(V, E) of sections of E is a module over ~r = C~(Graph(V, F)). Let G = Graph(V, F). Lemma 2. -- For any s e R, the equality (k (x) fo~k(7)f(y), k e CT(G), fe W"(V, E), y = s(y), *f) defines a representation of C~(G) in W'(V, E). Before we prove it, we have to explain the notation. Elements of C~~ are not quite functions but sections of the line bundle s*(G), where s : G -+ V is the source map and ~ the line bundle (trivial on V) of i-densities in the leaf direction. This gives meaning to the integral fo x k(7)f(y ) for scalar functions f. For sections of E one has to replace f(y) e E u by 7f(Y) e E, and then the integral is performed in E,. When dealing with Sobolev spaces which are not spaces of functions (i.e. s < o) the statement means that k. extends by continuity to W'. Proof. -- The definition of the Sobolev spaces Ws(V, E) is invariant under diffeo- morphism. More precisely given open sets V1, V 2 C V and functions ~i e C~~ with (support ~) C V~, any partial diffeomorphism tF : V 1 -~ V, covered by a bundle map defines by the formula T4 = q~x tF*(~, 4) a bounded operator in each of the W s. Hence (as in [I5] ) to show that k, is bounded in W ~ one may assume that k ~ C~(Gw) C C~(G) where W is a small open set in V (i.e. the foliation F restricted 293 ALAIN CONNES to W is trivial) and G w is the graph of the restriction of F to W. ]'hen one can write k 9 as an integral of operators of translation along the plaques of W and the statement follows (say by taking the local Sobolev norms to be translation invariant). [] Note that unless F= = T, for all x, the operators k. are not smoothing; they are only smoothing in the leaf direction. Lemma 3. -- Let D be a transversaUy elliptic pseudo,differential operator from E + to E- (both bundles are holonomy equivariant and the transverse symbol of D is holonomy invariant). Let Q.be any (x) pseudo-differential operator on V from E- to E + with order -- q (with q = order D) and transverse symbol aD 1. Let H + = W~ E+), H- -~ W~ E-) and F = [ ~ ~o ] " Then ( for any s 9 It) the pair (H, F) is a pre-Fredholm module over .~ = C~~ It is p-summable for any p > Codim F = dim V -- dim F. Proof. -- Let us first show that k(F 2 -- I) and (F ~ -- I) k belong to .~v(W') for anys, andp>na=CodimF. (We take n=dimV, n x -dimF, n2=-CodimF). Both DO..-- x and QD -- I are pseudo-differential operators of order o on V with vanishing transversal symbol, and we shall show that if S is such an operator, then kS and Sk are in .~v(W ~ for any k e C~~ It is enough, as in the proof of lemma 2, to prove it for k e C~~ where W is a small open set in V. This shows that the problem is local, and hence we may as well take for (V, F) the torus T" = T "2 � T"' (T = R/Z) with the foliation whose leaves are the T"' � {x}, x 9 T"'. Let a be the total symbol of S; then S is of the form (2=)-" f e '<~ ~' 6(x, ~) f(x -- s) Z(s) ds d~, (sf) = where s varies in R" (which acts by translations on T"), ~ in R. = (R")*, and X e C~~ ") is identically i near o. One has G=T"'� '~� and k 9 acts on functions by (,,) fk(xDyx, x,)f(yx, x~) dy x where x = (xx, x,) 9 T". (kf) To show that Sk 9 .oq~v(W~ it is enough to show that, given s, the .~V-norm of (I + A)~ ~- A) -'/2, k~,(x) = expi2~(~,x), does not grow faster than a poly- nomial in 0r = (0it, ~t, ~z) e Z"' +"' 4-,, Also since any k= as an operator is the product of a multiplication operator by a k=,, ~' of the form (-- ~, ~, o), it is enough to estimate [1(i + A)mSk=,(I + A)-'/2IIv, and as k=, commutes with A one is reduced to the case s = o. Finally it is enough to estimate Z ]l S~ k~, ]]~, where S~ has total symbol ~ independent of x: ~(~) = f e ~<~'=> a(x, ~) dx, ~ 9 Z". (x) For instance take Q, with symbol o(~) = (i --Z) ~ where Z ~ C~~ is equal to t on V C F "l" and p : T ~ --+ F is a linear projection. 294 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Now both Ss and k s, are diagonal in the basis e~,,,, (e~,,,(x) = exp i2~<0t", x>, 0~" ~ Z"). The operator Ss multiplies e,,, by (%. ~)(2~,t"), where ~ is the Fourier transform of X), andks, (~'= (--[~,~,o)) multiplies %, by o if at' 1' #[~ and by x if ~'1' = [~" Thus (ll lip) = * p. This is finite for p > n,, since by hypothesis one has for ~ (and hence a s 9 ~) an inequality I (x, <c(, + l[ lll)(i + [l xll + l[ ,ll) --a Since the same inequality holds for the partial derivatives with respect to x, one gets that the Cs's (for the o's) are of rapid decay in a, thus the conclusion follows. Let us check that iF, k] z .~P, p > n~, for any k e C2(G ). If P : W' -,,- W '-~ is a pseudo-differential operator of order k and its principal symbol vanishes on F x, we have kP and Pk in .oq~ ', W '-k) for any p > nz. This shows that to prove that if the principal symbol of P is holonomy invariant one has [P, k] e -LP v, one can assume that k ~ C2(Gw), W a small open set. One is then back to the above case where V = T"' x T"'. Applying again the above result one can now assume that P is exactly invariant under the action of the compact group T'. Now the action of k s C 2(G) in W' is of the form kf = f~, k, Ut(f) dt, where U t is the translation by t e T"' and k t is the multiplication by a smooth function of xeV (and teT"'). Thus [P,k] =f[P, ktlU tdt =fPtUtdt where P, is pseudodifferential with order --x. Using Fourier expansion one checks that any keC~(G) is of the form k=k 1,k2, thus [P,k] =[P,k dk~+k1[P ,k2] and both terms are in .L~ 'v by the above result, rq Remark 4. -- In the special case when the foliation (V, F) comes from a locally free action of a Lie group H (not necessarily compact), the graph of (V, F) is equal to V � H. The convolution algebra C~(H) becomes a subalgebra of C~~ (by composing f z G~(H) with the proper projection V � H -+ H). Thus given a transversally elliptic operator D for (V, F) one can restrict its n-dimensional character (n > dim V -- dim F) to G~~ If both D and a parametrix Q. are exactly H-invariant, then one can compute r~ from the distribution character Z of D. The easy computation gives this restriction "~, 9 = s'z, (. = 2m). The central distribution X is defined as in [2] by the equality x(f) --= Trace(action of fin Ker D) -- Trace(action of fin Ker D t) (eft [2], Remark, p. 17). In the simplest examples with H non compact, the distribution character Z of D is not invariant under homotopy. However, by the above results, its class in H'(C~'(H)) is stable. 295 8o ALAIN CONNES 9. Fredholm modules over the convolution algebra of a Lie group Given a Lie group G, we let ~ = C~(G) be the convolution algebra of smooth functions with compact support. The Mi~enko extension ([5o]) gives a natural construction of Fredholm modules over ~r = C~(G) parametrized by a representation 7: of the maximal compact sub- group K of G. In this section we shall show in the two examples G = 113 and G = SL(2, 11) that the corresponding modules are p-summable and go a good way in the computation of their Chern characters. For SL(2, 11) we shall find a precise link with the surface of triangles in hyperbolic geometry which is a standard 2-cocycle in the group cohomology with coefficients in C. This link appears very natural if one has the example of G = 113 in mind. The method that we use goes over to semi-simple real Lie groups of real rank one. For such groups, in the corresponding symmetric spaces G/K the angle under which one sees a given compact set B C G/K from a distance d tends to o as e-~d when d-+ oo. Thus the p.summability follows as in lemma I below using Russo's theorem ([63], p. 57). For groups of higher rank the problem of constructing natural p-summable Fredholm modules is open. The case G = 11s Let G ---- 112. We define a Fredholm module over the convolution algebra d ---- C~(G) as follows: H + ---- L2(R2), H- ---- L2(R 2) (with the action of d by left [ o o0 translation), and F--= where the operator D:H + -+ H- is the multipli- cation by the complex valued function ~(z) = z/Izl v z ~11, = c, z, o. but this is unimportant since only its class The function ~ is not defined at z = o in Lc~ 2) matters to define D. Lemma 1. -- The pair (H +, F) is a Fredholm module over d = C~~ It is p-summable for any p > 2. Proof. -- One has F 2 = I by construction. For fe C~(R 2) and ~ e H +, one has ~) (s) ~(s) ff(t) ~(s - t) dt -- ff(t) ~(s -- t) ~(s -- t) dt ([D,f] ds'. = ff(s -- s')(r -- ~(s')) ~(s') Thus it is the integral operator with kernel k(s, s') =f(s -- s') (~(s) -- ~(s')). Since fhas compact support one has k(s,s') =o if d(s,s')>C for some C<o% where d is the Euclidean distance. Also for [ s] large and d(s, s') ~ C, the term ~(s) -- ~(s') 296 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 8i -- P one has is of the order of I[] s ]. This shows that for any p > 2, with q p~, -'-'j(J[k(s,s')lqds)mds'<oo (and similarly for the kernel k* = k(s', s) ). So Russo's theorem ([63] , p. 57) gives the conclusion. [] We shall now compute the character x 2 of (H, F). By a straightforward compu- tation, as in section I, one gets "r2(f~ f2 ) = ~i~r f~o + ~" + ~'=o f~176 fi(si) f~(s~) c(s~ sl' s~) dsi ds~" where the function c(s ~ S 1, SO'), s~eR 2 is given by ~ s) s - s ~ s - sO - sq as c( o, sl, = f with ~(s ~ s) = I -- ~(s) -~ q~(s --s~ To get this, one just has to write the trace of ds. an integral operator as the integral fk(s, s) We shall compute c(s ~ s i, s 2) ; we try to prove the next lemma in such a way that the proof goes over to the case of hyperbolic geometry. Lemma 2. -- One has c(s ~ s l, s 2) = 2i=(s i A s2). Proof. -- Let us first simplify the integrand [~(s ~ s) [~(s i, s -- s ~ [3(s ~', s -- s o -- si). For that we consider the Euclidean triangle with vertices o, s ~ s o + s 1 (remember that s o + s i + s 2 -- o). Then $(s) -1 ~0(s -- s ~ = e ~ where 0~ = -~ (o, s, s ~ is the angle (between --~ and =) obtained by looking at the edge (o,s o ) from s. Thus ~(s ~ = i --e ~. Similarly ~(s 1,s-s o ) = i--e ~ where [~=-~(o,s--s o ,s 1) =~(s o ,s,s ~ 1) and [3(s 2,s-s ~ x) = I --e iv where y= <~ (o,s--s ~ 1,-s ~ 1) =<)~(s ~ 1,s;o). Since ~+} +y=o we get (I -- e i~) (I -- e ~) (I -- e iv) ---- -- e i~- e i~- e ~v + e -i~ + e -i~ + e -iv = -- 2i(sin o~ + sin [5 + sin 7)" Define ds. S(A, B, C) = f (sin <~ AsB + sin <~ BsC + sin <~ CsA) Then c(s ~ s 1, s ~) = -- 2iS(A, B, C), where A=o, B=s o , C=s~ i form the triangle A, B, C. The integrand is 0(Is] -3) for large s so that the integral is well defined. To prove that S(A,B, C) is proportional to the Euclidean area of the triangle (A, B, C), the main point is to show that S is additive for triangles Ti, T 2 such that T 1 u T, is again a triangle. Let us prove this. Let , be the symmetry around the straight line which contains three vertices, say B1, C~ ---- B e and C, and let A1 ---- As be the only vertex outside this line. Writing the integral defining S as a limit of integrals 11 8~ ALAIN CONNES over a-invariant subsets eliminates the terms of the form sin-~ B 1 sC1, sin -~ B s sC s and sin <~c B 1 sCs. Moreover one has sin -~ C1 sA1 = -- sin %t A s sB2. The equality S(T1 w Ts) = S(T1) A- S(Ts) is now clear. A 1 ~ A s ~t The next point is that S(A, B, C) ~ o if the triangle ABC is positively oriented (i.e. if the orientation ABC fits with the natural orientation of R s ---- C). To see that, consider the disk I) R with center A and radius R. Then D R is invariant under the symmetries around both sides AC and AB so that the integral expressing S reduces to f sin -~g BsC. Let cr be the symmetry around BC, then the complement of the line BC in DR has (for R large) two components D' and D" such that a(D') C D". As on D"\~(D') one has sin <~c BsC > o, one gets the answer. DR 298 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 83 Now we can define the functional S on all subsets C of R ~ which are finite unions of closed triangles. The collection ~ of such subsets is a compact class in the sense of probability theory and thus we see that S defines a translation invariant Radon measure on R 2. Thus S is proportional to the area, and the constant of proportionality is easy to check. Corollary 3. -- The 2-dimensional character ~:2 of (H, F) /s v~(fo, fl, ff) = fro dfl ^ dj~= where fo,fl,fz ~ C~(R ~) have Fourier transforms fi. The ease G = SL(2, R) Let G = SL(2, R). In fact we shall use the realization of G as SU(I, I) i.e. 2 by 2 complex matrices g= [~ ~], 10c12--1~]~=I. Asmaximal compact subgroup K, we choose K = e_~0 , 0 e R/2~Z , and we identify G/K with the unit disk U ~z+~ for zeU, geG. Tothe in the complex plane C, on which G acts by gz -- ~z + ~ [[e~~ ~ ])=e~~ corresponds an induced line bundle En character X, of K given by Z~[ o e -~~ on U whose sections correspond canonically to functions ~ on G such that ~(gk) = x~(k) -l~(g) for k eK, g eG. The tangent bundle of U considered as a complex curve corresponds to Z~ where x2(k) = e i2~ which is the isotropy representation of K. At each point p e U, p + o, there is a unit tangent vector ~(p) e Tp(U), i.e. the one-dimensional complex tangent space T = E2, such that o belongs to the half-line starting at p in the direction of q~(p). (We use the unique G-invariant metric of cur- vature -- I: 2(1 --]Z[2) -I Idz] as a Riemannian metric on U.) 299 84 ALAIN CONNES Being a section of E~ (on U/{o}), q~ can be considered as a function on G; given geG, g=[~ ~], gCK one gets q)(g) -- I I It has a simple interpretation in terms of the KA + K decomposition. One has ?' / teR and k,k'eK, aeA +. ~(kak') = Z_2(k'), where A = sh t ch ' ' For each n ~ Z we let D. be the operator of multiplication by ~0 from L2(U, E.) to L~(U, E.+~) (1). Lemma 4. -- a) For each n e Z, the pair (H., F.) is a Fredhotm module over d = C~ ~ (G), where H + = L~(U, E.), H~- = L~(U, E.+~), [; b) The direct sum (| H,, | Fn) is also a FredhoIm module and is p-summable for any p > i, as well as all (H,, F,). Proof. -- The algebra ~ acts by left convolution in H,. One has by construction F~ = i, F2= I (where F = | It is clearly enough to prove b). Now H + = | H + = L2(G) where ~ acts by the left regular representation; also H- = LZ(G), and the operator D = | D, is given simply by the multiplication by the function q~(g). As in the case of R z we get 4) (g) ([D,f] f f(gg'-~) ~(g') dg' -- f f(gg'-~) 9(g') ~(g') dg' = fk(g, g') ~(g') dg' where k(g, g') = (~(g) -- ~(g') ) f(gg'-~). We want to show that (H, F) is 2-summable, i.e. that f l k(g, g') lg dg dg' < or. Sincefhas compact support, it is enough to show (with d a left invariant metric on G) that f l v(g-1) _ ~(g,-1)12 dg < ov where (g, g ) < C < oo d ' (a) In this special case G = SL(2, R) we rely on the natural conformal structure of U = G]K, but the true nature of the construction is to take the Clifford multiplication by c? (cf. [5o]) for which one just needs an invariant spin c structure on G]K. 300 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 85 (more precisely f M(g) 2 dg < oo where M(g) = sup {[ q0(g -1) -- ~(g,-1)[, d(g, g') < C}). But by construction, if we let p = gK, p' = g'K e U, then [q~(g -1) -- q~(g,-1) [ is of the order of the angle .,~.pOp'. The basic formula in hyperbolic geometry ch c ~ ch a ch b -- sh ash b cos-~ C implies that i q~(g)-i _ q~(g,-1) [ is of the order of exp(-- d(o,p)). Since the area of the disk of center o and radius d = d(o, p) is of the order of exp d, one easily gets f M(g)2 dg < oo. [] As in the case ofR 2 we shall now compute the 2-dimensional character v~ of (H, F). (The computation of "r~ (for (H,, F,)) and its relation with characters of discrete series is postponed until part VII.) Note that obviously z2 = Y.z~. By a straightforward computation we get .~(fo, f~,f,) = 2i7: ~,~,g, =1 fO(gO) fX(g~) f2(g2) c(gO, g~, g,) dgl dg2 where the function c(g ~ gl, g~), gi e G, gO gl g2= I, is given by c(gO, gl, g~) = f ~(gO, g) ~(gl, (gO)-l g) ~(g2, (gO g,)-l g) dg with ~3(g ~ g) = i -- ~(g)-i ~((g0)-i g). We now relate c(g ~ ga, g2) to the 2-cocycle A(g 1, g~) which is given by the (oriented) area of the hyperbolic triangle (in the PoincarE disk U) with vertices o, (gl)-~(o), g~(o). Note the relations A(kg ~,g~) =A(g ~k,kg ~) =A(g ~,g~k) =A(g a,g2) VkeK and A(gO, gl) = A(g~, g~) __ A(g~, gO) for gO g~ gg. = I. Lemma 5. -- One has c(g~ 2) = 4i~A(g ~, g~) (where gOgXg~ = I). Proof. -- Let A = o, B = g~ C = g~ and let us consider the hyperbolic triangle T = (A, B, C) in the Poincar6 disk U. For g E SL(2, R) let p = g(o) 9 U. The value of q~(g)-i ~((gO)-lg) only depends on the three points A, p, B. Since ~((gO)-lg), considered as a function of g, is the section of the tangent bundle T(U) which to p e U assigns the unit tangent vector at p looking at g~ = B, we get ~(g)-~ ~((gO)-~g) = exp i~):ApB. Thus ~(g0, g) = i -- exp i-4:ApB. Also ~(gl, (gO)-lg) = I -- exp i~3, where ~ = -<~ (o, (g0)-i g(o), g'(o)) = <): (g~ g(o), gO gl(o)) = .~ BpC, and similarly one has ~3 (g2, (gO gl)-1 g) = I -- exp/'F, 7 = <): CpA. As in the Euclidean case one has ~r q- ~ + y = o so that the same computation as in lemma 9. ~ gives c(g ~ gX, gZ) = _ ~iS(A, B, C), where S(A, B, C) = fu (sin -<): ApB -k- sin <;t BpC q- sin <~ CpA) dp. 301 86 ALAIN GONNES Now the proof of lemma 9.2 is written in such a way that it goes over without changes to the hyperbolic case. For instance it is still true that the disk D R with center A is invariant under the symmetries around AC and AB while the complement of the line BC in D R has two components D', D" with aBc(D') C D". Thus as in lemma 9.2 one gets a G-invariant Radon measure on U so that S is proportional to the hyper- bolic area. Appendix x: Schatten classes In this appendix we have gathered for the convenience of the reader the properties of the Schatten classes SOP needed in the text. Let H be a separable Hilbert space, SO(H) the algebra of bounded operators in H and SO~176 the ideal of compact operators. For T ~ SO| we let ~,(T) be the n-th singular value of T, i.e. the n-th eigenvalue of [ T[ = (T* T) 1/~ (cf. [63] ). By definition, the Schatten class SOP(H) is, for p e [i, oo[, SOP(H) ---- {T ~ SO(H), Z~.(T)P < oo}. Proposition 1. -- a) SOP(H) is a two sided ideal in SO(H). b) SOP(H) is a Banach space for the norm IIT = (z~t,(T)P) lip. c) SOP(H) c SO (H) for p < r I I I d) Let p, q, r ~ [I, OO] with -=- +-. For any S ESOP(H), T~SOq(H), one r p q has ST eso'(H) and IISTll,< IlSll. llYllr Proof. -- See [63]. One could equivalently define SOP(H) starting from the trace on S~ which we consider as a weight, i.e. a map: SO(H) + -+ [o, oo] defined by Trace(T) = Z (T~,, ~,) for any orthonormal basis (~,) of H and any T e SO(H) +. (See [51] theorem 2.14. ) Proposition 2. -- a) SOP(H) = {T e SO(H), Trace IT] p < oo}. b) For T ~SOP(H) one has [ITllp = (Trace [TIP) 1/p. c) Trace(A* A) -- Trace(AA*) for all A e SO(H). d) The trace extends by linearity to a linear functional on SOX(H) and Trace(T) = Z <T~., ~,> for T E SOl(H) and any orthonormal basis (~,) of H. e) ]Trace(T)] ~ 117111 for T esol(H). 302 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Proof.- See [5 I] and [63]. The next theorem, due to Lidskii, expresses Trace T from the eigenvalues of T. Since LalC Laoo the eigenvalues of T e Lax form a sequence (~,), with ~, ---> o when n~oo (see [63]). Theorem 3. -- Let T 9 .El(H), then Z I X.(T) [ < oo and Trace T = ZZ,(T). Corollary 4. ~ Let A, B e La(H) be such that AB and BA belong to Lal(H). Then Trace(AB) = Trace(BA) (el. [63] , p. 5o). We shall now prove two results needed in part I. They are an easy modification of lemma 3.2, p. i58 in [30]. Proposition 5. -- Let p e [I, or[, S, T 9 La(H) and assume that [S, T] e LaP(H). Then: cr if f is an analytic function in a neighborhood of the spectrum of S, one has [f(S), T] e LaP(H); f~) if S is selfadjoint and if f is a C ~~ function on the spectrum of S, one has [f(S), T] e LaP(H). Proof. -- ~z) Let y be a simple closed curve containing the spectrum of S, with f analytic on y. Then, f(s) = (i/2i=) f.ff(x)(~, - s) -1 dx and hence, [f(S), T] ---- (I[2i~) fvf(k ) [(), -- S) -1, T] dL Now [(k -- S) -~, T] = (), -- S) -~ IS, T] (), -- S) -1, which implies that the map k ~f(),) [(k -- S) -1, T] is a continuous function from y to LaP(H). Thus the integral converges in LaP(H) and [f(S), T] e LaP(H). ~) Let us show that II[e"S,W]llp is O(Itl) when t~oo. For any U~La(H), with [U, T] e Lap one has, II[U",T]llp<nll[W,T]llpllWll o-1 Vn ~N. This shows that IIEe"S,W]llp is bounded on any bounded interval. Since, with U ---- e us, one gets, IIEe "'s, W]llp < n II[e "s, W]llp v n EN, it follows that II[e"S,T]llp<C(i + Itl) for all t. Then take f to be compactly supported, so that f=g with (1 + Itl) g eL~(R). This yields, lilY(S), W]llp < f lg(t)l II[e ''s, T]llpdt < C f lg(t)l(I + Itl) dt< oo [] 303 88 ALAIN CONNES Proposition 6. (Cf. [34]-) -- Let p e [I, oo] and P, Q, e s be such that I -- PO ~ .SfP(H), I -- QP e .~P(H). Then P is a Fredholm operator and for any integer n > p, one has, Index P = Trace(I -- Qp)n __ Trace(i -- PQ)". Proof. -- Since I -- QP and i -- PQ are compact operators, P is a Fredholm operator. Moreover I is an isolated point in K = { I } t3 Spectrum(i -- PQ) t3 Spectrum(I -- QP). Let y be the boundary of a small closed disk D with center I such that D n K = { i }. Set e=~ ifv dX -- xfv dX . Then e = e ~, f =fs; Ex = Range of e, F x = Range off are finite dimensional, and admit respectively E 2 = Kere, F 2 = Kerr as supplements in H. For any ~z ~C one has, Q(~ - PQ) = (~ - QP) Q. Thus, for any XCK, one has (X -- (I -- Qp))-I Q= Q(x- (I - pQ))-l. This shows that Qf = eQ and similarly that Pe =fP. Thus, P(Ex) C F1, P(Es) C V,, Q(F1) C E,, Q(F2) C E,. Let Pj (resp. Q~) be the restriction of P (resp. Q) to Ej (resp. Fj), j = I, 2. By cons- truction the restrictions of QP to E~ and of PQ. to F~ are invertible operators, and hence, a) Index P ----- dim E x -- dim Fx, b) Trace(IE, -- Q2 P2)" : Trace(IF. -- P~ Q.2)" v n > p. The spectrum of IE, -- Q1 P1 and of IF, -- P1 Qa contains only { I }, thus, c) Trace(IE, -- Q1 P1)" -- Trace(IF, -- Pt Q1) n = dim E 1 -- dim F 1. Combining a), b), c), one gets the conclusion. [] Appendix 2: Fredholm modules The notion of Fredholm module is due to Atiyah [3] in the even case, and to Brown, Douglas, Fillmore [I i] and Kasparov [42] in the odd case. Their definitions are slightly different from the definition below and our aim is to clarify this point. Let X be a compact space and A-= C(X). An element of Atiyah's Ell(X) is given by two representations, a+ : A -+ s176 a- : a -+ .W(H-) 304 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY of A in Hilbert spaces H +, H- and a Fredholm operator P : H + -+ H- with para- metrix Q, which intertwines a + and a- modulo compact operators, Po +(a) Q-a-(a) cot ~ Va 9 The typical example is obtained when X = V is a smooth compact manifold, H + = L2(~ +) are Hilbert spaces of square integrable sections of bundles ~ over V, a~: are the obvious actions of C(V) by multiplication, and P is an elliptic operator of order o from ~+ to ~-. With Q a parametrix of P, let F e ~(H + | H-) be given by .[; One has, ~) H is a Z/2 graded *module over A= C(X), ~) [F,a] 9 Va 9 y) F 2- I 9 Note that in general F 2 4= I since P is not invertible. Definition 1. -- Let ~1 be a Z/2 graded algebra over C. Then a pre-Fredholm module over d (resp. Fredholm module over d) is a pair (H, F) where I) H is a Z[2 graded Itilbert space and a graded left d-module, 2) Feff(H), Fs=--r [F,a] E~ Va 9 3) a( F~ -- I) 9 ~f V a 9 ~ (resp. F ~ = I). We shall now show how to associate canonically to each pre-Fredholm module a Fredholm module. Let Hr be the Z[2 graded Hilbert space Hg =C and let H=HQHr be the graded tensor product ofHby H 0. One has H+ =H +| H- =H-| +. We turn H into a graded left d-module by a(~ (~ ~) = a~ Q e~ Va~d, ~eH, "~eH c, ext, w,th .__[; we define where e c .~e(Hc) , e['o :1 P '-PQ] 0.= -Qy (Qy - 2) o_..1' -vQ One checks that = I, QP = I, so that ~2= i. P~7). ~~ Proposition 2. -- Let (H, F) be a pre-Fredholm module over d. a) (H, F) is a Fredholm module over d. b) Let H 0 be the Hilbert space H with opposite Z[2 grading and o-module structure over ~1, then (H0, o) is a pre,Fredholm module over dt. 12 ALAIN CONNES 9 ~ c) One has ~ = H r H 0 as a Z/2 graded sg-module, and a(ff--F| e~ Vae~r Proof. -- a) Since F ~ = I, conditions I) and 3) are verified. Let us check that IF, a] e.~e', Va e~r One has by hypothesis [F,a] eOff, a(F ~-- I) e.~ and hence (1 ~- I) a~r Let us first assume that a is even. One has Pa -- aP a(i -- PQ,)] e .Y{'. Ptz -- aP = (I -- Q P) a Let now a be odd and Since 0. = ~-' one gets 0." -- ,,~ ~,X", hence IF, a] e .X'. LP(H) be the corresponding operator in H. By hypothesis one has [:. 1 (QP - i) al~ e ~, a~t(QP -- I) e gr (PQ-- x) a,t e ~", at2(PO _- I) e ~. The action of a in ~=H+| is given by the matrix T = T' where T'= t T"= [o [,: :1 ?: :1 One has to check that T" ~ + 0.T' e ~ and T' i~ + fiT" e or. This follows easily. from our hypothesis. b) Obvious. [o ~ [: o] [o :] Then F'= with P'= , Q' = c) Let F'=F| P' o With a even one has, a(~ P') = o a(Q-Q') =[a(2-QP) Q-aQo a(I-QP)] e~'o The odd case is treated similarly. [] Let p e [ I, oo[. We shall say that a pre-Fredholm module (H, F) over d is p-summable when, a) IF, a] e.~'(H) for aed, 13) a(F ~- I) e.LaP(H) for aed. Proposition 3. -- Let (H, F) be a p,summable pre-Fredhotm module over d. Then (ffI, F) is a p-summable Fredholm module. Proof. -- In the proof of proposition 2 one can replace 3g" by any two-sided ideal. [] We shall now discuss the index map associated to a Fredholm module. 306 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Lemma 4. -- Let s/ be a Z/2 graded algebra, (H, F) a Fredholm module over ~/. a) Let ff = s/ | C be obtained from ~t by adjoining a unit. Let ~ act in H by (a+),I)~=at +?~ for ar ),eC. Then (H,F) is a Fredholm module over ~. b) Let H,~=H| F.----F| and M.(.~r =~r174 act in H,, in the obvious way. Then (H,, F,) is a Fredholm module over M,(~). The proof is obvious. Let us now assume that ~ is trivially graded. Proposition 5. -- Let (H, F) be a Fredholm module over d. There exists a unique additive map ~ : K0(~' ) -+ Z such that for any idempotent e e M,(d), q~([e]) is the index of the Fredholm operator from ell+, to eI-'I-; given by T~=eF,~ V~eeH +. Proof. -- One checks that T is a Fredholm operator with parametrix T' where eF, ~ for ~q e eHn, and that the index of T is an additive function of the class T' ~q of e in K0(~ ). Finally we shall relate the above notion of Fredholm module with the Kasparov A _ B bimodules, we recall (cf. [42]). Definition ft. -- Let A and B be C*-algebras. A Kasparov A -- B bimodule is given by l) a Z/2 graded C*-module 8 = ~+ | o ~- over B, 2) a 9 homomorphism 7: of A in .L~'B(8), 3) an element F of .oq~B( e) such that Va cA, a(V - r') VaeA, Va~A. v) a( i) (Cf. [42] for the notions of endomorphism (-9~B(8)) of 8 and of compact endomor- phism (~g'n(r Let us take B = C. Then a Kasparov A--C bimodule is in particular a pre-Fredholm module over A. Conversely, Proposition 7. -- Let A be a C*-aIgebra and (H, F) a * Fredholm module over A. Let F' = F [F[-1. Then (H,F') is a Kasparov A-- C bimodule. Moreover, for each t, F l = F 1F [- t defines a * Fredholm module (H, Fj). Proof. -- One has [F*F,a] e~Y', VaeA, thus [IFl',a] cog" for seR, aeA. Let F = J A be the polar decomposition of F, with A ---- I F ]. One has F- 1 _-- A- 1 j-1 which gives the right polar decomposition of F ---- F -1, thus J -j-1 and J -j* -- F', so that (H, F') is a Kasparov A -- C bimodule. Finally jAj -1 = A -1 and hence j As j-1 _~ A- s for any s e R, so that J A s is an involution for any s. It follows that (H, F,) is a * Fredholm module. [] 307 9o ALAIN CONNES Appendix 3: Stability under holomorphic functional calculus Let A be a Banach algebra over C and d a subalgebra of A, A and ,~ be obtained by adjoining a unit. Definition 1. -- d is stable under holomorphic functional calculus if and only if for any hen and aeM, CM,(A) one has, f(a) M.(d) for any function f holomorphic in a neighborhood of the spectrum of a in M,(A). In particular one has GL,(d ~) = GL,(.~) n M,(~7), hence if we endow GL,(~ ~) with the induced topology we get a topological group which is locally contractible as a topological space. We recall the density theorem (cf. [4], [4o]). Proposition 2. -- Let d be a dense subalgebra of A, stable under holomorphic functional calculus. a) The inclusion i : d ~ A is an isomorphism of Ko-groups i,: Ko(d ) ~ Ko(A ). b) Let GL~(~ be the inductive limit of the topological groups GLn(~7). Then i. yields an isomorphism, ~k(GL~(~7)) ---> =k(GL=(A)) --- Kk+I(A ). Let now (H, F) be a Fredholm module over the Banach algebra A and assume that the corresponding homomorphism of A in .'Ae(H) is continuous. Proposition 3. -- Let pe[i, oo[ and d={aeA,[F,a]e.s Then ~ is a subalgebra of A stable under holomorphic functional calculus. Proof. -- One has IF, ab] = IF, a] b + a[F, b] for a, b e A. Thus as .s is a two-sided ideal in .oq~ ~r is a subalgebra of A. Let n ~N, (H,, F,) be the Fredholm module over M,(A) given by lemma 4 b) of Appendix 2 and ~, the corres- ponding homomorphism: M,(A) ---> .ge(H,). One has, M,(d) = {a e M,(~,), IF,, 7:,(a)] e .o'AeP(H,)}. Moreover Sp(n,(a)) C Sp(a), and since =, is continuous, ~:,(f(a)) =f(~,(a)) for anyf holomorphic on Sp(a). The conclusion follows from proposition 5 of Appendix I. [] Let now A be a C*-algebra and ~r a dense 9 subalgebra of A stable under holo- morphic functional calculus. Proposition 4. -- Let (H, F) be a 9 Fredholm module over .~. Then the corresponding 9 homomorphism ~ of~in .s is continuous and extends to a 9 homomorphism ~ of A in .Lf(H) yielding a * Fredholm module over A. 308 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Proof. -- We can assume that d and A are unital and ~(i) = i. Let a e d, then the Spectrum of a* a in .~ is the same as its Spectrum in A. Thus the norm of a in A is ]] a ]] = pl/~ where 0 is the radius of the Spectrum of a* a in ~f. One has Spectrum (=(a* a)) C Spectrum(a* a), thus [] =(a)][* = Spectral radius of~(a* a) < p = ][ a][ 2. This shows that rc is continuous. Let ~ be the corresponding 9 homomorphism of A in .o~~ For a e A, I-F, ~(a)] is a norm limit of I-F, =(a,)], a, e d, which are compact operators by hypothesis, thus [l v, ~(a)] e 3/" for all a e A. [] 309 II. -- DE R.HAM HOMOLOGY AND NON COMMUTATIVE ALGEBRA In part I the construction of the Chern character of an element of K-homology led to the definition of a purely algebraic cohomology theory H~(d). By construction, given any (possibly non commutative) algebra ~r over C, H~(~r is the cohomology of the complex (C~, b) where C~ is the space of (n + 1)-linear functionals q~ on d such that qo(a 1,...,a",a ~ ~-(-- I)"~(a ~ Va ~ed and where b is the Hochschild coboundary map given by (bqg) (a 0, ..., a n+l ) = ~ (-- i)/r 0, ..., aJa j+l, ..., a n+l ) j=0 + (-- I)"+lT(a"41a ~ ...,a"). Moreover H~(d) turned out to be naturally a module over H~((1) which is a polynomial ring with one generator r of degree 2. In this second part we shall develop this cohomology theory from scratch, using part I only as a motivation. We shall arrive, in section 4, at an exact couple of the form H*(d, d*) H~(d) > H~(~) relating H~(d) to the Hochschild cohomology of ~ with coefficients in the bimodule of linear functionals on d. This will give a powerful tool to compute H~(d) since Hochschild cohomology, defined as a derived functor, is computable via an arbitrary resolution of the bimodule d (cf. [I3] [47]). For instance, if one takes for d the algebra C ~ (V) of smooth functions on a compact manifold V and imposes the obvious continuity to multilinear functionals on d, one arrives quickly at the equality (for arbitrary n) H"(d, ~*) = space of all de Rham currents of dimension n. (This will be dealt with in section 5. The purely algebraic results of sections I to 4 easily adapt to the topological situation.) The operator I o B : Hn(~ r d*) -+ Hn-l(d, d*) coincides with the usual de Rham boundary for currents, and the computation of H~(~) will follow easily (cf. section 5). In particular we shall get H*(~ r = Ordinary de Rham homology of V, 310 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY where H*(d) is defined as in part I by H'(d) = H;,(d) As another application we shall compute H*(d) for the following highly non commutative algebra, Fix 0 ~ R/Z, 0 r Q./Z. Then ~10 is defined by d 0 ~- { Y~an,. U" V'; (a., ,~)., m ~ Z sequence of rapid decay }, where VU = (exp i2n0)UV gives the product rule. The algebra d0 corresponds to the " irrational rotation C*-algebra " studied by Rieffel [58] and Pimsner and Voiculescu [55]. It arises in the study of the Kronecker foliation of the 2-torus [16]. In section I we introduce the following notion of cycle over an algebra ~ which is crucial both for the construction of the cup product H~(~) � H~(~) -~ H~ +'(~ | ~) and for the construction of B : Hn+l(d, ~*) ---~H~(~). By definition a cycle of dimension n is a triple (~, d, f) where ~ = 0 ~i is a j-0 algebra, d is a graded derivation of degree I such that d ~ = o, and f : ~" -~ (l graded is a closed graded trace. A cycle over an algebra d is given by a homomorphism p: ~r _+ f~0 where (f2, d, f) is a cycle. In part I we saw that any p-summable Fredholm module over .& yields such a cycle. Here are some other examples. 1) Foliations Let (V, F) be a transversally oriented foliated manifold. Using the canonical integral of operator valued transverse differential forms [i4] of degree q = Codlin F, we shall construct in Part VIa cycle of dimension q over the algebra d= C~~ F)). 2) C* dynamical systems Given a C* dynamical system (A, G, ~) (el. [I9] ) where G is a Lie group, the construction of [I9] associates a cycle on the algebra A ~ of smooth elements of A to any pair of an invariant trace -r on A and a closed element v ~ A (Lie G). 3) Discrete groups In part V we shall associate a cycle on the group algebra C(F) to any group cocycle co e Z"(F, C) and obtain in this way a natural map of the group eohomology Hn(I ", C) to H[(C(r)). Given a cycle of dimension n, ~r ~ ~ over d, its character is the (n + I)-linear functional ~(a~ ..., a") = rp(a ~ dp(a ... dp(a"). 311 9 6 ALAIN CONNES We show that x 9 Z[(d) =- C~(d) r~ Ker b, that any clement of Z~(d) appears in this way and that the elements of B~(d) = bC~-l(d) are those coming from cycles with fl ~ flabby. (See [40] for the definition of a flabby algebra). Then the straightforward notion of tensor product of two cycles gives a cup product HI(d) | H~'(~) -+ H~ +'~(d | ~). We then check that H~(C) is a polynomial ring with a canonical generator a of degree 2 and we define at the level of cochains the map S : HI(d) -+ H[+*(M) given by cup product by a. In section 2 we show that the standard construction of the Chern character by connexion and curvature gives a pairing of H~V(d) with the algebraic K theory group K0(~r ) and of H~d(d) with K x. The invariance of this pairing naturally yields the group H*(d) = H~(~) | C, inductive limit of the groups H~(d) with map S. We then discuss the invariance of H~(d) under Morita equivalence. In section 3 we show that two cycles over ~ are cobordant (cf. 3 for the definition of cobordism) if and only if their characters xl, x, differ by an element of the image of B, where B is a canonical map of the Hochschild cohomology Hn+l(d, d*) to H~(d) defined as follows: = 2; (0 yEF where F is the group of cyclic permutations of {o, ..., n}, ~ ..., a = ..., r is the signature and (B0-r)(a ~ ='r(~,a ~ ~) § (-- ~)"r(a ~ '~,~) for all g a d. Thus defined at the level of cochains, B : C"(~', .~/*) + C"-l(d, ~*) commutes (in the graded sense) with the Hochschild coboundary b, which yields the basic double complex of section 4. The above result yields a new interpretation of H*(d) as H*(~) = (Cobordism group of cycles over a~r | a ~ (I which is completed in section 4 thanks to the exact triangle H*(d, ~*) H~(d) + H~(~ r where I is induced by the inclusion map from the subcomplex C[ to C n. This exact triangle gives in particular the characterization of the image of S which was missing in part I (cf. theorem 16): z 9 S if and only if 9 is a Hochschild coboundary. It also proves that H~(~/) is periodic with period 2 above the Hochschild dimension of ~r 312 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY By comparing the above exact triangle with the derived exact sequence of o ~ C~ -+ C" ~ C"/C~ -~ o we prove that there is a natural isomorphism H"(C/Cx) ~ H~-1(~r We then show that the cohomology of the double complex C"' " = C"-'(d, d*), d~ = b, d~ = B is equal to HeY(d) for n even, and H~162 for n odd. The spectral sequence associated to the first filtration ( 2~ C"") does not converge in general, and in fact has initial n>p term E2 always o. This and the equality S = bB -x are the technical facts allowing to identify the cohomology of this double complex with H*(~r The spectral sequence associated to the second filtration ( ~ C"") is always ra>q convergent. It coincides with the spectral sequence associated to the above exact couple and i) its initial term E 1 is the complex (H"(d, d*), I o B) of Hochschild cohomology groups with the differential given by the map I o B; 2) its limit is the graded group associated to the filtration F"(H*(d)) by dimensions of cycles. Finally we note that in a purely algebraic context the homology theory (which is dual to the cohomology theory we describe here) is more natural. All the results of our paper are easily transposed to the homological side. However, from the point of view of analysis, the cohomology appeared more naturally and, for technical reasons (non Hausdorff quotient spaces), it is not in general the dual of the homology theory. This motivates our choice. Part II is organized as follows: CONTENTS x. Definition of H~(~ r and cup product ...................................................... 97 2. Pairing of H~(~r with Kl(~r i = o, ~ ................................................... xo7 3- Cobordism of cycles and the operator B .................................................... x 14 4. The exact couple relating H~(~r to Hochschild cohomology ................................. tx 9 5- Locally convex algebras ................................................................... 125 6. Examples ................................................................................ x27 I. Definition of Hi(d ) and cup product By a cycle of dimension n, we shall mean a triple (f~, d, f) where is a graded algebra, d is a graded derivation of degree i with d z = o and : ~" --> C is a closed graded trace. 13 98 ALAIN CONNES Thus one has: I) f~� ~+~ Vi, j 9 iWjs 2) dta' c '+I, a(toto') = (ato) to' + (-- to #to,, d" = o; Given two cycles fl, fl' of dimension n, their sum f~| is defined by (~")~ = f~ @ fU, (tox, to~) (to~, to~) = (to~ to,, to~ to~), d(to, to') = (dco, dto') and Given cycles f~, f~' of dimensions n and n', their tensor product f~" = ~ | f~' is the cycle of dimension n + n' which as a differential graded algebra is the tensor product of (f~, d) by (~', d'), and where f(to| fto' V~ 9 co'eft'"'. For example, let V be a smooth compact manifold, and let C be a closed current of dimension q (<dimV) on V. Let f~i, i 9 ...,q} be the space &~ AIT'V) of smooth differential forms of degree i. With the usual product structure and dif- ferentiation f~ = O f~i is a differential algebra, on which the equality f to = (C, to ), i~0 for to 9 f~q, defines a closed graded trace. In this example f~ was graded commutative but this is not required in general. Now let ~1 be an algebra, and f~(~r be the universal graded differential algebra associated to aar ([I] [39]). Proposition 1. ~ Let .r be an (n + I )-linear functional on d. Then the following conditions are equivalent: I) There exists an n-dimensional cycle (~, d, f) and a homomorphism p: ~ -+ ~o such that v(a ~ a') p(a ~ d(p(aX)) .. d(pCa")) V a ~ ., e d. 9 . .~ -~- f " . . a n 2) There exists a closed graded trace T of dimension n on f~(~) such that 9 .,~ 9 ., a n "r(a ~ a") =T(a ~ t .. da n ) V ~ a ., 9 d. 9 "*, ~ "''9 "~ an 3) One has r(a 1, a", a ~ (-- x)" .r(a ~ a") for a ~ , 9 ~t and (-- I) i'~(a0, ..., a iai+l ..., a n+l ) + (-- I) "+l"r(a "+1 a ~ ..., a") = o for a ~ ..., a "+1 9 d. Proof. ~ Let us first recall the construction of the universal algebra f~ (d) (It] [39])- Even if d is already unital, let ff be the algebra obtained from d by adjoining a unit: ff={a+)~x;aed,), 9 For each heN, n~ i, let fP(d) be the linear space -+ o-(d) d. 314 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 99 The differential d:~n-+ ~n+1 is given by d((a ~ + X ~174174 | = ~|176174 | 9 By construction one has d 2 ---- o. Let us now define the product ~i x E~ i -* f~ i+i. One first defines a right d-module structure on El" by the equality (~~174174 | n) a-- ~ (-- I)n-i~~174174174 j=O Let us check that (toa) b = to(ab) 'v' to 9 fV', a, b 9 ~r One has n+l ('d'O| ... |174 aJ+l| ... |174 n+l) a n+2 = ~_a Cj, k O~j,k k~O where e~, k = o ifj = k, e~,~ = (-- i) "+k-1 if j< k and cj, k : (-- I) n-k if k <j = = a~174 |174 |174 | 2. while % k ak, j ......... Thus if one expands ((,~o| ... | a "+1) a "+2, one gets twice the term ~j,k for j, k 9 { o, I, ..., n} and with opposite signs: (-- I) "-j (-- r) "+l-k and (-- I) "-k (-- I) ~-~. Thus ((~o|174 ~)a.+l)a.+2= ~ (_ i),-~r j=O _--_ (~o| ... | a,,+~). This right action of ~r on f~" extends to a unital action of ~ One then defines the product: f~ x ~ -4 f~i+j by to(~~174174 | = to~o|174 | v co 9 ~. It is then immediate that the product is associative. With to=~~174 l|174 . 9 one has, for a 9162 d(toa) = ~ (-- I)n-iI|176174 |174 | j=O n+l (do))a= ~ (-- I)"+l-iI|176174174174174 j=O = (-- I) "-1 to da + d(toa). Thus (~, d) is a differential graded algebra, and the equality ~O da 1. . . da ~ = ~o | a l | . . . | a,, shows that it is generated by d. One checks that any homomorphism ~r -~ f~,o of ~r in a differential graded algebra (f2', d'), d'2= o, extends to a homomorphism ~ of (f~(d), d) to (ta', d') with -p(.~o dat ... da") = p(a ~ a'(p(al)) d'(o(a2)) ... d'(p(a")) + X ~ d'(p(a')) ... d'(p(a")) for a i 9162 ~o 9 ~-o = (a o,xo). 315 loo ALAIN CONNES Thus I) and 2) arc obviously equivalent. Let us show that 3) =~ 2). Given any (n + i)-linear functional q~ on d, define ~ as a linear functional on ~"(d) by ~((a ~ + x ~ ,) | (1' | | a") = ~0((1 ~ (1', ..., (1"). By construction one has r = o for all co e s Now, with ": "satisfying 3) let us show that "~ is a graded trace. We have to show that ~(((1o d(11 ... d(1~)((1,,+, d(1~+2 ... a(1,+,)) = (-- ,)k("--k)'~((ak+' dak+2.., dan+')(a~ dak)). Using the definition of the product in ~(~/) the first term gives Z (- ,)~-~ ~((10, ..., (1j (1j+,, ..-, ~,+,), and the second one gives n--k Z (-- I) k(n-k)+n-k-j Z((1 TM, ..., (1k+l+j (1k+t+j+t, ..-, ak). j=0 The cyclic permutation ),, )~(l) ---- k + x + t, has a signature equal to (-- i) "(k+t) so that, as .:x = r x by hypothesis, the second term gives n+l - Z (- ,)~-j~((1o, ..., (1j(1j+,, ...,(1,+~). k+t Hence the equality follows from the second hypothesis on ":. Let us show that I) :~3). We can assume that ~----s ~ One has ,,r O, (ll, ..., an) = f ((1O d(11)(d(l?. . " d(1n) = (__ i)n-l f (d(12 . . . da")(a~ da 1) = (- ,)"f(a(1~... da"a(1 ~ a' = (-- ,)".:((1', ..., (1", (1o). To prove the second property we shall only use the equality fat~ = f,oa for o) egl", a e.~. From the equality d(ab) = (da) b + a db it follows that (da a . .. da '~) a'~+ ' = ~Z (_ i),,-J da ~ . . . d(ai aj+a) ... da,~+l J-1 + (-- i)"aada~.., da"+', thus the second property follows from (Note that the cohomology of the complex (El(d), d) is o in all dimensions, including o since s176 ---- d.) Let us now recall the definition of the Hochschild cohomology groups H"(d, Jr') of ~r with coefficients in a bimodule ,/4 ([i3]). Let ~ ---- d | ~0 be the tensor 316 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY I01 product of ~r by the opposite algebra. Then any bimodule ~' over ~r becomes a left ~r module and by definition: H"(~r162 = Ext~,(~g,../t'), where ~qr is viewed as a bimodule over ~r via a(b) c =abc, V a, b, c ~ ~/. As in [I3] , one can reformulate the definition of H"(~, ~r using the standard resolution of the bimodule ~r One forms the complex (C"(~r ..it'), b), where a) C"(~qr ..r is the space of n-linear maps from ~r to .,r b) for T e C"(,~r162 bT is given by (bT) (a 1, ..., a "+l) = a 1T(a 2, ..., a "+l) +,~(-- ,)'T(al,...,a'a '+l, ...,a "+l) + (-- i) "+1T(a', ..., a n ) a "+1 Definition 2. -- The Hochschild cohomology of a~/ with coefficients in ~r is the cohomo- lo~y H"(d, ..Of) of the complex (C"(d, ..r b). (Note the close relation of the ~'~(~r with the standard resolution and the use of the bimodules ~r in the process of reduction of dimensions: see for instance [36], p. 8). The space d* of all linear functionals on d is a bimodule over ~ by the equality (a~b) (c) = ~(bca), for a, b, c ~ ~. We consider any T E C"(.~, .~r as an (n + 1)- linear functional -r on ~r by the equality x(a ~ 1,...,a") =T(a 1,...,a")(a ~ Va ~d. To the boundary bT corresponds the (n q- 2)-linear functional by: (b'~) (a ~ ..., a "+l) = v(a ~ a l, a ~, ..., a "+l) 9 . "'', --., a ~ ...,a n ). + ~l(-- I)i'r(a O, aia i+l, a n+l) + (-- I)n+l,r(a n+l Thus, with this notation, the condition 3) of proposition r becomes a) "r v = r "r for any cyclic permutation "I" of {o, I, ..., n}; b) b'r = o. Now, though the Hochschild coboundary b does not commute with cyclic permutations, it maps cochains satisfying a) to cochains satisfying a). More precisely, let A be the linear map of C"(d, d*) to C"(d, d*) defined by (Ao) = Z r ~-e, where I' is the group of cyclic permutations of { o, I, ..., n }. Obviously the range of A is the subspace C~(d) of C"(d, d*) of cochains which satisfy a). One has Lemma 3. -- boA=Aob' where b':C"(d,d*) ~C"+1(~r r is defined by the equality (b'o)(x ~ ..., x "+l) = ~ (-- I) iq~(x ~ ..., xJx j+l, ..., x"+l). j=0 317 I02 ALAIN CONNES Proof. -- One has ((Ab') ~)(x~ ..., x "+1) = Y,(-- i) ~+I"+1)* q~(x~,..., x~+~x~+~+l, ..., x ~-1) where o<i<n, o<k<n+ i. Also ((ba) @)(x ~ ..., x "+1) = ~ (-- I) j (A?)(x ~ ..., x j x j+l, ..., x "+1) j=0 + (-- I)"+l(A?)(x"+lx~ ..., Xn). For j~{o,...,n} one has (A~)(x ~ ..., x jx j +1, ..., x,+l) = ~ (__ I)"k 9(x k, ..., x~x ~+1, ..-,xk-1) k=0 n+l + Z (- I)"(k, ~1 ~(x k, ..., x "+1, x ~ ..., x~ x ~+~, ..., x~-~). k=j+2 (Aq~)(x "+1 x ~ ..., x") = ~0(x n+l x O, ..., X n) Also, + ~: (- ilJ ~(~J, ..., x", ~"+~x ~ ..., xi-1). In all these terms, the xJ's remain in cyclic order, with only two consecutive x~'s replaced by their product. There are (n + I) (n + 2) such terms, which all appear in both bA~ and Ab' ~. Thus we just have to check the signs in front of Tkj (k +j + i) where Tk.j=~(x u, ...,x~x ~+1, ...,xk-1). For Ab' we get (--I) ~+l"+l)k where i=j--k(modn+~) and o<i<n. For bA we get (-- I) j+"~ if j~k and (-- I) ~+"ln-1) if j< k. When j~ k one has i =j--k thus the two signs agree. When j<k one has i~-n+2--k+j. Then as n + 2 -- k +j + (n + I) k =j + n(k -- I) modulo 2 the two signs still agree. [] Corollary 4. -- (C~(off), b) is a subcomplex of the Itochschild complex. We let H~(~) be the n-th cohomology group of the complex (C~, b) and call it the cyclic cohomolog~ of the algebra off. For n = o, H~(d) = Z~ is exactly the linear space of traces on d. For off=C one has H~=o for n odd but H~=C for any even n. This example shows that the subcomplex C~ is not a retraction of the complex C", which for off = C has a trivial cohomology for all n > o. To each homomorphism p : off -+ ~ corresponds a morphism of complexes: p* : G~(~) -+ C~(off) defined by (p* ~p)(a ~ ..., a") = ~p(p(a~ ..., p(a")) and hence an induced map p*:H~(~) -+H~(off). 318 NON-COMMIYrATIVE DIFFERENTIAL GEOMETRY ~o3 Proposition 6. -- I) Any inner automorphism of ~r defines the identity morphism in H[(~). 2) Assume that there exists a homomorphism 0 : ~-+ a~r and an invertible element X [:o] [:o] of M~(ad) (here we suppose ~r unitaI) such that X X -t = for a 9 sJ. Then H[(~) is o for all n. p(a) p(a) Proof. -- i) Let a ~ ,~/ and let 8 be the corresponding inner derivation of d given by 8(x) =ax--xa. Given ~eZ~(d) let us check that +, 4(a ~ . a") - ~: ~(a ~ ~(a'), ... a"), is a coboundary, i.e. that 4 9 B~(d). Let 4o(a ~ ..., a "-t) = ~(a ~ ..., a "-1, a) with a as above. Let us compute bA4o = Ab' 4o. One has: tl--1 (b'4o)(a ~ ...,a n ) = Y, (-- I)~r ~ ...,a~a i+x, ...,a n ,a) i=0 = (b~?)(a ~ ..., a n , a) -- (-- I) n q~(a ~ ..., a n-l, aria) - (- x) "+1 ~(aa ~ ..., a "-1, a"). Since b~ = o by hypothesis, only the last two terms remain and one gets Ab' 40 -- (-- i)" 4. Thus 4 = (-- I) n Ab' 40 = b((-- i) n A4o ) E B~(~). Now let u be an invertible element of d, let ~ e Z[(~) and define 0(x) -- uxu -1 for x 9 d. To prove that q~ and q~ o 0 are in the same cohomology class, one can [: o] replace d by M~(d), u by v= and q~ by q% where, for a~ed and b ~ 9 M~(s u-1 92(a ~174 b ~ a t| b t, ..., a n| b") = q~(a ~ ..., a n) Trace(b ~ ... b"). Now v=v lv~ with vl= , v2 = . One has [: :][: :][:, o I [ o :1 I --I 7~ v~ = exp a~, a~ =~ v~, thus the result follows from the above discussion. 2) Let T 9 Z[(~r and ~0~. be the cocycle on M~(~r defined in the proof of i). For a e,~, let e(a)= p(a) o(a) . By hypothesis e and ~ are homomorphisms of ,~ in M~(,ff) and, by i), ~. ~ and ~. ~ are in the same cohomology class. From the definition of ~ one has q~2(~(a~ ..., ~(a")) = 9(a ~ ..., a") + e?(p(a~ ..., p(a")), ~(~(a~ ..., ~(a")) = ,?(~(a~ ..., ~(an)). [] Following Karoubi-Villamayor [4x], let (3 be the algebra of infinite matrices (a~i)~,~e~ with a o e s such that 0~) the set of complex numbers {%) is finite, [~) the number of non zero a~'s per line or column is bounded. 319 xo 4 ALAIN CONNES Then C satisfies condition ~) of proposition 5, taking ~ of the form p(a) = Diag(a, o, a, o, ...). The same condition is satisfied by ~1 | C for any ~1, thus: Corollary 6. -- For any ~ one has H[(C~r = o where C~1 = C | s/. We are now ready to characterize the coboundaries B~ C Z~ from the corresponding cycles, as in proposition i. For convenience we shall also restate the characterization of Z~. Definition 7. -- We shall say that a cycle is vanishing when the algebra f~o satisfies the condition ~) of proposition 5 ([41]) 9 f~o, Given an n-dimensional cycle (f~, d, f) and a homomorphism ~: ,~r we shall define its character by ,~(a ~ ..., a") = f p(a ~ d(o(aX)) ... d(o(a")). ,I Proposition 8. -- Let -c be an (n + i)-linear functional on ~(; then x) v e Z~,(~r if and only if .r is a character; ~) x e B](.~r if and only if x is the character of a vanishing cycle. Proof. -- ~r is just a restatement of proposition I. p) For (fl, d, f) a vanishing cycle, one has H~(f~ ~ = o, thus the character is a coboundary. Conversely if -r ~B~(d), "r = b~ for some d/e C~-l(d), one can extend + to Cd = C | d in an n-linear functional d~ such that ~(I | a ~ ..., I | a n-x) = d?(a ~ ..., a n-l) for all a ~ e ~r and such that ~x = r ~ for any cyclic permutation ~, of {o, ..., n --x }. (Take for instance ~(b ~ ..., b "-1) = +(o~(b~ ..., a(b"-l)) where ~(b) = bl~ ~ d for any b= (b~) eccl.) Let p:d~Cd be the obvious homomorphism o(a) = l| Then "r' = b~ is an n-cocycle on Cd and 9 = p* "r' so that the implication 3) ~ 2) of proposition 1 gives the desired result. [] Let us now pass to the definition of the cup product H~(~) | H~'(~) -~ H~ +'(d | ~). In general one does not have f~(d | ~) = ~(~/) | f~(~) (where the right hand sid is the graded tensor product of differential graded algebras) but, from the universale property of ~(.~/| we get a natural homomorphism 7: : f~(~/| -+ f~(~/) | f~(~). Thus, for arbitrary cochains (p ~ Cn(~/, Ja/*) and ~b ~ Cm(~, ~*), one can define the cup product 9 # ~ by the equality (v +)^ = | ;)o 320 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY xo 5 To become familiar with this notion, let us compute 9 # %5 where 9 ~ C"(~r d*) is an arbitrary cochain, and where %5 ~ CI(C, C) (so that ~ = C) is given by d~(i, I) = r. Here ~r174162 so that 9# %5~C"+1(~1, a~r One has (9 # +)(~0, ..-, ~,+,) = (~| ~)(~(~o| ~)d(~'| ~)... d(~"+'| ~)). One has rcd(al| =dal| +aa| As r"= I one gets I(dI) ~ ~-o thus the only component of bidegree (n, r) of (rc(a~174 ~) d(a*| ~) ... d(a"+~| t)) is (a~ da ~ ... da") a"+~| ~ d~. Hence we get # %5 = ~ (- ~?+" 9(~~ ~'~+', ..., a "+') --(-~)" b' 9 with the notation of lemma 3- Theorem 9. -- I ) The cup product 9, q~ ~ ? # ~ defines a homomorphism H~(d) | H~'(8) ~ H~ + "(.~ | 8). 2) The character of the tensor product of two cycles is the cup product of their characters. Proof. -- First, let 9 e Z~(d), q~ e Z~'(8); then ~ (and similarly ~?) is a closed graded trace on f~(d), thus 7~ | ~ is a closed graded trace on f~(~r174 f~(8) and 9 # %5 e Z~+m(d | 8) by proposition x. Next, given cycles f~, f2' and homomorphisms p : d -+ f~, O' : 8 ~ f~', one has a commutative triangle a(~r | a(8) ~| Thus 2) follows. It remains to show that if 9 e B~,(~r then 9 # %5 is a coboundary: 9 # + ~ B~ +'(~ e 8). This follows from 2), proposition 8 and the trivial fact that the tensor product of any cycle with a vanishing cycle is vanishing. [] Corollary 10. -- i) H~(C) is a polynomial ring with one generator ~ of degree z. 2) Each Hi(d ) is a module over the ring H*x(C ). Proof. -- I) It is obvious that H~(C) = o for n odd and H~(C) -- C for n even. Let e be the unit of 13; then any 9 e Z~(13) is characterized by 9(e, ..., e). Let us 14 I06 ALAIN CONNES compute 9 # q~ where q~ 9 hb 9 Z~"'(C). Since e is an idempotent one has in ~(C) the equalities de = ede + (de) e, e(de) e = o, e(de)' = (de)' e. Similar identities hold for e| e and ~(e @ e) 9 f~(C)| f~(C) and one has ~((e| d(e| d(e| = edede| + e| (m + m')! Thus one gets (9 # ~b) (e, ..., e) -- m! m'! 9(e, ..., e) +(e, ..., e). We shall choose as generator of HI(C ) the 2-cocycle 6 6(I, I, I) = 2i~. ~) Let 9~Z~(~r Let us check that 6#9=9#6 and at the same time write an explicit formula for the corresponding map S :H~(a~r ~ H~+a(d). With the notations of I) one has 2~---~ (9 e 6) (a ~ "" "' a"+") = ~| 2z--~x ~ (aO| | ... d(a"+'| = ~(~0 a~ a, ~... ~.+,) + ~(~o ~(~ a~) da'.., da "+') + ... + ~(a" d~'.., d~'-~(~' a '§ d~+~.., da "§ +... + ~(a ~ I ... da"(a "+x a"+2)). The computation of 6 # 9 gives the same result. For 9 9162 let S?=6#~=q~#6 9 By theorem 9 we know that SB~(a~r C B~+'(off) but we do not have a definition of S as a morphism of cochain complexes. We shall now explicitly construct such a morphism. Recall that ~ # ~b is already defined at the cochain level by (9 # +) ^ = (~ | ~) o re. Lemma 11. -- For any cockain 9 9162 let $9 9 be defined by $9---- A(6#~); then n+3 a) -- A(6 4. ~) = 6 4* 9 for 9 9 Z~(~), so S extends the previously defined map. n+3 n+I b) bS9 =--Sb9 for 9 9 n+3 Proof. -- a) If 9 9 Z~,(~') then (6 4. 9) x = ~(X) 6 4. 9 for any cyclic permu- tation X of {o, I, ..., n + 2 }. b) We shall leave to the reader the tedious check in the special case + =- 6 of the equality (bAg) ~ hb = hA(9 4. hb) for 9 9 G"(a~r a~r It is based on the fol- lowing explicit formula for A(? 4# 6). For any subset with two elements s = {i,j}, i<j, of {o,~,...,n+2}=Z/(n+3) one defines ~(s) : q~(a ~ ..., ~'-~, ~ d +~, ..., ~J ~+~, ..., ~"+~). 322 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY xo7 In the special case j=n+~ one takes ~(s) ---- ~(a "+~ a ~ ..., a'd +~ .... , a "+t) if i < n + x, ~(s) = q~(a '*+x a "+~ a ~ ..., a") if i -~ n -k- x. Then one gets A(~#~) = Y, (-- ~)~+~(n+3--~i)+~ where, forn even, one has i=l +~ = ~,({o, i)) + ~({ ~, i + x }) + ... + ~({n + ~, i - ~ }), and for n odd +, = ~({o, i}) + ... + ~,({n + 2 - i, n + 2}) --~({n+2--i+I,o})...--~{n+2, i--x}. [] We shall end this section with the following proposition. One can show in general that, ff ~ e Z"(~, ~*) and ~b e Z'(~, ~?') are Hochschild cocycles, then r # + is still a Hochschild cocycle ~ # + e Z" + m(~ | M, ~* | M*) and that the corresponding product of cohomology classes is related to the product v of [I3], p. ~x6, by (n -t- m)! [r # +] -- n! m! [~] v [+]. Since a e Z~(C, C) is a Hochschild boundary one has: Proposition 12. -- For any cocycle ~ ~ Z[(~), S~ is a Hochschild cobounda~y: S~ ~ b+ where +(a ~ .. a "+1) = 2i~ Z (- ~)J~(a~ . daJ -~) ~(a~J +~ . aa")). "~ $--1 .... Proof. -- One checks that the coboundary of thej-th term in the sum defining ~ gives ~(a~ 1 ... da ~-1) a j a j ~l(daJ+~ ... da"+2)). [] 2. Pairing of H~(~/) with I~.(.~r i ---- o, Let d be a unital (non commutative) algebra and K0(.~r Kl(d ) its algebraic K-theory groups (of. [i6]). By definition K0(M ) is the group associated to the semi- group of stable isomorphism classes of finite projective modules over d. Also Kx(d) is the quotient of the group GL~(~r by its commutator subgroup, where GL~(~r is the inductive limit of the groups GL,(d) of invertible elements of M,(~'), under In this section we shall define by straightforward formulae a pairing between H[~(~) and Ko(~ ) and between H~"(d) and K~(d). The pairing satisfies (S~,e)= (~,e), for ~eH~(~), eeK(d) and hence is in fact defined on H*(d) ~- H[(~) | C. As a computational device we shall 323 xo8 ALAIN CONNES also formulate the pairing in terms of connexions and curvature as one does for the usual Chern character for smooth manifolds. This will show the Morita invariance of H~(d) and will give in the case d abelian, an action of the ring K0(d ) on H~(,~). Lemma l& -- Let 9eZ~(d) and p, qeProjM~(d) be two idempotents of the form p = uv, q = vu for some u, v e Mk(d ). Then the following cocycles on = {x e Mk(a~r xp ~- px ~- x} differ by a coboundary +l(a ~ ..., a") = (~ # Tr)(a ~ ..., a"), +~(a ~ ..., a") ---- (? # Tr) (va ~ ..., va"u). Proof. -- First, replacing d by Mk(d ) one may assume that k = I. Then one can replace p, q, u, v by , , , and hence assume the existence ['o o] [: o][: :l [: :1 o q of an invertible element U such that UpU- 1 = q, u ---- pU- 1 = U- a q, v = qU ----- Up [' ' "]) ta e U = . Then the result follows from proposition 5-I. [] v i--q Recall that an equivalent description of Ko(d) is as the abelian group associated to the semi-group of stable equivalence classes of idempotents e ~ Proj Mk(a~ ). Proposition 14. -- a) The following equality defines a bilinear pairing between K0(~t ) and H~V(,~): ([el, [~]) = (2i~) -m (m!) -x (~? # Tr) (e, ..., e) for e ~ Proj Mk(d ) an b) One has ([e], IS,] ) = ([e], [9] ). Proof. -- First if ~ r B~m(d), ~ ~ Tr is also a coboundary, , # Tr = b+ and 2rn hence (~#Tr)(e,...,e) =bt~(e,...,e)----- ~3 (-- I) ~+(6 ...,e) = +(e, ...,e) = o, i=0 since +x = _ ~b. This together with lemma 13 shows that (~ # Tr) (e, ..., e) only the result, one gets the additivity and hence a). 2m b) One has --:-Sq~(e, ..., e) =  ~(e(de) ~-1 e(de) "-j+1) and, since e~= e, one 21~ j=l has e(de) e = o, e(de) ~ = (de) ~ e, so that 2i~ S~(e, ..., 8) ---- (m + I) ~(e, ..., e). [-] We shall now describe the odd case. 324 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY xo9 Proposition 15. -- a) The following equality defines a bilinear pairing between K1(~r ) and Hx~ (,~r : < [~], b] > = (2i~)-" ~-(~'+') (m -- I/2) .., I/2 (~p#Tr)(u -1-I,u-x,u -1-1,...,u-t) where ~ ~ Z~'-l(d) and u ~ GLk(d ). b) One has < [u], [$9] > = < [u], [9] >. Proof. -- a) Let ~ be the algebra obtained from d by adjoining a unit. Since ~r is already unital, ,~r is isomorphic to the product of ~r by 13, by means of the homo- morphism p:(a,Z) ~(a+),I,k) of~to a~r215 Let ~eZ~(~) be defined by the equality '~ 0 ~((a, x~ ..., (a", ~")) = ~(a ~ ..., a"), V (a', X') ~ Let us check that b~' = o. For (a ~176 ..., (a,+1,),,+1) e ff one has ~ q~((a, o k~ ..., (a i, ),) (ai+t, Xi+ 1), --., (a,+l,)`,+ 1)) = 9(a ~ ia i+1, ...,a n+l ) +X ig(a ~ ...,a i-l,ai+ 1, ...,a "+1) 'd+z, a-+l) +)~+19( a~ ..., . ~ 0 . (a,+l,)`,+l)) Thus bq~((a , X~ .. , _ )`0 q~(a l, a,+l) + (_ i),-i )`0 q~(a,+l, .... -- . .. ~ a 1, a n) o. Now for u e GLi(.~r ) one has q~(U -1 -- 1, U -- I, ..., /Z -I -- I, U -- 1) = (7~ p--l)(~--I if, "'', ~--I, ~) where ~ = (u, I) e ~qr X 13. Thus to show that this function Z(u) satisfies )~(uv) = )~(u) + Z(v) for u, v e GLx(~), 9 a n- 1) a i one can assume that q~(I, a ~ = o for E ~r and replace Z by z(u) = ~(u -1, u, .., u -~, u). Now one has with U= [: o] , V= [; ;1 Z(uv) = (cp # Tr) (U -a, U, ..., U -1, U), Z(u) +Z(v) = (9#Tr)(V -1,v,...,v -l,v). Since U is connected to V by the smooth path lCOS t sin -- cos t sin tJ 325 IIO ALAIN CONNES it is enough to check that (? # Tr) (U~ -a, Ut, ..., Ut) = o. Using (U~-I) ' = -- U~ -1 U~ U~ -1 the desired equality follows easily. We have shown that the right hand side of 15 a) defines a homomorphism ofGLk(d ) to C. The compa- tibility with the inclusion GL k C GL k, is obvious. To show that the result is o if ? is a coboundary, one may assume that k = i, and, using the above argument, that ~ =b~ where 4/eC~ -1, d~(i,a ~ ...,a ~-~) =o for a ~. (One has b~= (b+) ~ for +eC~-l.) Then one gets b+(u -~, ...,u -1,u) =o. b) The proof is left to the reader. [] Definition 16. -- Let H'(~ 0 = H~(~r | Here H~(C), which by corollary IO I) is identified with a polynomial ring C[a], acts on C by P(a) ~ P(I). This homomorphism of H~(C) to C is the pairing given by proposition 14 with the generator of K0(C ) = Z. By construction H*(a~r is the inductive limit of the groups H~(d) under the map S : H~(.~ 0 -+ H~+~(~r or equivalently the quotient of H~(.~) by the equivalence relation ~0 ~ S% As such, it inherits a natural Z/2 grading and a filtration: F" H*(~g) = Im H~(d). We shall come back to this filtration in section 4. Corollary 17. -- One has a canonical pairing between H~ and Ko(~r and between H~d(d) and K~(d). The following notion will be important both in explicit computations of the above pairing (this is already clear in the case ~ = C~~ V a smooth manifold) and in the discussion of Morita equivalences. Definition 18. -- Let d ~ fl be a cycle over ~, and 8 a finite projective module over ~. Then a connexion V on @ is a linear map V : ~' --> g | ~1 such that V(~.x) = (V~) x q- ~ Q dp(x), V~e#, xed. Here o ~ is a right module over d and ~x is considered as a bimodule over d using the homomorphism ~ : z~ -> fl0 and the ring structure of ~*. Let us list a number of obvious properties: Proposition 19. ~ a) Let e e End~r be an idempotent and V a connexion on ~; then -> (e| i) V~ is a connexion on eg. b) Any finite projective module g admits a connexion. c) The space of connexions is an affine space over the vector space Hom~(g, g| ~1). 326 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY i i i d) Any connexion V extends uniquely to a linear map of ~ 8 | ~ into itself such that V(~| =(V~)o)+~| V~eg, o) e~. Proof. -- a) One multiplies the equality I8 by e | I (on the left). b) By a) one can assume that 8 = Ca| for some k. Then, with (~')~=1 ..... the canonical basis of g, put Note that, if k = i (for instance), then ~r| = O(i) ~1 and Va = p(I) dp(a) for any a e d since .~r is unital. This differs in general from d, even when p(I) is the unit of ~0. c) Immediate. d) By construction ~is the finite projective module over ~ induced by the homo- morphism p. The uniqueness statement is obvious since V~ is already defined for ~ e 8. The existence follows from the equality V(~a) o) + ~a | do) := (V~) ao) + ~ | d(ao)) for any ~eg, ae~r and o) e~'l. [] We shall now construct a cycle over End,c(g). We start with the graded algebra Enda( ~ (where T is of degree k if Tg ~ C g"j+k for all j). For any T e Enda( ~ of degree k we let ~(T) --~VT-- (-- I) kTV. By the equality d) one gets V(~O)) = (V~) O) -~- (-- I) deg ~ do) for ~ e ~, o) e ~, and hence that 8(T) e Enda(~, and is of degree k + I. By construction 8 is a graded derivation of Endo( ~. Next, since o~is a finite projective module, the graded trace ,[: ~"--)-C defines a trace, which we shall still denote by f, on the graded algebra Enda(~. nmma zo. - one f = o for T F.nd.( Y) of degree n- ,. Proof. -- First, if we replace the connexion V by V'= V + F, where F e Hom~,(@, g| ~), the corresponding extension to ~ is V' : V + ~, where eEnda(g ~ and is of degree i. Thus it is enough to prove the lemma for some connexion on g. Hence we can assume that g = e.~ k for some e eProj Mk(.~ ) and that V is given by 19 a) from a connexion V 0 on ~r Then using the equality 8(T) ---=e 80(T) e for T e End ,~C End ~0 (go = .~tk), as well as ~o(T) = 8o(eTe) = ~o(e) T + 8(T) + (-- i) ~ T bo(e), one is reduced to the case g = ~r with V given by 19 b). Let us end the computation say with k = I. Let e ---- 9(I). One has ~= eft, Endn(~) := e~2e, 8(a) = e(da) e. Thus f S(a) = f (a(e e) - (de) a--(-- I)~176 ade) =o. [] 327 II2 ALAIN CONNES Now we do not yet have a cycle over End~,(g) by taking the obvious homomorphism of Endd(d ~ in Endn(~), the differential ~ and the integral . In fact the crucial - f property 82= o is not satisfied: Proposition 21. -- a) The map 0 = V ~ of ~ to ~ is an endomorphism: 0 ~ Endn(~'~ and 8~(T) =0T--T0 for all T~Endn(g ). b) One has ( [d~], [v] ) = ~.~ If (0/~) "m , when n is even, n = era, where [~] e K0(.~ ) is the class of ~', and .r is the character of f~. Proof. -- a) One uses the rules V(~) = (V'~) co + (-- I) "~ ~ d~ and d' = o to check that V~(~) = V~(~) ~. b) Let us show that f0 '~ is independent of the choice of the connexion V. The result is then easily checked by taking on d ~ = e~ ~ the connexion of proposition ~ 9- Thus let V' = V + P where r is an endomorphism of degree ~ of ~. It is enough check that the derivative of f 0~ is o where 0 t corresponds to V~ = V + tP. Also to it is enough to do it for t=o. We get: dldt 07= 0, 0, k=0 f -'f As (dot) :FV+VF=~(I')one has \a~ l 1=o 0 m mfS(o "-1 = o. [] f = ") Thus, while M * o, there exists 0 ~ ~' = Endn(~ ) such that M(T)-----0T--T0, VTE~)'. We shall now construct a cycle from the quadruple (~', 8, 0, f). Lemma 22.- Let (~1', 8, O, f)bea quadruple such that ~' is a graded algebra, ~ a graded derh~ation of degree I of ~' and 0 ~ ~'~ satisfies 8(0) = o and 82(~) = 0o~ -- ~o0 for o~ ~ ~'. Then one constructs canonically a cycle by adjoining to ~' an element X of degree I with dX = o, such that X ~ = 0, ~l Xc~ = ~ V ~ ~ f~'. Proof. -- Let ~1" be the graded algebra obtained by adjoining X. Any element of ~" has the form co = on + wl, X + Xo21 + Xc0,, X, co~j ~ 9V. Thus, as a vector space, t)" coincides with M2(~'), the product is such that 328 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY If3 and the grading is obtained by considering X as an element of degree i; thus [o)ij] is of degree k when con is of degree k, oh2 and co~1 of degree k -- i and co** of degree k -- 2. One checks easily that ~" is a graded algebra containing ~'. The differential d is given by the c6nditions do) : 8(r + Xo) --(-- I) deg~ o)X for ca e~'C ~", and dX=o. One gets to) l -- -- J + O)21 r J _ o0 'o]. One checks that the two terms on the right define graded derivations of f~" and that d2=o. Finally one checks that the equality defines a closed graded trace. [] Putting together proposition 2I a) and lemma 22 we get: Corollary 23. -- Let d ~ ~ be a cycle over d, 8 a finite projective module over M' and ~' = Endue(g). To each connexion V on ~ corresponds canonically a cycle dt' P-~ ~' over d/'. One can show that the character 1:'~ Z~(d') of this new cycle has a class [x'] ~ H~(d') independent of the choice of the connexion 7, which coincides with the class given by lemma 13. One can then easily check a reciprocity formula which takes care of the Morita equivalence. Corollary 24. -- Let d, ~ be unital algebras and ~ an d, ~ bimodule, finite projective on both sides, with d = EndB(g), 5~ = End~(g). Then H~(d) is canonically isomorphic to H~(~). Finally when d is abelian, and one is given a finite projective module 8 over ~, then one has an obvious homomorphism of ~r to d' = End~,(8). Thus in this case, by restriction to d of the cycle of corollary 23 one gets: Corollary 25. -- When d is abelian, H~(d) is in a natural manner a module over the ring Ko(~ ). To give some meaning to this statement we shall compute an example. We let V be a compact oriented smooth manifold. Let ,~f = C~176 and ~ be the cycle over given by the ordinary de Rham complex and integration of forms of degree n. Let E be a complex vector bundle over V and 8 = Coo(V, E) the corresponding finite 15 Ix 4 ALAIN CONNES projective module over ~r = C~176 Then the notion of connexion given by deft- nition I8 coincides with the usual notion. Thus corollary 25 yields a new cocycle 1: e Z[(d), ~r = C~~ canonically associated to V. We shall leave as an exercise the following proposition. Proposition 26. -- Let co k be the differential form of degree ~k on V which gives the component of degree 2k of the Chern character of the bundle E with connexion V : ~k ---- i/h! Trace ~ , where 0 is the curvature form ([x 7]). Then one has the equality where "~ ~ Z[-~(,ff) is given by ~ o f ^.. ^ v/ed=C (V), ok(f, " = fO dfl . df.-~ and where .c is the restriction to d = Coo(V) of the character of the cycle associated to the bundle E, the connexion V, and the de Rham cycle of ~t by corollary 23. 3. Cobordism of cycles and the operator B By achain of dimension n +I we shall mean a triple (fl, 0h, f) where ~ and 9~2 are differential graded algebras of dimensions n + i and n with a given surjective morphism r : ~ ~ 0~ of degree o, and where f : L2" + 1 ___> C is a graded trace such that fdr Vcoe~" such that r(r =o. By the boundary of such a chain we mean the cycle 0~, where for o~' e (0~)" one (f') takes co'= dr for any r e~n with r(o~) -----o~. One easily checks, using the ; f surjectivity of r, that f' is a graded trace on O~ which is closed by construction. Definition 27. -- Let d be an algebra, and let d o_~ ~, ~ p~ ~, be two cycles over d (el. proposition I). We shall say that these cycles are cobordant (over d) if there exists a chain ~" with boundary ~ @ ' (where ~' is obtained from ~' by changing the sign of f) and a homo- morphism p" : ~ --> ~" such that r o p" ~ (p, 0'). Using a fibered product of algebras one checks that the relation of cobordism is transitive. It is obviously symmetric. Let us check that any cycle over ~r is cobordant to itself. Let ~0 _-- Coo([o, i]), ~1 be the space of C ~ x-forms on [o, I], and d be the usual differential. Set 0~ = C | C and take f to be the usual integral. Then taking for r the restriction of functions to the boundary, one gets a chain of dimension i with boundary (C| r ~(a, b) = a-- b. Tensoring a given cycle over ~ by the above chain gives the desired cobordism. 330 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY ~5 Thus cobordism is an equivalence relation. The main result of this section is a precise description of its meaning for the characters of the cycles. We shall assume throughout that the algebra .~r is unital. Lemma 28. ~ Let xt, % be the characters of two cobordant cycles over a~t. Then there exists a Hochschild cocycle 9 e Z"+a(d, ..r such that v 1 -- % ----- B o % where (Bo ~)(a ~ ..., a") = ~(i, a ~ ..., a") -- (-- ~)"+~ ~(ao, ..., a", I). Proof. -- With the notation of definition 27, let q~(a ~ ..., a "+1) -= fp"(a ~ dp"(a 1) ... dp"(a"+l), V a i e d. Let r = p"(a ~ dp"(a t) ... dp"(a n) e n"n. Then by hypothesis one has (~t - ~)(~0, ~, ..., ~,) = f d,o. Since p"(a ~ ~-- p"(I) p"(a ~ one has do.) = (d{9"(I)) [~"(a O) d~"(G 1) ,,, d[~"(a n) + ~"(I) d[:}H(a O) ,,, d~}H(an), Using the tracial property of f one gets fdo~ = (-- I) n q)(a 0, a x, ..., a", I) + r a ~ ..., a"). the tracial property of f one checks that q~ is a Hoschchild cocycle. [] Using again Lemma 29. ~ Let -cl, v 2ez~(d) and assume that Vl--V2~--Boq~ for some e Z"+I(~ r ~'). Then any two cycles over ~ with characters "~x, "~ are cobordant. Proof. ~ Let ~r ~ ~ be a cycle over d with character -~. Let us first show that it is eobordant with (~(d), ?). In the above cobordism of ~ with itself, with restriction maps ro, q, we can consider the subalgebra defined by rx(r ) e ~', where ~' is the graded differential subalgebra of ~ generated by p(d). This defines a cobor- dism of ~ with ~'. Now the homomorphism "~ : ~2(d) -+ ~' is surjective, and satisfies ~'*f =-~. Thus one can modify the restriction map in the canonical cobordism of (~(d), -2) with itself to get a cobordism of (~2(a~), "~) with ~'. Let us show that (~(,.qg), "~a) and (~(~), ~) are cobordant. Let ~ be the linear functional on ~"+l(d) defined by I) ~(G 0 da 1 ... da "+1) ---- ~(a ~ ..., an+l), 2) ~.(da 1 ... da "+i) -- (Bo ~) (a 1, ..., a"+l). Let us check that ~ is a graded trace on ~(d). We already know by the Hoch- schild cocycle property of q~ that ~(a(bo~)) = ~((br a), V a, b e d, r e ~,+l. 331 xx6 ALAIN CONNES Let us check that ~t(am) = ~t(o~a) for o~ = da 1 ... da TM. The right side gives n+l ~z( Z (-- I)"+l-Jda ~ ... d(a~a ~+1) ... da'~+lda) + (-- I) "+' ~(a 1 daZ.., da) n+l = Y. (-- l)n+a-~(Bo ~)(a 1, ..., a~a ~+I, ..., a "+1, a) =1 + (-- 1) TM q~(a 1, a z, ..., a "+1, a) -- a n + ~', a). -- (-- I)" ((b' B 0 r -- ~0) (a 1, a s, ..., Now one checks that for an arbitrary cochain $ 9 C"+1(~r ~qr one has Bob~ + b, Bo 9 = 9 _ (_ ~),+1 9x, where X is the cyclic permutation X(i) = i- i. Here q~ is a cocycle, bq~ = o and b' B 0 q~ -- q~ ----- (-- I) n q~x so that v@oa) ---= q~(a, a l, ..., a "+1) = ~(ao~). It remains to check that for any a 9 d and co ~ ~" one has ~((da) ~) = (-- I) n ~(co da). For ~ 9 df~"-1 this follows from the fact that B o ~ 9 C~ (recall that B 0 q~ = x 1 -- -@. For co = a ~ da 1 ... da" it is a consequence of the cocycle property of B o ~p. Indeed one has bB 0 ~ =- o, hence b' B o 9(a ~ a 1, ..., a", a) = (-- ~)" B o r ~ a t, ..., a") and since b'B oq~=q~-(-I) "+lq~x we get ~(a ~ ..., a", a) -- (-- I) "+1 q~(a, a ~ ..., a") = (-- I)"+l(Boc?)(aa ~ a 1, ..., a"), i.e. that ~((da) a ~ da 1 ... da") = (-- I)" vt(a ~ da 1 ... da n da). To end the proofoflemma 29 one modifies the natural cobordism between (~(d), ~1) and itself, given by the tensor product of~(d) by the algebra of differential forms on [o, I], by adding to the integral tile term ~ o rx, where r 1 is the restriction map to { I } C [o, I]. [] Putting together lemmas 28 and 29 we see that two cocycles vx, x2 9 Z~(d) correspond to cobordant cycles if and only if ~1- "~ belongs to the subspace Z~(d) n B0(Z"+l(d, ~')). We shall now work out a better description of this subspace. Since Az ---- (n + i) x for any x ~ C[(~), where A is the operator of cyclic antisymmetrisation, the above subspace is clearly contained in the subspace Z"(d) c~ B(Z"+I(~r ~r where B = ABo: C "+1 ~ C". Lemma 30. -- a) One has bB=--Bb. b) One has Z[(d) c~ Bo(Z"+l(d, d*)) = BZ"+I(d, a"). Proof. -- a) For any cochain 9 e C"+l(d, M*), one has B 0 b~ + b' B 0 q~ = ~ -- (-- I) n+l q~X, 332 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY xI7 where ~ is the cyclic permutation X(i) = i- I. Applying A to both sides gives AB o br + Ab' B o ~ = o. Thus the answer follows from lemma 3 of section I. b) By a) one has BZ"+I(~ r d*.)C Z~,(d). Let us show that BZn+l(,r d*) C Bo Zn+l(d, ~r Let ~eBZn+l(d,d*), so that ~3=Bq~, r We shall construct in a canonical way a cochain + e Cn(d, d*) such that I I --~-~Bo(~o--b+). Let 0-=B or --~. By hypothesis A0=o. Thus there n+I n+i exists a canonical ~b such that ~b -- r +x = O, where X is the generator of the group of cyclic permutations of {o, I, ...,n}, X(i) = i--I. We just have to check the equality B0 b~b = 0. Using the equality B o b+ + b' B o ~b = + -- r qx, we just have to show that b' B o + = o. One has ..... , a n-l) ..., a n-l, x) Bo +(a ~ ., a "-l) ___= +(,, a ~ -- (-- i) n +(a ~ = (__ i)n--1 (+__ ~(~.) +X) (aO, ...,an-l, I) = (-- I) n-10(a 0, ...,a n-1 , I) = (_ ~)n-1 (~(~, a o, ..., an-l, ~) _ (_ ~)n+l ~(a o, ..., an-l, ~, ~)) -3 l- ~ (-- I) n ~(a 0, ..., a n-1 , I). The contribution of the first two terms to b' B o +(a ~ ..., a n) is n--1 (-- I) n-1 2~ (-- I) ~ (r a ~ ..., aJa j+l, ..., a n, i) j=o + (_ ~)n ~(a o, ..., aJ a ~+l, ..., a n, i, i)) = (- I) n (b~0, a ~ ..., a n, ~) - ~(a ~ ..., a", ~)) -- (be~(a O, ..., a n , I, I) -- (-- I) n r O, ..., a n , I)) = 0 since b~? = o. The contribution of the second term is proportional to n--1 2~ (-- I)J~(a ~ ..., aJa j+t, ..., a n, I) ---- b~(a ~ ..., a n, I) = O. [] j=0 Corollary 31. -- x) The image of B:Cn+l-+ C" is exactly C~. 2) B~(d) c Bo zn+l(~r ~r Proof. -- I) =~ 2) since, assuming i), any b% ~ e C[ +l is of the form bB+ ---- -- Bbd/ and hence belongs to BZn+l(~r d*) so that the conclusion follows from b). To prove i) let q~ e C[. Choose a linear functional ~0 0 on d with ~0(I) = i, and then let +(a ~ ..., a "+~) = ~o(a ~ ~(a l, ..., a n+l) + (- ~)" ~((a ~ -- ~o(a ~ ~), a l, ..., a n) ~o(a" +l). 333 ~,8 ALAIN CONNES One has +(~,a ~ =r ~ ...,a") and q,(a ~ ..., a", ~) = v0(~ ~ ~(a ~, ..., a", ~) + (-- ~)" v(a ~ ..., a") + (-- ~)"+~ ~0(a ~ ~(~, a ~, ..., a") = (-- ~)" ~(a ~ ..., a"). Thus B 0+--2r and (p~ImB. [] We are now ready to state the main result of this section. By lemma 4 a) one has a well-defined map B from the Hochschild cohomology group H"+~(~r ', ~r to H[(~). Theorem 32. -- Two cycles over .~t are cobordant if and only if their characters za, z~ e H~(d) differ by an element of the image of B, where B: H"+X(~ r ..~'*) -+ H[(~r It is clear that the direct sum of two cycles over ~/is still a cycle over ~r and that cobordism classes of cycles over d form a group M*(~/). The tensor product of cycles gives a natural map: M*(~q/) � M*(~)-->M*(~/| Since M*(C) is equal to H~(C) : C[~] as a ring, each of the groups M*(~') is a C[(~] module and in particular a vector space. By theorem 32 this vector space is H~,(~r B. The same group M*(~) has a closely related interpretation in terms of graded traces on the differential algebra ~(d) of proposition i. Recall that, by proposition i, the map ": ~ ? is an isomorphism of Z~(d) with the space of closed graded traces of degree n on ~(d). Theorem 33. -- The map v ~ ? gives an isomorphism of H~(~)/Im B with the quotient of the space of closed graded traces of degree n on Q(~r by those of the form d t tL, ~ a graded trace on ~(~t) (of degree n + I). Proof. -- We have to show that, given ": e Z[(d), one has ? = d ~ V for some graded trace ~ if and only if .reImBDB~,. Assume first that -~=d t~. Then as in lemma 28, one gets -~----B 0 cp where (p e Z"+1(d, d*) is the Hochschild cocycle ~(a ~ 1, ...,a "+1) : ~(a ~ da 1... da"+l), V a ~ e ~r Thus v---- AB 0~EImB. n-j-I Conversely, if n:eImB, then by lemma 30 b) one has ~:B 0T for some ~0 e Z"+l(d, d*). Defining the linear functional ~z on ~"+l(d) as in lemma 29 we get a graded trace such that ~(da ~ da 1... da") ----'~(a ~ ...,a"), Vaie i.e. ~(a~) = ~(~), v ~ ~ ~"(d). [] Thus M*(d) is the homology of the complex of graded traces on ~)(d) with the diffe- rential d t. This theory is dual to the theory obtained as the cohomology of the quotient of the complex (~(d), d) by the subcomplex of commutators. The latter appears 433 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY zI 9 independently in the work of M. Karoubi [39] as a natural range for the higher (]hem character defined on all the O uillen algebraic K-theory groups K~.(~r Thus theorem 33 (and the analogous dual statement) allows: I) to apply Karoubi's results [39] to extend the pairing of section 2 to all K~(~r 2) to apply the results of section 4 (below) to compute the cohomology of the complex (~(~')/[ , ], d). 4- The exact couple relat/ng H~(d) to Hochschild cohomology By construction the complex (C~(d), b) is a subcomplex of the Hochschild complex (C"(d, d*), b), i.e. the identity map I is a morphism of complexes and gives an exact sequence: o -+ c~ ~ c- -+ c"/c~ -~ o. To this exact sequence corresponds a long exact sequence of cohomology groups. We shall prove in this section that the cohomology of the complex C/C~ is H"(C/C 0 = H"-I(C~). Thus the long exact sequence of the above triple will take the form o ~ H~ I , -> H~ d') -> H;1(~r --> Hk(d) -> S (d, d') -+ H~ -+ n~(d) -+ ... H"(d) i H"(d, ~g*) -+ H~-'(zr -+ H~+'(~r ~ H"+~(g, d') -+ ... On the other hand we have already constructed morphisms of cochain complexes S and B which have precisely the right degrees: S : H~-~(d) -+ H~+'(M), B: H"(d, d*) -+ H~-I(~). We shall prove that these are exactly the maps involved in the above long exact sequence, which now takes the form B I H~(d) ~ H"(d, M*) -+ H~-I(.~g) L H~+I(,~) ---> ... Finally to the pair b, B corresponds a double complex as follows: C n,m = C"-'(d, d*) (i.e. (]"'' is o above the main diagonal) where the first diffe- rential dl:C~'m-+ (],+l.m is given by the Hochschild coboundary b and the second differential d2: (]~"~-+ C n'm+l is given by the operator B. By lemma 3 ~ of section 3 one has the graded commutation of dl, d~. Also one checks that B2= o so that d~ = o. By construction the cohomology of this double complex depends only upon the parity of n and we shall prove that the sum of the even and odd groups is canonically isomorphic with H~(d) (~) C =- H'(,~') H~lC) (where H:(C) acts on C by evaluation at : := i). 335 I oO ALAIN CONNES The second filtration of this double complex (F q = Y, C"") yields the same m>q filtration of H*(d) as the filtration by dimensions of cycles. The associated spectral sequence is convergent and coincides with the spectral sequence coming from the above exact couple. All these results are based on the next two lemmas. Lemma 34. -- Let + eC"(d, ~*) be such that b+ eC[+l(~). Then B~b eZ~-l(~) and SB+ = 2i~zn(n + I) b~b in H~,+l(d). Proof.- One has B+ e C~ -1 by construction, and bB+-------Bb~b-----o since b+eC~ +1. Thus B+eZ~ -1. In the same way b+eZ[ +1. Let 0----B+, by proposition x2 of section I one has SO = b+' where ~b'(a~ "'" an) = j-1 ...... (- i) j-1 ~(a~ d~J-') aJ(d~ +1 d*)). It remains to show that +' -- e(X) +'x = n(n + i)(+" -- ~(X) +"x) where X(i) =i-- i for ie{o,x,...,n+ i} and ~b"--~beB". Let us first check that (+' - ~(x) +'~) (a ~ ..., *) = (- i) "-1 (n + ~) o(a" a ~ al, ..., a,-1). One has n-1 ~b'X(a ~ ..., a") =  (-- I) j ~((da ~ ... da ~-l) aJ(da j+l ... da n-1 ) a"). j=0 Let % = a~ 1 ... da i-i) a~(da ~+1 ... da "-1) a". Then d% = (da ~ ... da j-l) aJ(da ~+1 ... da "-1) a" + (-- I) j-la~ I ... da "~ ... da "-1) a" -~ (-- I) n a~ I ... da ~-1) ai(da ~+1 ... da"). Thus for je{i,...,n-- i} one has (-- I) j-1 ~(a~ ... da j-l) a~(da~+l ... da")) -- E(~)(-- I) j ~((da 0 . ,. aa j-l) aJ(da j+l ,.. da n-l) (l n ) = (- I) "-1 o(a" a ~ a 1, ..., a"-l). Taking into account the cases j -- o and j = n gives the desired result. Let us now determine ~b", ~b" -- ~b e B"(~, ~*) such that (+" ~(x) +"~)(a ~ ...,a") (- I)"-1 __ -- 0(a" a ~ ..., a"-l). Let 0----B 0~b and write 0=01+03 with A01=o, 03eC~-l(ad) so that 03=-0. Since A01 = o there exists ~b I e C "-1 such that 01 = D~b 1 where D~b I = +1 -- r ~bl x. 336 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY I2I Parallel to lemma 3 of section ~ one checks that D o b ---- b' o D and hence D(b+l ) = b' 0 i. Let t~"=+--b+i. As D=B 0b+b'B 0 we get Dt~=b'B 0t~=b'0i+b'0, hence I b' Dd/'----b'0~.=n ~" Finally since bq~----o one has b'~ = (-- ~)"-l~?(a"a~ ...,a"-~). [] As an immediate application of this lemma we get: Corollary 86. -- The image of S:H~-~(~) -+H~+t(~) is the kernel of the map : I-I +l(se) This is a really useful criterion for deciding when a given cocycle is a cup product by a e H[(C), a question which arose naturally in part I. In particular it shows that ifV is a compact manifold of dimension m and if we take d =- C~(V), any cocycle -~ in H~(d) (satisfying the obvious continuity requirements (of. Section 5)) is in the image of S for n>m-~dimV. Let us now prove the second important lemma: Lemma 36. -- The obvious map from (ImB n Ker b ) [b (Im B) to (KerB n Ker b ) /b (Ker B) is bijective. Proof. -- Let us show the injectivity. Let ~ e Im B c~ Ker b, say ~ s Z~+l(M), and assume ~ eb(Ker B). Then the above lemma shows that q~ and So = o are in the same class in H~+I(M) and hence 9 ~ b(Im B). Let us show the surjectivity. Let ? e Z"+l(d, d*), Bq~ = o and + e C"(,~/, d,), -- e(X) d/x -= B o q0. As in the proof of lemma 3 ~ of section 3 one gets B 0 b+ = B o ~. This shows that 9' ---- q0--b+eZ~(,~) since D 9'--B 0b 9' +b'B 09' =o. Let us show that B+ebG~ -2. Since t~--~(X)+X=B0b~b one has b'B o+-=o. One checks easily that b '* = o and that the b' cohomology on C"(~, ,~*) is trivial (if b' 91 = o one has b'~l(a ~ ",I) ~--o i.e. ~i=b'~?2 where ~ ..., a "-1) = (-- I) ~ ..., a "-1, Thus B 0+=b'0 for some 0eG "-~ and B~=Ab'0=bA0ebG~ -~. Thus since G~ -2 = ImB one has BO/----= bB01 for some 01 eG "-I i.e. +b0 ieKerB and b+eb(KerB). As ~--bd/eZ~ this ends the proof of the surjectivity. [] Putting together the above lemmas 34, 36 we arrive at an expression of S:H~-I(~) ~H~+l(d) involving b and B: S = ~ir~n(n + I) bB -1. 16 122 ALAIN CONNES More explicitly, given ~ eZ~-l(d) one has ~ elmB, thus ~ = B+ for some +, and this determines uniquely b+ e (Ker b n Ker B)/b(Ker B) = H[+I(~). To cheek I I that b+ is equal to 2ire n(n + i~ S~ one chooses + as in proposition i2: ~(a~ .... ' an) = n(n -~ I~) $~1 ~(a~ "" " daJ-1) a'?(daJ+l " " " dan))" As an immediate corollary we get: Theorem 37. -- The following triangle is exact: H'(d, ~*) HI(~ r > H~(d) Proof. ~ We have already seen that Im S-~ Ker I. By the above description ofS onc has KerS :ImB. Next BoI =o since B is equal to o on C x. Finally if q~eZ'(~r and Bq~GB[ -1, B~:bB{} for some {}~C "-1 so that +b0eKerBnKerbCImI+b(KerB) by lemma 36 . Thus KerB=ImI. [] We shall now identify the long exact sequence given by theorem 37 with the one derived from the exact sequence of complexes o -+Cx -+C -~ C/Cx -+o. Corollary 38. -- The morphism of complexes B:C/Cx ~ C induces an isomorphism of H"(C/Cx) with H[-l(d) and identifies the above triangle with the long exact sequence derived from the exact sequence of complexes o --> Cx -+ C ~ C/Cx -+ o. Proof. ~ This follows from the five lemma applied to H"(Cx) > H"(C) , >, H"(a/az) > H"+I(Cx) > H"+~(C) H[(,~) : s > H"(d,d*) 13> H[-'(~) > H~+~(~r > H"+~(s], d*) [] Together with theorem 32 of section 3 we get: Corollary 39. ~ a) Two cycles with characters vl, v2 are cobordant if and only if Sv 1 = Sv 2 in H~(~r b) One has a canonical isomorphism M*(~r | C = H*(~r (of. definition ~6). Z~*(c) e) Under that isomorphism the canonical filtration F" H*(~r corresponds to the filtration of the left side by the dimension of the cycles. 338 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY xo 3 Proof of b). -- Both sides are identical with the inductive limit of the system (H~(~r S). [] Let us now carefully define the double complex C as follows: a) C"" = C"-'(.~, ~*), V n, m E Z; b) for q~eC"", d x~= (n--m+ I) bq~EC"+"m; c) for oEC"'", d~-- --Bq~C "'"+1 (if n=m, the latter is o). n--m Note that dld,=--d2dl follows from Bb =--bB. Theorem 40. -- a) The initial term E 2 of the spectral sequence asociated to the first filtration F vC =  C "''~ is equal to o. n~p b) Let FqC = ~] C"'' be the second filtration, then HP(FqC) = H~(~r for ra>_q n =p-- 2q. e) The cohomology of the double complex C is given by H"(C) = HeV(.~ r if n is even and H"(C) = H~ r if n is odd. d) The spectral sequence associated to the second filtration is convergent: it converges to the associated graded Y,F q H'(~)/F q+x H*(d) and it coincides with the spectral sequence associated with the exact couple. In particular its initial term E 2 is Ker(I o B)/Im(I o B). Proof. -- a) Let us consider the exact sequence of complexes of cochains o -+ Im B ~ Ker B -+ Ker B/Ira B --~ o where the coboundary is b. By lemma 3 6 the first map: Im B ~ Ker B becomes an isomorphism in cohomology, thus the b cohomo- logy of the complex Ker B/Im B is o. b) Let q~(F qC) v= Y~ C"'', satisfy dq~=o, where d=dl+d2. m>>q,n+m~p By a) it is cohomologous in F qC to an element d/ of C p- q'~. Then d~=o means q~ ~ Ker b c~ Ker B, and ~ ~ Im d means ~ E b(Ker B). Thus using the isomorphism (Ker b n Ker B)/b(Ker B) = H~,-2~(~/) (lemma 3 6) one gets the result. c) By the above computation of S as dl d2 -x we see that the map from HP(F q C) to HP(F q-1 C) is the map S from HxP-2q(~a r to H~-2q-'2(d) ; thus the answer is immediate. d) The convergence of the spectral sequence is obvious, since C'"~= o for m > n. Since the filtration of H"(C) given by H"(F q C) coincides with the natural 339 ALAIN CONNES ~2~t filtration of H*(off) (cf. the proof of c)), the limit of the spectral sequence is the associated graded Z F q H~V(off)/Fq+l HeY(off) for n even, F qH~ q+l H~ for n odd. It is clear that the initial term E 2 is Ker I o B/Im I o B. One then checks that it coincides with the spectral sequence of the exact couple 9 [] We shall end this section with several remarks. Remarks. ~ a) Relative theory. Since the cohomology theory Hi(d ) is defined from the cohomology of a complex (C~, b), it is easy to develop a relative theory HI(Off, ~), for pairs off -~ N of algebras, where r~ is a surjective homomorphism. To the exact sequence of complexes o -~ c~(~) --, ~* c-(d) + c-(off, ~) = c-(off)/c-(~) + o corresponds a long exact sequence of cohomology groups. Using the five lemma, the results of this section on the absolute groups extend easily to the relative groups, provided that one also extends the Hochschild theory H*(off, off*) to the relative case. b) Action 0fH*(off, off). Using the product v of [I3] H"(off,./t'l) | H"(off,./t'2) ~ H"+m(off,~r | one sees that H*(off, d) becomes a graded commutative algebra (using off | off = off, as d bimodules) which acts on H*(Off, d ~ (since off Qd d*= d*). In particular any derivation 8 of d defines an element [8] of Hi(Off, off). The explicit formula of [i 3] for the product v would give, at the cochain level 9 .., a TM) . . (~ v 8) (a ~ a', = ~(~(a "+') a ~ a', ., a"), V ~ ~ Z"(Off, off'). One checks that at the level of cohomology classes it coincides with (~ # 8) (a ~ a', ..., a "+') I n+l - -- Z (- i)~ ~(a~ aa j-~) ~(aJ)(aa~+~... ga"+~)), n+Ij=x v <p ~ Z"(off, off*). With the latter formula one checks the equality 8" ~ ---- (I o B) (8 v G) q- 8 v ((I o B) ~) in H"+I(off, off*,) ft (where ~*v(a ~ =~q~(a ~ for all a ~eoff). This is the natural extension of the basic formula of differential geometry Ox = dix + ix d, expres- sing the Lie derivative with respect to a vector field X on a manifold. 340 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY ~5 c) Homotopy invariance of H*(~). Let d be an algebra (with unit), ~ a locally convex topological algebra and ~ e Z[(~) a continuous cocycle (of. section 5). Let Pt, t e [o, i], be a family of homomorphisms Pt : ~r -+ ~ such that for all ae~, the map te[o,I] ~pt(a) e~ is of class C x. Then the images by S of the cocycles p~ ~ and p~ ~ coincide. To prove this one extends the Hochschild cocycle ~ # ~ on ~ | C~([o, i]) giving the cobordism of ~ with itself (i.e. +(fo, fl)_= f:fo dr1, Vf' eCl([o, x])) to a Hochschild cocycle on the algebra Cl([o, I], ~) of Cl-maps from [o, i] to ~. Then the map p : ~/--->C1([o, I], ~), (p(a))t = p~(a), defines a chain over ~ and is a cobordism of p~ ~ with p~ ~. This shows that if one restricts to continuous cocycles, one has p~ = p; : H*(~) ---> H*(d). 5" Locally convex algebras Before we begin with the examples we shall briefly indicate how sections x to 4 adapt to a topological situation. Thus we shall assume now that the algebra d is endowed with a locally convex topology, for which the product d � d ---> d is conti- nuous. In other words, for any continuous seminorm p on ~ there exists a continuous seminormp' such that p(ab) ~ p'(a) p'(b), V a, b e ~/. Then we replace the algebraic dual d* of d by the topological dual, and the space C"(d, d*) of (n + x)-linear functionals on ~ by the space of continuous (n + 1)-linear functionals: ~ e C" if and only if for some continuous seminorm p on d one has Ir ~ ..., a") [ ~_ p(a ~ ... p(a"), V d e s/. Since the product is continuous one has b~ e C "+1, V c? e C". Since the formulae for the cup product of cochains only involve the product in d they still make sense for continuous multilinear functions and all the results of sections i to 4 apply with no change. There is however an important point which we wish to discuss: the use of reso- lutions in the computation of the Hochschild cohomology. Note first that we may as well assume that ~r is complete, since C" is unaffected if one replaces d by its comple- tion, which is still a locally convex topological algebra. Let ~ be a complete locally convex topological algebra. By a topological module over ~ we mean a locally convex vector space .//g, which is a ~-module, and is such that the map (b, ~) ~ b~ is continuous (from ~ � ~r to ~). We say that .~r is topologically projective if it is a direct summand of a topological module of the form ~' ~- ~ 6,= E, where E is a complete locally convex vector space and ~. means the projective tensor product ([29]). In particular ~ is complete, as a closed subspace of the complete locally convex vector space r 341 ALA[N CONNES It is dear then that if~t' a and .,/g= arc topological N-modules which are complete (as locally convex vector spaces) and p : d( x ~ ~= is a continuous ~-linear map with a continuous (;-Linear cross-section s, one can complete the triangle of continuous N-linear maps T .."" 1 l- for any continuous N-linear map f: .1(4 ~ ..r 2. Definition 42. -- Let ..14 be a topological N-module. By a (topological) projective resolution of .K we mean an exaa sequence of projective N-modules aM N-lincar continuous maps which admits a C-linear ~ontinuous homotopy s~ :~/t~->~+i b,+ts,+s~_lb~=id, Vi. As in [36] the module ~r over ~ = d ~,~ ~ (tensor product of the algebra .~r by the opposite atgebra z~ r given by (a | b ~ c = acb, a, b, c ~ M admits the following canonical projective resolution: i) .At = N~.E. (as a N-module), with E. = zr ... @,,M" (n factors); 2) r162 is given by z(a| ~ =ab, a,b ~t; 3) b.(I|174174 = (a~|174174174 n--1 + Y, (-- I) ":I |174 ... | .-- | j=l + (-,)" | a ~ | (al | | a._ ,). The usual section is obviously continuous: Sn((a~b ~174174174 ) = (I| ~174174174174 ) . Comparing this resolution with an arbitrary topological projective resolution of the module ~r over ~ yields: Lemma 43. -- For any topological projective resolution (~r b.) of the module z~ over N = d @~ ,~r the Itochschild cohomology H"(~r zr coincides with the cohomology of the complex -+ Hom~,(..lt a, .~*) -+ ... Homa(..//~ ..~') ~: (where Hom~ means continuous N linear maps). Of course, this Iemma extend~ to any c~mplete topological b~module over aa/. Let us now pas~ to the examples. 342 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY ~7 6. Examples i) d = Coo(V), V a compact smooth manifold. We endow C~176 with its usual Frechet space topology, defined by the semi- norms sup [O'f[ =p,(f) using local charts in V. I~l_<_, As a locally convex space, Coo(V) is then nuclear ([29]) and one has coo(v) c o (v) = coo (v x v). Thus ~ = ~r ~,, ~10 is canonically isomorphic to C=(V � V) and the module over ~ corresponds to the diagonal A: Vfe Coo(V � V), e(f) ----- A*f. Let us assume for a while that the Euler characteristic of V vanishes. The general case will be treated by crossing V with S 1. Let E k be the complex vector bundle on V � V which is the pull back by the second projection pr, : V � V -+ V of the exterior power A * T;(V) of the complexified cotangent bundle of V. By construction, the dual E~ of E 1 is the pull back by pr, of the complexified tangent bundle. We let X(a, b) be a section of E~ such that: a) for (a, b) close enough to the diagonal, X(a, b) coincides with the real tangent vector expg-l(a) (where expb: Tb(V) -+ V is the exponential map associated to a fixed affine connexion); b) X(a,b) 4=o when a4:b. By hypothesis, the Euler characteristic of V vanishes so that there exists on V a real nowhere vanishing vector field Y, with the help of which one easily extends the germ of X around the diagonal to a section of E~ satisfying b). (Use Y as a purely imaginary component.) Lemma 46. -- The following is a continuous projective resolution of the module Coo(V) over Coo(V � V) (with the diagonal action): A* ix ix Coo(V) ~ Coo(V � V) ~- Coo(V 2, El) 4-- ... <--- Coo(V 2, E,) ~ o (n---= dim V) where i x is the contraction withX. Proof. -- Each of the modules ..r k = Coo(V � V, Ek) is finite projective and hence also topologically projective. Obviously i:~ = o. To show that one has a topological resolution it remains to construct a continuous linear section. Let Z,x'eC~176 be such that : X(a, b) = expg-t(a), V (a, b) ~ Support Z'; Z' = I on the support of Z and X -=-- i is a neighborhood of A. 343 x",8 ALAIN CONNES Let ta' be a section of E lsuchthat <X, co')= i on the support of I --Z. Put ~t(a, b) = exp,(tX(b, a)) for (a, b) close enough to A and let s(ta) = z' 7 + - z) ta. By construction s is C~(V)-hnear in the variable a. Fixing a and taking normal coor- dinates around a = 0 one gets q~t(b) = tb, X(o, b) = -- b, so that one can easily check the equality (cp; di x cox) + i x (~; dta~) -/- = q~;(0x c~ ? = tat for any differential form ta x vanishing off the support of Z and satisfying tat(a, a) = o. Applying this with tax = Zta shows that s/x + i x s = id. [] We are now ready to prove: Lemma 4g. m Let V be a compact smooth manifold, and consider ~ = C~~ as a locally convex topological algebra, then: a) The continuous Hochschild cohomology group H~(d, d*) is canonically isomorphic with the space of de Rham currents of dimension k on V. To the (k + I)-linear functional ~0 is asso- ciated the current C such that < C,f0 dr1 ^ ... ^ dfk> = y, ~(~) ~(fo, fo(1),fo(2), ...,f~ a effik b) Under the isomorphism a) the operator I o B : Hk(d, ~q~*) ~ H~-X(~, d') /s the de Rham boundary for currents and the image orb in H~-X(~q~) is contained in the space of totally antisymmetric cocycle classes. Proof. -- a) One just has to compare the standard projective resolution of ,~1 with the resolution of lemma 44, applying lemma 43. Note that (cf. [33]) given any commu- tative algebra ,~/ and bimodule ~4-/, the map T ~ ~ ~(~) T ~ where T e C*(M, ~r ae~k and T~ ..., a ~) = T(a~ a~ transforms Hochschild cocycles in Hochschild cocycles and its kernel contains the Hochschild coboundaries. Next, if q~eZk(d,d*) and q~~ for ~e(Sk, with ~=C~(V), then (under the obvious continuity hypothesis) there exists a current C on V such that <C,fO dfl ^ ... ^ dfk> = ~(fo, fa, ...,fk), V f' ed. Indeed q~ now satisfies the condition ~?(fo, fx f2 fs, . . .,fk +x) = q~(fo fx,f2,f~, ... ,f~ +1) -k- ~(fo f2,fx, fs, . . .,fk+x) for fie C~~ which shows that, as a distribution on V k+x, its support is contained in the diagonal Ak + x = { (x, x, ..., x) e V k + x, x E V }. Thus the problem of existence 344 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY ~a9 of C is local and easily handled say with V = T" or also using local coordinates. Let O k be the space of currents of dimension k on V. Define ~ : -@k -+ Hk( d, d*) by ~(C) (fo, fl, ...,fk) = (C,fO dr1 ^ ... A dfk), V f' ~ C~(V); then the map ~ has a left inverse e given by e(9) = C, where (C,f~ dfl ^ ... ^ dfk) = Ilk!  ~(~r) q~(fo, fo(~),...,fo(k)). a ~b k To check that ~ o ~r = id we may replace V by V � S ~, since the homomorphism p:&----- C ~(V� 1)-+d = C ~(V) given by evaluation at apoint p~S 1 induces a p* split injection Hk(d, d*) ---> t-Ik(&, &*). Thus we may as well assume that the Euler characteristic of V is o. Let X be a section of E~ as above. Let then (.ACE, b~,) be the projective resolution of C~~ given by lemma 44: ..r -= &~ Ek) , 0 k = i x. By lemma 43 the Hochschild cohomology Hk(C~176 (C~~ *) coincides with the cohomology of the complex Homc~o(v,)(..gt'~, C~(V)*). One has a natural isomorphism c~(v 2, Ek) | C~(V) ~ &~ V, A* Ek) and since A* E k is by construction the exterior power AkT;(V), one has a natural isomorphism of Homc~(v,)(.~, C~(V) ") with the space ~k of k-dimensional currents on V. More explicitly, to T e Homc| *) corresponds the current C given by the equality (C,~)=T(to')(i), V(o'~.~r Mr162 Since the restriction of X to the diagonal A is zero we see that the coboundary operator i~ is zero and hence that Hk(.~r .~1") = ~k. To write down explicitely the isomorphism we just need a chain map F of the resolution ,g' to the standard resolution ("gk = (~1 6,~ ~10) 6,~ ~r 6,~ ... 6,~ ~1) above the identity map "go -+ -'g0. Here ~r &~ VXVXV k) and we take (Fto) (a, b, x 1, ...,x k) = (X(x ~, b) ^ ... ^ X(x k, b), ~(a, b) ), V a, b, x i ~ V and co e.//~ = C~~ 2, Ek). One has 9 .., 9 .., X k-l) (b k Feo) (a, b, x 1, x k-') = (Feo) (a, b, a, x t, k--1 -- 3", (-- i)iF~(a, b, x 1, ..., x j, x j, ..., x ~-1) j=l + (-- i) ~ Fta(a, b, x t, ..., x ~-~, b) = (X(a, b) ^ X(x t, b) ^ ... ^ X(x k-t, b), o(a, b)). This shows that b~ Fco = Fix to, V ~, so that bk F = Fb~, and F is a chain map. 17 ~3 o ALAIN CONNES Let q~ e Zk(d, zr be a Hochschild cocycle, the corresponding element of Homc~olv.)(dlk, ~r is given by the equality ,~( ( f | g) | ft | | fk) ( fo) = r f, fl, . . .,fk), /, g,f, E ~t. Let us compute the k-dimensional current corresponding to ~ o F. One has <C,/0 df 1 ^ ... ^ dfk> = ~0 F(o t) (I), where ~, =f0 ol A ... ^ oh, or b) = drY(b) ~ T](V). One has Fo'(a, b, x 1, ..., x k) = <X(x 1, b) ^ ... A X(x k, b), co'(a, b)> =f~ 1-l<X(x', b), a E~k 1 This shows that to compute ~ o F one may replace q~ by the total antisymme- trization r I y. r ~?o on the last k variables. As the differential of the function k l~ k x~<X(x,b),df(b)> at the point x=b is equal to df(b), we conclude that the k-dimensional current corresponding to ~ o F is C = kI 0t(q~) and hence that ~ is an isomorphism. b) Let C e 9k be a k-dimensional current, and q~ the corresponding Hochschild cocycle: ~(fo, fl, ...,fk) =_ <C,fO dr1 ^ ... ^ dfk>. Then Bo q~(fo, ...,fk-1) = q~(i,fo, ...,fk-x) = <C, df~ ... A dfk-t> = <bC,f~ ... ^ dfk-t>. As an immediate corollary, we get: Theorem 16. -- Let d = C~(V) as a locally convex topological algebra. Then: x) For each k, I-t~(z~') is canonically isomorphic to the direct sum Ker b (C ~k) | C) | , C) | ... (where Hq(V, C) is the usual de Rham homology of V). 2) t-I*(d) is canonically isomorphic to the de Rham homology H,(V, C) (with filtration by dimensions). Proof. -- x) Let us explicitly describe the isomorphism. Let q~ a H~(d). Then the current C = ,(I(q~)) given by I ~ q~(fo, f,Ct)...,lOCk)) < C,f~ djl ^ ... ^ dfk > = o is closed (since B(I(q~)) = o), so that the cochain ~(fo, fl, ...,fk) = < C, fO dfl ^ ... ^ df~> 346 NON-COMMUTATIVE DIFFEREN'IIAL GEOMETRY 13x belongs to Z~(d). The class of ~ -- ~ in H~(d) is well determined, and is by cons- truction in the kernel of I. Thus by theorem 37 there exists + ~H~-2(d) with S+ = ~ -- ~, and + is unique modulo the image of B. Thus the homology class of the closed current a(I(~b)) is well determined. Moreover by lemma 45 b) the class of + -- ~ in H~-~(d) is well determined. Repeating this process one gets the desired sequence of homology classes r ~Hk_2~(V , C). By construction, q~ is in the same class (in H~(~)) as C + ~, Si ~j (where for any closed current mj in the class one takes ~ = ~ "~j(fo, fl, ...,fk-2j) = <(aj,fo dft ^ ... ^ df~-21>). This shows that the map that we just constructed is an injection of H~(~ r to Ker b(C ~fk) | Hk-2(V, C) @... | Hk_2,(V, C) | The surjectivity is obvious. ~,) In x) we see by the construction of the isomorphism, that S : Hxk(~) ~ H~+2(M) is the map which associates to each C e Ker b its homology class. The conclusion follows. [] Remarks 47. -- a) In this example the spectral sequence of theorem 39 d) is dege- nerate and the E 2 term is already the de Rham homology of V (with differential equal to o). b) Let 9 ~ H~(C~ Then theorem 46 shows that q~ is in the same class as + ~ S j ~j where the current C is well defined and the homology classes oj are also j=l well defined. One can prove that, once an affine connection V on V has been choosen, one can associate canonically a sequence c0j of closed currents to any ~ ~ Z~(C~ whose support (in V k+l) is close enough to the diagonal A ={(x, ...,x),x~V}. Moreover if ~ is local, i.e. if its support is contained in A, then the germ of oj around any x ~ V only depends upon the germ of ~ around x and the connexion V. This is proven by explicitly comparing the resolution of lemma 44 and the standard one. It remains valid without the hypothesis z(V) = o. c) Let W C V be a submanifold of V, i* : C~~ --> C~ the restriction map, and o -+ Ker i* ---> C~ ---> C~ ~ o the corresponding exact sequence of algebras. For the ordinary homology groups one has a long exact sequence H~(W) ~ H,(V) -~ Hq(V, W) --. H~_,(W) ~ ... where the connecting map is of degree -- I. Since H~, is defined as a cohomology theory, i.e. from a cochain complex, the long exact sequence --, HI(Ca(W)) ~ HI(C~(V)) -+ HI(O~ C~ -->~ H~+I(C~(W) ) ~... 347 ALAIN CONNES x32 has a connecting map of degree -t- 1. So one may wonder how this is compatible with theorem 46. The point is that the connecting map for the long exact sequence of Hochschild cohomology groups is o (any current on W whose image in V is zero, does vanish), thus Im(0) C SH~-~(C~176 d) Only very trivial cyclic cocycles on C~(V) do extend continuously to the C*-algebra C(V) of continuous functions on a compact manifold. In fact for any compact space X the continuous Hochschild cohomology of d = C(X) with coefficients in the bimodule d* is trivial in dimension n ~ 1 (cf. [35]). Thus by theorem 37 the cyclic cohomology of ~/is given by H~"(~r = H~(d) and I-~x"+l(~) ---- o. This remark extends to arbitrary nuclear C* algebras [51]. Example2.-- d=d0, 0elR/Z. (Cf. [16] [I9] [55] [58] 9 ) Let X= exp2~i0. Denote by St(Z*) the space of sequences (a,, ,,),,,, 9 z, of rapid decay (i.e. (In[ -t- [m[) q [a,,,~l is bounded for any q 9 Let d 0 be the algebra whose generic element is a formal sum Y~a,,,, U~ U~, where (a,,,,) e ~9~ *) and the product is specified by the equality U2 U1 = XUx U,. For 0 9 O this algebra is Morita equivalent, in the sense of corollary 24, to the commutative algebra of smooth functions on the 2-torus. Thus in the case 0 e Q,, the computation of H*(d0) follows from theorem 46. We shall now do the computation for arbitrary 0. The first step is to compute the Hochschild cohomology H(d0, ~r where of course d 0 is considered as a locally convex topological algebra (using the seminorms p,(a) = Sup(1 + In [ -}- [m [)q [ a,,,,[). Let us describe a topological projective resolution of d 0 viewed as a module over = do @,, -~0. Put Mr i = ~ | fii where fl --= fl 0 @ t-I x (9 fi2 is the exterior algebra over the 2-dimensional vector space fi x = C * with canonical basis ex, e,. For j = 1, 2 let bj :.~j--*~+, be the ~-linear map such that bl(I| = I| j= 1,2. b2(I| l^e~)) = (U,|174 (XUx| 1@U ~174 As usual, let r o be given by r =ab for a,b 9162 Lemma 48. -- a) (~/t~, hi) is a projective resolution of the module ~r b) H'(d0, d~) =o for i> 2. Proof. -- For v= (na,n,) 9 let U *--U~'U~' 9 X ~=-U *| 9 and yv = i | IJ * e SY. Then X * and Yr commute for any ,~, v' and any element of ~ is of the form x = Y~a~,~, X ~ Y~', where the sequence (a,,r is an arbitrary element of 5g(Z4). One has X ~x r "'"Ix ~+v, Y~Y~' =X "i'Y~+r 348 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Let us check that Ker ~ = Imbl. The inclusion Im b 1C Ker ~ is clear. For x = ~]a~.r X ~ Y~', ~(x) -- o implies Za~,v X ~ X v = o i.e. x = Y~a~. v X'(Y v -- X"). Using the equality o o (I | V~') (I | U~*) -- (VT'| i)(U~'| I) rt i -- 1 o o = (i | u?)( Z u{| up -~-j) (i | u, - u,| i) nj -- i -J- (U~'@ !) ( Z U2 j~ U~'-I-J) (i ~) U 2 -- U 2e i), we see that the left idealKer r is generated by I| l| and I| 2-U~| and hence is equal to Im b 1. Next, one checks that bib z:o. Given x=x l| 1-x s| seKerbl, one o o has Xi(l|174174174 ). To prove that xeImb 2 it is enough to find y e~ such that x 1 :y(Ug| I -- X| With Z----},Us174 U2 one first proves that xl( Z k) : o, using the relation --oo o __ o ~ o Xi( ~, Zk)(I ~Vl Vl~ I ) : xl(I | 1 Vl| I ) ~ (V21| -oo -oo = x~o(I | s Us| ) ~ (U21| = O. --oo Then writing xl = Y.ak Z k, where (aJ is a sequence of rapid decay of elements of the closed subalgebra of ~ generated by Ux| x, i | U1, Us| i, one gets k--1 x~ = Za~(Z ~ - ~) = Za~( Z ZJ)(z- i) =y~(Z- ~). Finally the injectivity of bs is immediate. [] Using this resolution one easily computes H~(~0, do*). We say (of. [32]) that 0 satisfies a diophantine condition if the sequence Ix --k"[ -1 is O(n ~) for some k. Proposition 49. -- a) Let 0 r Q. One has H~ do* ) = C. b) If 0 r Q satisfies a diophantine condition, then Hi(d0, do*) /s of dimension 2 for j = I, and of dimension i for j = ~. c) If 0 r Q does not satisfy a diophantine condition, then H 1, H s are infinite dimensional non Itausdorff spaces. (Recall that by theorem 46, Hi(.~e, ~r is infinite dimensional for j ~ 2 when 0~Q.) Proof. -- We have to compute the cohomology of the complex (Hom~(.Aci, d~), b~). The map T e Hom~(~, d~) ---> T(I) e ~; allows to identify Hom~(c/t'i, d~) with 349 ALAIN CONNES x34 do*| ~-. Moreover, using the canonical trace 9 on d0, v(Y.a~ U ~) = al0,0 / one can identify do* with the space of formal sums 9 = Za U where (a~)~e z, is a tempered sequence of complex numbers (la.,,.I ~ C(In I + [ml) ~ for some C and ~). The linear functional is given by (% x)= "r(gx ) for x e ~. With these notations, the above complex becomes do* do* 9 do* 2- do* o where at(9) = ((U 1 ~ -- 901), (U s ~o -- ?Us) ) and as(gi, (~S) = Us 91 -- Xgl Us -- (~kUl 9s -- 9s U1). Since )~ r O one easily gets Ker a i = C, which gives a). For (gx, gs) eKcras, onc has Us91--XgtUs=XUi9s--gsUl and the cocfficicnts a v of 9 = ~a, U ~ arc uniqucly determined by the conditions a(0. 0) ~ o, Ui 9 -- ~Ua = 9~, Us 9 -- 9Us = cPs. -- -- a I and (X" -- I) a.,, .,_ t = a~,., For Indeed one has (I ~n,) an t-l'n* n,, nt these conditions to be compatible one needs 1 a s 1 S [-Anx an,, n, +IV~ -- i) -1 a., o = o Vn; 0,. = o Vn; Z')- anll + 1, n.(I -- for n 14=o, n s4=o. From the hypothesis as(gt , 92) = o one gets (x a 1 - (I - x') a 2 v nt, -- n I +t, n s -- nx, nl +I Thus the compatibility conditions are: a t a s 1,0 ~ O, 0,1 ~ O. If 0 satisfies a diophantine condition, the sequence (a~) is automatically tempered, which shows that H t (do, do* ) = (I 2. If 0 does not satisfy a diophantine condition, then by choosing say the pair (~Pt, o) where 91 ----- 2~ Ut U~, one checks that the compatibility conditions are fulfilled but n#0 that (a,) is not tempered. This proves b), c) for Ht; the proofs for H 2 are similar. [] At this point, it might seem hopeless to compute H*(d0) (cf. definition 16) when 0 is an irrational number not satisfying a diophantine condition, since the Hochschild cohomology is already quite complicated. We shall see however that even in that case, where H*(d0, do*) is infinite dimensional non Hausdorff, the homology of the complex (Hn(d0, do*), I o B) is still finite dimensional. The first thing is to translate I o B in the resolution used above. Before we begin the computations we can already state a corollary of proposition 49 and theorem 37: Corollary 80. -- (0 r Q,). One has H~ = C and the map I : Hi(d0) -+ Ht(d0, d;) is an isomorphism. (Thus in particular any I-dimensional current is closed.) 350 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Proof. -- By proposition 49, a) one has t-~x(~r =H~162 *) = C. By theorem 37 the following sequence is exact: o ~ H~(~r i H~(~r ' ~r ~ t-Px(~r L H~(,aco). Since the image by S of the generator -r of I~x(~r is non zero (it pairs non trivially with I Proj~r one gets B =o. [] Lemma 51. -- Let ~ ~ ~r ~ = I-I2(~r d0") ; then (I o B) (~) ~ Ht(~r ~r = Ker ~/Im ,,, is the class of (~, ~) where (q)l)n,. = -- x--l( 1 -- x[n--1}m) (I -- ~.n--1)--I e~., .+1 and (cp2).,, . = X--I(I -- X n(m-1)) (I -- Xrn--1) -1 q)n+l,m" Proof. -- To do the computation we first have to compare the projective resolution oflemma 4 8 with the standard resolution (.,It' k' ---- ~ ~, ~10 ~k ... ), i.e. to find morphisms h : ~r ~ ,//' and k : ,~r -+ ~ of complexes of ~-modules which are the identity in degree o. Recall that b~(I| 1|174 = (a l| 2|174 n--1 + W, (-- i)Ji| ajaj+t| | | n -31- (-- I)n(I| ~174 t|174 n_,). The module map h 1 is determined by ht(I | which must satisfy Vh,(x eel) = bl(~ *es) = ~ | -- Uj| i; thus we can take hl(I| = I| i. One determines in a similar way (but we do not need it for the lemma) &(i * (e,^ e,)) = ~ * U2| U1 - x~ | Ul| U2. The module map k 1 : & ~,, d o --+ B @ f~t is determined by kl(I | O v) (v ----- (nl, n2)) which must satisfy bl(kl(I @ U~)) = b~(I @ U ~) = U~@ I -- I | (U~) ~ As in the proof of lemma 4 8 we take kl(I | ~) ----- A~Qe I + B~Qe 2 where o o o 1 o o __ nl nt A~ -~ U~(U~ ~ -- U~) (Ut U1)- , B~ = U t (U 2 -- U~') (U9 -- U2) -1 where to simplify o o notation we omit the tensor product signs (i.e. Ui, U i mean Uj | I, i | Ui). Now the module map /~:~2' ~-'gcz is uniquely determined by the equality b zk~=k lb~ since ~r176 A tedious but straightforward computation gives: &(I | U~| U 0 = U~' X~'"~ O~ -- X -"~"~o LI~ O;, -- x", ~r3~" I3~',| (el ^ e~). X" U~ -- X-" U 1 Uz -- XU2 351 I36 ALAIN CONNES In fact we shall only need the special cases a) v---- (i, o), ~ arbitrary, b) v arbitrary, Vt = (o, o), c) ~ arbitrary, Vt= (i,o); one may as well check directly that k s = o in cases a), b) (compute k 1 b~) and that o o k2(IQU~QU1) = U 1 " (Us " --)" U~') (U s XUs)-I | (e 1 ^ e~). We thus have determined the morphisms h and k. They yield the morphisms k~ : Homa,(..g~, ds* ) ~ Hom~(.g/~, ~), h~ : Hom~(..lg~, do) ~ Hom~(..gl, ~), and we want to compute the composition Let q~ e ~q~ and let ~ be the corresponding element of Homar(.Wg2, ~): '~(a | b ~ | el ^ ez) (x) = ~(bxa) V a, b, x e ,.~t o. Let +=k~'----~ok s. One has +(x ~ x~, x~) = ~(k~(~ | x~ | ~')) (xO) v xO, x~, x~ ~ do. Let ~b l= (IoB)+. One has by definition, for x ~ led| +I(X O, X 1) = +(I, X O, X 1) -- +(X O, X 1, I) -- +(I, X 1, X O) --~ +(X 1, X O, I). Using b) one gets that +(x ~ 1,I) =o for x ~ le~r thus +l(X O, X 1) = +(I, X O, X 1) -- +(I, X 1, X O) V X O, X 1 6~0. Let then ,r162 = (*l, q~s). One has ,r162 = h; +t; thus Let us compute oj(U*), ,J ----- (nl, n~), j = x, 2. Using a), we have ~(u ~) = +(i, u ~, u~). Using c) we have ~(o ~) = +(~, u ~, o~) = ~(k,(~ | O~| U,))(,) __{: -- ' - ~kcn' +l)nt) (I - )k(nt +1,)-1 q)(U~' U~.-1) ifn~ 4: o, if n~ = o. The knowledge of ~l(U~), V v e Z 2, determines the coefficients a~ of ~l = Xa~ U" by the equality 1 X"~" ~l(U-~). a,~ -~- Hence we get ~.,~ = x.~(~ _ x(.-~)-) (~ _ x-(.-~) -~ x.I---, ~.,~+~, where q~ -~ Xa~ UL The computation of ~?~ = Za~ U ~ is done in a similar way. [] 352 NON-COMMIYrATIVE DIFFERENTIAL GEOMETRY x37 We are now ready to determine the kernel and the image of I o B. Let e d*/Im 0~ e H2(d0, do*) be such that (I o B) ~ e Im aa. Let thus ~b = Zb, U * ~ ~r with oq(d?) = (I oB) @. Then I) (I -- ~kn,) b.x_l,n, = _ ~--l(i -- ~(., --1) n,) (I -- ~.,--1)--1 ant,n,+1 ' 2) (~ - x",) b~,.._~ = - x-'(~ - x-,c.,-.) (~ - x~-')-' ~..+l,~. So the image (I o B) q~ e Ht(d0, do*) is o if and only if the following sequence is tempered: c.,,. = (i -- ~k rim) (I -- Xn) -1 (I -- X") -1 a.+~,.,+l. One has ~ e Im ~ if and only if one can find tempered sequences (c{,r.), j = I, ~, such that (X" I)r .+l, ~ ra + ( x'~ -- I) ~ V n, m. This is equivalent -- 6n, m+ 1 = an+l,m+l, to a~,~ = o and the temperedness of the sequence (Ix"-- i[ + IX'-- i[) -a a,+t,,,+t. Thus the next lemma shows that in all cases the kernel of I o B is one-dimenslonal. Lemma 82. --For any O ~ Q, and (n, m) e Z ~, (n, m) 4:(o,o), one has (Ix"- ~[ + Ix ~- ~ I)-~<_ Iml + I~ -x"~[ I~ - x"l-~l ~ -x'l -~ X = ~0. w/th Proof. -- For n = o, [ (I -- X'") (i -- X")- a ] is equal to [ m [ > I SO that the ine- quality is obvious. Thus we may assume that n 4: o, m 4: o. If [ I -- X""[ > [ I -- X" [ the inequality is again obvious, thus one can assume [I -- X'"[ < ]I -- X" [. With X"=e ~, xe[--~,~[, onehas [I--e~'~[<[I--ei~[ with m4=o, thus Imo~[>n, I,~" -- ~ l ~_ 21m. [] Let us now look for the image of I o B in Hi(d0, do*) ~--- Ker ~,/Im ~1. Any pair (@1, ~,) eIm(I oB) + Imp1 satisfies a], 0 = o, ~,l = o (using lemma 5i). Conversely, if a~, 0:a0s,l=o, let us find ~ed0* (~?:~a~U ~) and ~be~o* (hb = ~b~ U ~) so that, with the notation of lemma 5 I, one has (r ~2) = ~1(+) "4- (I o B) q~o This means: I) a~, m = (I -- X m) b._l, m- X-I(I -- X ("-l)m) (I -- ~.-1)-l~,m+l, 2) ~,~ = (x" - x) b.,~_~ + x-~(~ - x "c~-~)) (~ - x~-~) -~ a.+~,~. Since ~(~1, ~) ---- o by hypothesis, one has (X" -- I) a~+l. = ---- (I -- X') a~,m+ 1. Thus one can find sequences b, a satisfying the above equalities with Ib.,.I = la.+,,.+,l =(~ + I I -x'l I I - x"l-' I ~ - x'l-')-' ~+"" i--X m 4,.+, where for m=o and e4=o the right term is replaced by I--X" " 18 ~3 8 ALAIN CONNES By lemma 5 2, lb.,~l < (i + Iml)(Ix"--I I- q- I~.m--I l)I a.~--~,~ I [(X -- Xm)-~l -- (~ + Iml)(I a.~+x,ml + 1~,~+~[). Thus a, b are tempered and we have shown that (q~x, q~) belongs to the image of I o B in Ha(~o, a'o*). Theorem 53. -- a) For all values of O, HOb(do) -~ C 2 and H~162 ~ C s. b) The map (q~t, e?2) ~ Ker ~ ~ (qh(U~-~), q~2(U~-~)) ~ s 2 gives an isomorphism of H~162 = Ht(~r a'o*)/Im(I o I3) with C 2. c) One has H~162 = H2(~r it is a vector space of dimension 2 with basis Sv ('r the canonical trace) and the functional ~ given by ~(,o, e, ,s) = ,o(8,(,~) 8,(x') - 8s(x~) 8~(,~)) v x' ~ do. In the last formula, 81, 82 are the basic derivations of do: 81(U *) = 2nin 1 U", ~s(U ~) = 2reins U'. Proof. -- Since H"(~r do*) = o for n > 3, one has by theorem 37 an equality H~ = I-I~(d0) = H~(d0)/Im B. By corollary 5 ~ one gets H~(d0)/Im B ---- H~(do, do*)/Im(I o B). Thus b) follows from the above computations. In the same way, one has H~V(do)----I~x(d0) , and the exact sequence o -+ H~(d0) _~s i_~(~,0 ) L H2(d0, do* ) -+ Hi(d0). With 0 r Q. one has H~(d0) = G with generator v, and using corollary 5 ~ and the computation of Ker(I o B), we see that the image of I in the above sequence is the one-dimensional subspace of H2(d0, d0* ) = do*/Im oc 2 generated by U 1 U s (i.e. the functional x ~ v(xU 1 Us), V x e do). Let us compute the image I(q~) of the c? e I-Px(do) given by 53 c). Let ~ Hom~(.//~, ~O*) be given by ~((a Q b 0) Q X 1 ~ X 2) (x 0) = ~0(bx 0 a, X 1, X 2) V a, b, x i e do, with the notations oflemma 5 I. Under the identification of H2(d0, do* ) with do*/Im aa, I(~) corresponds to the class of ~' o/6. One has ~(h2(I | e~ ^ e2) ) (x ~ = q~(x ~ Us, U1) -- Xq~(x ~ Ux, U2) = - 2x(2~i)" ~(xO u~ u~). This shows that H~(do) is generated by Sz and ~. [] We can now determine in this example the Chern character, viewed (as in section 2) as a pairing between Ko(do) and H~162 With the notations of theorem 53, we take S'r and 9 as a basis for He'(do). From the results of Pimsner and Voiculescu [55] 35d NON-COMMUTATIVE DIFFERENTIAL GEOMETRY I39 and of [I9] lemme i and th6or6me 7 the following finite projective modules over d 0 form a basis of the group K0(J~'0) ----ZZ: i) d 0 as a right d0-module. 2) 5~(R), (the ordinary Schwartz space of the real line), with module structure given by: (4. u1) (s) = ~(s + o), (4 us) (s) = e ~' ~(s), v s ~ R, ~ ~ ~(R). We shall denote the respective classes in Ko(~r by [I] and [5r Lemma 54. -- The pairing of K0(d6) with HeY(d0) is given by: a) ([I], Stag) = I, ([~o], 8%') = 0 ~]o, i] b) ([I], ~p) ---- o, ([5~ ~p) ---- I. Proof. -- a) One has x(i) ---- i. We leave the second equality as an exercice. b) Since 8j(1) = o the first equality is clear. The second follows from [I9] thfior~me 7, noticing that the notion of connexion used there is the same as that of definition i8 above relative to the cycle over d 0 which defines ~p namely: where A 1, A s are the exterior powers of the vector space (]3, dual of the Lie algebra of R s (which acts on d 0 by 81, 82). (Cf. [I9] definition 2.) Corollary 55. -- For 0 r 0 the filtration of Hev(,~0) by dimensions is not compatible with the lattice dual to K0(d0). We shall see in chapter 4 that any element of this dual lattice is the Chern character of a 2 + ~ summable Fredholm module on d 0. Problem 56. -- Extend the result of this section to the " crossed product " of C~176 1) by an arbitrary diffeomorphism of S 1 with rotation number equal to 0 [32]. 355 Terminology (references to part II) Chain, section 3 Character of a r introduction and proposition t Cobordism of cyd*s, section 3 Cup product of cochains, section x Cycle,, introduction and section x Cyclic cohomology, section I, corollary 4 Exact couple, section 4 Filtration by dimension, section 2, definition x6, section 4, corollary 39 Flabby (algebra), introduction, section t, corollary 6 and [13] Hochschild cohomology, section x, definition 2 Hoehechild coboundaty, introduction Homotopy invariame, section 4, remark c. Irrational rotation algebra, section 6 Pairing with K-the0ry, section 2 Relative theory, section 4, remark a. Stabilized cyclic cohomology, section 2, definition 16 Suspension map, section x, lemma It Tensor product of cycles, section i Topological projective module, section 5 Universal differential algebra, section t, proposition I and [I] [I4] Vanishing cycle, section i, definition 7 List of formulae in Part II bA = Ab' b "~ = o~ b r'~ ~ o Db = b" D B o b + b' B o = D bB ----- -- Bb B s= o SB = 2ixn(n + I) b n+3 n+t bS = -- Sb n+3 Z~ ~ B o Z n+a = BZ n§ Im B = C~ bC~, C Bo Z ~+1 [9 @ ] = (n+m)! n.t m ! [~] v [+] e(de) e = o e(de) s = (de) ~ e 356 Notation used in part II d, ,~ algebras over C cn(d, ~*) space of n + I linear forms on r ~ ..... a n) = 9(a "r(~ .... , a "Y(n)) V r ff Cn(d, d*), y permutation of { o, I .... , n } and aJ E d ~, . ~ c.(d, w') ~p(ao ..... an+X) = ~0 (- x)J r 0 ..... aJ aJ +1 .... , a n+l) + (-- x) n+x tp(a n+l a ~ ..... a n) J- zn(d, d*) = Ker b, Bn(d, d*) =Im b, Hn(d, ~r = Zn/B n C~(d) = { 9 E Gn(d, M*)}, q~X = r ~ V), cyclic permutation z~(~,) = c~(d) c~ Ker b B~(~') = bC~-'(d) H~(d) = Z~(d)/B~(d) algebra obtained from mr by adjoining a unit /1(~r universal graded differential algebra 9 ~(a ~ da I ... da n) = x(a ~ a 1 .... , a n) (proposition i) ~r = ~r | ~0, ~r = opposite algebra of ~r A 9 = ~ e(y) q~v, I ~ = group of cyclic permutations -t~F b' ~ = ~ (-- z)i ~(x ~ ..., xJ xJ +x, ..., x n+x) V ~ e C"(~r .~') r~:~(~C| --~ ~(~)| VxJE.~ r # ~ = (~| o~ e Z~(C), a(,, ,, ,) = 2i~ S~ = ~ # ~ V ~ e z~(~r H*(aq) = 1~ (H~(ar S) F n H*(~r = Im H~(~') B 0 r o ..... a n-l) = r a ~ ..... a n-') -- (-- i) n ep(a ~ ..... a n-l, I), V ~p ~ cn(~, .~r M*(~r Cobordism group of cycles over I : morphism of complexes (C~, b) ~ (C n, b) Dcp = 9--r ~), Vq~ ~ Cn(~, ~1"), ), canonical generator of cyclic group I" g~ = (n--m+ ,)~ V,~ ~C"," = c"-m(.~,~ *) d~q~= x i~ e Cn,m+ x V~ ~cn, m n--m ~" ~(a ~ ..., a") = ~ ~(a ~ ..... ~(ai), ..., a n) V a i ~ ~, q~ ~ cn(~, Mat') and 8 derivation of .~. i-1 357 BIBLIOGRAPHY [I] W. 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KA~o,~rm, Connexions, courbures et classes caract6ristiques en K-th6orie algdbrique, Canadian Math. Soc. Proo., Vol. 2, part I (x982), '9-27. [40] M. KAROUm, K-theory. An introduction, Grundlehren der Math., Bd. 226 (i978), Springer Verlag. [4 I] M. KAROtraI et O. VmLAMAYOR, K-thdorie alg~brique et K-th~orie topologique I., Math. Scand. 28 (x97Q, 265-307 9 [42] G. KASPAROV, K-functor and extensions of C*-algebras, Izv. Akad. Nauh SSSR, Ser. Mat. 44 (198o), 571-636. [43] G. KmPARGV, K-theory, group C*-algebras and higher signatures, Conspectus, Chcrnogolovka (1983). [44] G. KASPAROV, Lorentz groups: K-theory of unitary representations and crossed products, preprint, Chernogolovka, x983. [45] B. KOSTANT, Graded manifolds, graded Lie theory and prequantization, Lecture Notes in Math. 570, Berlin- New York, Springer (I975). [46] J. L. LObAr and D. Q UILL~N, Cyclic homology and the Lie algebra of matrices, C. r. Acad. Sci. Paris, Sdrie I, 296 (~983), 295-297. [47] S. MAC LA~E, Homology, Berlin-New York, Springer (1975). [48] J. MILNOR, Introduction to algebraic K-theory, Annals of Math. Studies, 72, Princeton Univ. Press. [49] J. MILNOR and D. STASHEFF, Characteristic classes, Annals of Math. Studies 76, Princeton Univ. Press. [5 o] A. S. MIw Infinite dimensional representations of discrete groups and higher signatures, Math. USSR Izv. 8 (1974), 85-xi2. [51 ] G. I~DERSEN, C*-algebras and their automorphism groups, New York, Academic Press (I979). [52] M. PENINGTON, K-theory and C*-algebras of Lie groups and Foliations, D. Phil. thesis, Oxford, Michaelmas, Term., x983. [53] M. PENINOTOZ~ and R. PLYMEN, The Dirac operator and the principal series for complex semi-simple Lie groups, J. Funct. Analysis 58 (1983) , 269-286. [54] M. P~MSNEI~ and D. VOICULESCU, Exact sequences for K-groups and Ext groups of certain cross-product C*-algebras, J. of operator theory 4 (I98O), 93-I I8. [55] M. PIMSNER and I). VOICULESGU, Imbedding the irrational rotation C* algebra into an AF algebra, f. of operator theory 4 (I98O), 2ox-2I ~. [56] M. PX~SNE~ and D. VOICULESCU, K groups of reduced crossed products by free groups, J. operator theory 8 (~) (a982), 131-156. [57] M. I~ZZD and B. SLoop, Fourier Analysis, Self adjointness, New York, Academic Press (1975)- [58] M. Rm~F~L, C*-algebras associated with irrational rotations, Pac. J. of Math. 95 (2) (x98x), 415-429 . [59] J. ROS~N~G, C*-algebras, positive scalar curvature and the Novikov conjecture, Publ. Math. LH.E.S. 58 (x984), 4o9-424 9 [6o] W. RUDIN, Real and complex analysis, New York, McGraw Hill (~966). 359 x44 ALAIN CONNES [6t] I. SEOAL, Q.uantized differential forms, Topology 7 (x968), x47-x72. [62] I. S~OAL, O uantization of the de Rham complex, Global Analysis, Proa. Syrup. Pure Math. 16 (I97O), 2o5-2io. [63] B. StMON, Trace ideals and their applications, London Math. Sor Lecture Notes 35, Cambridge Univ. Press (i979). [64] I. M. 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Coll&ge de France, I I, place Marcelin-Berthelot, 75231 Paris Cedex o 5 et Institut des Hautes l~,tudes scientifiques, 35, route de Chartres, 9144 ~ Bures-sur-Yvette Manuscrits refus le I er flnvier 1983 (Part I) et le 1 er avril 1983 (Part II). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Non-commutative differential geometry

Connes, Alain
Publications mathématiques de l'IHÉS , Volume 62 (1) – Aug 30, 2007
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Springer Journals
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Copyright © 1985 by Publications Mathématiques de L’I.É.E.S.
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Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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Abstract

by ALAIN Introduction This is the introduction to a series of papers in which we shall extend the calculus of differential forms and the de Rham homology of currents beyond their customary framework of manifolds, in order to deal with spaces of a more elaborate nature, such as, a) the space of leaves of a foliation, b) the dual space of a finitely generated non-abelian discrete group (or Lie group), c) the orbit space of the action of a discrete group (or Lie group) on a manifold. What such spaces have in common is to be, in general, badly behaved as point sets, so that the usual tools of measure theory, topology and differential geometry lose their pertinence. These spaces are much better understood by means of a canonically associated algebra which is the group convolution algebra in case b). When the space V is an ordinary manifold, the associated algebra is commutative. It is an algebra of complex-valued functions on V, endowed with the pointwise operations of sum and product. A smooth manifold V can be considered from different points of view such as o~) Measure theory (i.e. V appears as a measure space with a fixed measure class), 8) Topology (i.e. V appears as a locally compact space), y) Differential geometry (i.e. V appears as a smooth manifold). Each of these structures on V is fully specified by the corresponding algebra of functions, namely: ~) The commutative von Neumann algebra L~ of classes of essentially bounded measurable functions on V, 8) The &-algebra G0(V ) of continuous functions on V which vanish at infinity, y) The algebra G~~ of smooth functions with compact support. CO~N-~q'ES ALAIN CONNES It has long been known to operator algebraists that measure theory and topology extend far beyond their usual framework to A) The theory of weights and yon aVeumann algebras, B) C*-algebras, K-theory and index theory. Let us briefly discuss these two fields, A) The theory of weights and yon Neumann algebras To an ordinary measure space (X,$) correspond the von Neumann algebra L~~ ~) and the weight ~: r =fxfd VfeL~ X, ?t) +" Any pair (M, ~) of a commutative von Neumann algebra M and weight q~ is obtained in this way from a measure space (X, ~). Thus the place of ordinary measure theory in the theory of weights on von Neumann algebras is similar to that of commutative algebras among arbitrary ones. This is why A) is often called non.commutative measure theory. Non-commutative measure theory has many features which are trivial in the commutative case. For instance to each weight q~ on a yon Neumann algebra M corresponds canonically a one-parameter group a T e Aut M of automorphisms of M, its modular automorphism group. When M is commutative, one has aT(x ) = x, V t e R, V x e M, and for any weight ~ on M. We refer to [i 7] for a survey of non-commutative measure theory. B) C*-algebras, K.theory and index theory Gel'land's theorem implies that the category of commutative C*-algebras and .-homomorphisms is dual to the category of locally compact spaces and proper conti- nuous maps. Non-commutative C*-algebras have first been used as a tool to construct von Neumann algebras and weights, exactly as in ordinary measure theory, where the Riesz representation theorem [6o], Theorem 2.14, enables to construct a measure from a positive linear form on continuous functions. In this use of C*-algebras the main tool is positivity. The fine topological features of the " space " under consideration do not show up. These fine features came into play thanks to Atiyah's topological K-theory [2]. First the proof of the periodicity theorem ofR. Bott shows that its natural set up is non-commutative Banach algebras (cf. [7I]). Two functors K0, K 1 (with values in the category of abelian groups) are defined and any short exact sequence of Banach algebras gives rise to an hexagonal exact sequence of K-groups. For A = C0(X), the commutative C~ associated to a locally compact space X, Kj(A) is (in a natural manner) isomorphic to KJ(X), the K-theory with compact supports of X. 258 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Since (cf. [65] ) for a commutative Banach algebra B, Kj(B) depends only on the Gel'fand spectrum of B, it is really the C*-algebra case which is most relevant. Secondly, Brown, Douglas and Fillmore have classified (cf. [rx]) short exact sequences of (]*-algebras of the form o ---> Yf ~ A -+ C(X) ~o where ~ is the C*-algebra of compact operators in Hilbert space, and X is a compact space. They have shown how to construct a group from such extensions. When X is a finite dimensional compact metric space, this group is naturally isomorphic to KI(X), the Steenrod K-homology of X, cf. [24] [38]. Since the original classification problem of extensions did arise as an internal question in operator and C*-algebra theory, the work of Brown, Douglas and Fillmore made it clear that K-theory is an indispensable tool even for studying C*-algebras per se. This fact was further emphasized by the role of K-theory in the classification of C*-algebras which are inductive limits of finite dimensional ones (cf. [IO] [26] [27] ) and in the work of Cuntz and Krieger on C*-algebras associated to topological Markov chains ([22]). Finally the work of the Russian school, of Mi~enko and Kasparov in particular, ([5 ~ [42] [43] [44]), on the Novikov conjecture, has shown that the K-theory of non- commutative C*-algebras plays a crucial role in the solution of classical problems in the theory of non-simply-connected manifolds. For such a space X, a basic homotopy invariant is the F-equivariant signature cr of its universal covering X, where F = r~l(X ) is the fundamental group of X. This invariant a lies in the K-group, K0(C*(F)) , of the group C* algebra C*(F). The K-theory of C*-algebras, the extension theory of Brown, Douglas and Fillmore and the Ell theory of Atiyah ([3]) are all special cases of Kasparov's bivariant functor KK(A, B). Given two Z/2 graded C*-algebras A and B, KK(A, B) is an abelian group whose elements are homotopy classes of Kasparov A-B bimodules (cf. [42] [43]). For the convenience of the reader we have gathered in appendix 2 of part I the defi- nitions of [42] which are relevant for our discussion. After this quick overview of measure theory and topology in the non-commutative framework, let us be more specific about the algebras associated to the " spaces " occurring in a), b), c) above. a) Let V be a smooth manifold, F a smooth foliation of V. The measure theory of the leaf space " V/F " is described by the von Neumann algebra W*(V, F) of the foliation (cf. [I4] [I5] [I6]). The topology of the leaf space is described by the C*-algebra C*(V, F) of the foliation (cf. [i4] [I5] [66]). b) Let F be a discrete group. The measure theory of the (reduced) dual space is described by the von Neumann algebra ~,(F) of operators in the Hilbert space t*(F) which are invariant under right translations. This von Neumann algebra is the weak closure of the group ring CF acting in t2(F) by left translations. The topology of the 259 44 ALAIN CONNES (reduced) dual space F is described by the C*-algebra C,*(F), the norm closure of CF in the algebra of bounded operators in t~(F). b') For a Lie group G the discussion is the same, with C~(G) instead of CF. r Let F be a discrete group acting on a manifold W. The measure theory of the " orbit space " W/F is described by the von Neumann algebra crossed product L~(W) >q F (cf. [5I]). Its topology is described by the C*-algebra crossed product c0(v) r (of. [51]). The situation is summarized in the following table: Space V V/F F G W/F Measure theory L~~ W*(V, F) ;~(F) ),(G) Lc~ )~ F Topology Co(V ) C*(V, F) C;(F) C;(G) Co(W ) >~ F It is a general principle (cf. [5] [18] [7]) that for families of elliptic operators (Du)ve Y parametrized by a " space " Y such as those occurring above, the index of the family is an element of K0(A), the K-group of the C*-algebra associated to Y. For instance the F-equivariant signature of the universal covering X of a compact oriented manifold is the F-equivariant index of the elliptic signature operator on X. We are in case b) and , e K0(C;(F)). The obvious problem then is to compute K.(A) for the C*-algebras of the above spaces, and then the index of families of elliptic operators. After the breakthrough of Pimsner and Voiculescu ([54]) in the computation of K-groups of crossed products, and under the influence of the Kasparov bivariant theory, the general program of computation of the K-groups of the above spaces (i.e. of the associated C*-algebras) has undergone rapid progress in the last years ([16] [66] [52] [53] [68] [69] ). So far, each new result confirms the validity of the general conjecture formulated in [7]. In order to state it briefly, we shall deal only with case c) above (x). By a fami- liar construction of algebraic topology a space such as W/F, the orbit space of a discrete group action, can be modeled as a simplicial complex, up to homotopy. One lets I" act freely and properly on a contractible space EF and forms the homotopy quotient W � EP which is a meaningful space even when the quotient topological space W/I' is patho- logical. In case b) (F acting on W = {pt}) this yields the classifying space BF. In case a), see [I6] for the analogous construction. In [7] (using [16] and [18]) a map ix is defined from the twisted K-homology K.,,(W � EF) to the K group of the C'-algebra C0(W) >~ F: ~z : K,,,(W � EF) ~ K,(Co(W ) N V). The conjecture is that this map ~ is always an isomorphism. At this point it would be tempting to advocate that the space W � EF gives a sufficiently good description of the topology of W/F and that we can dispense with (x) And we assume that I" is discrete and torsion free, cf. [7] for the general case. 260 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY C*-algebras. However, it is already clear in the simplest examples that the C*-algebra A = G0(W ) :~ F is a finer description of the " topological space " of orbits. For instance, with W = S z and F = Z, the actions given by two irrational rotations R0, , R0, yield isomorphic C*-algebras if and only if 01 = 4- 02 ([54] [55]), and Morita equivalent C*-algebras if and only if 0 z and 02 belong to the same orbit of the action of PSL(2, Z) on PI(R) [58]. On the contrary, the homotopy quotient is independent of 0 (and is homotopic to the 2-torus). Moreover, as we already mentioned, an important role of a " space " such as Y = W/F is to parametrize a family of elliptic operators, (Dy)yey. Such a family has both a topological index Indt(D), which belongs to the twisted K-homology group K,.~(WXrEI`), and an analytic index Ind,(D) = tz(Ind~(D)), which belongs to K,(C0(W ) � F) (cf. [7] [20]). But it is a priori only through Inda(D ) that the analytic properties of the family (Dr)re Y are reflected. For instance, if each Dy is the Dirac operator on a Spin Riemannian manifold M u of strictly positive scalar curvature, one has Inda(D) ---= o (cf. [59] [2o]), but the equality Indt(D ) = o follows only if one knows that the map Vt is injective (cf. [7] [59] [2o]). The problem of injectivity of is an important reason for developing the analogue of de Rham homology for the above " spaces ". Any closed de Rham current C on a manifold V yields a map q~c from K*(V) to C q~c(e) = (G, che) VeeK*(V) where ch:K*(V) -+ H*(V, R) is the usual Ghern character. Now, any " closed de Rham current " G on the orbit space W/l` should yield a map q~o from K,(G0(W ) xl l') to G. The rational injectivity of tz would then follow from the existence, for each co e H*(W � EF), of a " closed current " G(co) making the following diagram commutative, K,,,(W � Er) ~--+ K,((Co(W) ~ r) ob, ---~C H,(W � El', R) Here we assume that W is F-equivariantiy oriented so that the dual Chern character ch, : K,,, ~H, is well defined (see [2o]). Also, we view co eH*(W � EF, C) as a linear map from H,(W � El-', R) to C. This leads us to the subject of this series of papers which is i. The construction of de Rham homology for the above spaces; 2. Its applications to K-theory and index theory. The construction of the theory of currents, closed currents, and of the maps q% for the above " spaces " requires two quite different steps. 261 46 ALAIN CONNES The first is purely algebraic: One starts with an algebra M over C, which plays the role of C~ and one develops the analogue of de Rham homology, the pairing with the algebraic K-groups K0(d), Kl(d), and algebraic tools to perform the computations. This step yields a eontravariant functor H~ from non commutative algebras to graded modules over the polynomial ring C(~) with a generator a of degree 2. In the definition of this functor the finite cyclic groups play a crucial role, and this is why H~ is called cyclic cohomology. Note that it is a contravariant functor for algebras and hence a covariant one for " spaces ". It is the subject of part II under the title, De Rham homology and non-commutative algebra The second step involves analysis: The non-commutative algebra d is now a dense subalgebra of a C*-algebra A and the problem is, given a closed current C on d as above satisfying a suitable conti- nuity condition relative to A, to extend q0 c : K0(M ) ~ C to a map from K0(A ) to C. In the simplest situation, which will be the only one treated in parts I and II, the algebra d C A is stable under holomorphic functional calculus (el. Appendix 3 of part I) and the above problem is trivial to handle since the inclusion M C A induces an isomorphism K0(d ) m K0(A ). However, even to treat the fundamental class of W/P, where I" is a discrete group acting by orientation preserving diffeomorphisms on W, a more elaborate method is required and will be discussed in part V (cf. [2o]). In the context of actions of discrete groups we shall construct C(co) and ~c(,~} for any cohomology class co e H*(W � EF, C) in the subring R generated by the following classes: a) Chern classes of F-equivariant (non unitary) bundles on W, b) F-invariant differential forms on W, c) Gel'fand Fuchs classes. As applications of our construction we get (in the above context): ~) If x e K,,~(W � EF) and (ch, x, co) # o forsome o~ in the above ring R then ~z(x) # o. In fact we shall further improve this result by varying W; it will then apply also to the case W = {pt), i.e. to the usual Novikov conjecture. All this will be discussed in part V, but see [2o] for a preview. ~) For any o~ e R and any family (Du)ue ~ of elliptic operators parametrized b2 Y = W/F, one has the index theorem: ~c(Ind=(D)) = (ch. Ind,(D), co). When Y is an ordinary manifold, this is the cohomological form of the Atiyah-Singer index theorem for families ([5]). 262 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY It is important to note that, in all cases, the right hand side is computable by a standard recipe of algebraic topology from the symbol of D. The left hand side carries the analytic information such as vanishing, homotopy invariance... All these results will be extended to the case of foliations (i.e. when Y is the leaf space of a foliation) in part VI. As a third application of our analogue of de Rham homology for the above " spaces " we shall obtain index formulae for transversally elliptic operators, that is, elliptic operators on those " spaces " Y. In part IV we shall work out the pseudo- differential calculus for crossed products of a C'-algebra by a Lie group (cf. [19]), thus yielding many non-trivial examples of elliptic operators on spaces of the above type. Let A be the C* algebra associated to Y, any such elliptic operator on Y yields a finitely summable Fredholm module over the dense subalgebra ~ of smooth elements of A. In part I we show how to construct canonically from such a Fredholm module a closed current on the dense subalgebra .~/. The title of part I, the Chern character in K-homology is motivated by the specialization of the above construction to the case when Y is an ordinary manifold. Then the K homology K.(V) is entirely described by elliptic operators on V ([9] [18]) and the association of a closed current provides us with a map, K,(V) --> H,(V, C) which is exactly the dual Chern character ch,. The explicit computation of this map oh, will be treated in part III as an intro- duction to the asymptotic methods of computations of cyclic cocycles which will be used again in part IV. As a corollary we shall, in part IV, give completely explicit formulae for indices of finite difference, differential operators on the real line. IfD is an elliptic operator on a " space" Y and C is the closed current C = oh, D (constructed in part I), the map ~c : K,(A) ---> s makes sense and one has 9c(E) ---- (E, [D] ) ---- Index D E V E e K,(A) where the right hand side means the index of D with coefficients in E, or equivalently the value of the pairing between K-homology and K-cohomology. The integrality of this value, Index D E e Z, is a basic result which will be already used in a very efficient way in part I, to control K,(A). The aim of part I is to show that the construction of the Chern character ch. in K homology dictates the basic definitions and operations---such as the suspension map S--in cyclic cohomology. It is motivated by the previous work of Helton and Howe [3o], Carey and Pincus [12] and Douglas and Voiculescu [25]. There is another, equally important, natural route to cyclic cohomology. It was taken by Loday and Q uillen ([46]) and by Tsigan ([67]). Since the latter's work is independent from ours, cyclic cohomology was discovered from two quite different points of view. 263 48 ALAIN CONNES There is also a strong relation with the work of I. Segal [6I] [62] on quantized differential forms, which will be discussed in part IV and with the work of M. Karoubi on secondary characteristic classes [39], which is discussed in part II, Theorem 33. Our results and in particular the spectral sequence of part II were announced in the conference on operator algebras held in Oberwolfach in September i98i ([2I]). This general introduction, required by the referee, is essentially identical to the survey lecture given in Bonn for the 25th anniversary of the Arbeitstagung. This set of papers will contain, I~ The Ghern character in K-homology. II. De Rham homology and non commutative algebra. III. Smooth manifolds, Alexander-Spanier cohomology and index theory. IV. Pseudodifferential calculus for C* dynamical systems, index theorem for crossed products and the pseudo torus. V* Discrete groups and actions on smooth manifolds. Foliations and transversally elliptic operators. VI. VII. Lie groups. Parts I and II follow immediately the present introduction. 264 I. -- THE CHERN CHARACTER IN K-HOMOLOGY The basic theme of this first part is to " quantize " the usual calculus of differential forms. Letting ~ be an algebra over (3 we introduce the following operator theoretic definitions for a) the differential df of any f ~ ~, b) the graded algebra ~ = @ ~q of differential forms, c) the integration co --~ f co ~ (3 of forms co E ~n, df = i[F,f] = i(Ff -- fF) V f ~ M', nq = { xf o df 1 ... dfq, fJ }, ~". f~ = Trace(,o)) V co The data required for these definitions to have a meaning is an n-summable Fredholm module (H, F) over d. Definition 1. -- Let d be a (not necessarily commutative) Z/2 graded algebra over G. An n-summable Fredholm module over s~ is a pair (H, F), where, z) H = H+ ~ H - is a Z[2 graded Hilbert space with grading operator ~, ~---- (-- z) a'gr for all ~ eH +, 2) H is a Z/2 graded left d-module, i.e. one has a graded homomorphism rc of d in the algebra .9~(H) of bounded operators in H, 3) F~.s F 2= I, F~=--~F andforany a~d one has Fa -- (-- I) d~ a aF e .%f"(H) where ,Lf"(H) is the Schatten ideal (cf. Appendix 1). When d is the algebra C~176 of smooth functions on a manifold V the basic examples of Fredholm module over d come from elliptic operators on V (cf. [3]). These modules are p-summable for any p > dim V. We shall explain in section 6, theorem 5 how the usual calculus of differential forms, suitably modified by the use of the Pontrjagin classes, appears as the classical limit of the above quantized calculus based on the Dirac operator on V. The above idea is directly in the line of the earlier works of Helton and Howe [3o], Carey and Pincus [z2], and Douglas and Voiculescu [25]. The notion of n-summable Fredholm module is a refinement of the notion of Fredholm module. The latter is due to Atiyah [3] in the even case and to Brown, Douglas and Fillmore [i i] and Kas- 265 5o ALAIN CONNES parov [4 2] in the odd case. The point of our construction is that n-summable Fredholm modules exist in many situations where the basic algebra d is no longer commutative, cf. sections 8 and 9. Moreover, even when d is commutative it improves on the previous works by determining all the lower dimensional homology classes of an extension and not only the top dimensional "fundamental trace form ". This point is explained in section 7. Let then d be a not necessarily commutative algebra over C and (H, F) an n-summable Fredholm module over ~. We assume for simplicity that d is trivially Z/2 graded. For any a e~r one has da = i[F, a] es For each q e N, let f~q be the linear span in .L~/g(H) of the operators (a ~ da ida 2 ... da q, a ted, X ~C. Since .~/ql � .W,/q~ C .W "/Iqs + q~) (el. Appendix I) one checks that the composition of operators, f~q~ � flq~ ~ f~q~ +q~ endows ~ = + f~J with a structure of a graded j=0 algebra. The differential d, do~ = i[F, to] is such that d 2 = o, d(o l = (do 0 + (-- I) d~ do2 V c0z e Thus (f~, d) is a graded differential algebra, with d 2 = o. Moreover the linear func- tional f: f~" -+ C, defined by = Trace(coo) V to e f~" has the same properties as the integration of the trace of ordinary matrix valued diffe- rential forms on an oriented manifold, namely, for any o~je~V, j= 1,2, ql+q2=n. Thus our construction associates to any n-summable Fredholm module (H, F) over d an n-dimensional cycle over d in the following sense. Definition 2. -- a) a cycle of dimension n is a triple (f~, d, f) where ~ = + ~i is j=0 graded algebra over C, d is a graded derivation of degree i such that d ~ = o, and f : ~" ~ C is a closed graded trace on ~. and b) Let d be an algebra over C,. Then a cycle over ~r is given by a cycle (f~, d, f) a homomorphism p : d --~ ~o. As we shall see in part II (cf. theorem 32) a cycle of dimension n over d is essentially determined by its character, the (n + i)-linear function % 9 (a ~ ..., a") = fp(a ~ d(p(al)) d(p(a~)) ... d(p(a")) V a t e~r 266 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 5I Moreover (cf. part II, proposition i), an n + I linear function -r on ~r is the character of a cycle of dimension n over ~qr if and only if it satisfies the following two simple conditions, ~) "r(a ~, a ~, ..., a n, a ~ = (-- I)" "r(a ~ ..., a") V a ~ e ,.4, ~) ~ (-- I) ~'t'(a 0, ..., a ~a j+l, ..., a n+l ) + (-- I) n+lz(a n+la 0, a 1, ..., a n ) = o. There is a trivial manner to construct functionals -: satisfying conditions ~) and ~). Indeed let C~(~) be the space of (p + x)-linear functionals on ~ such that, 9(a 1,...,a v,a ~ = (-- I) ~(a ~ ~) Va ~e~r Then the equality, bq~(a~ "'" ap+l) ---- ~0 (- I)j q)(a0' "" "' aj aJ+X' "" "' ap+l) + (-- I) '+l~(a '+ta ~ ...,a ~) defines a linear map b from C~(d) to C~+I(J~) (cs part 11, corollary 4). Obviously conditions 0:) and ~) mean that v ~ C~ and bx = o. As b ~" = o, any b9, q~ ~ C~-1(.~), satisfies a) and ~). The relevant group is then the cyclic cohomology group H~,(,~) ={': e C~,(,~), b'r = o)/{&p, ~O e C~-1(.~)}. The above construction yields a map ch*: {n summable Fredholm modules over ~r I-I~(ad). Since ~r is trivially Z[2 graded, the character -r e C~(~r of any n summable Fredholm module over d turns out to be equal to o for n odd. Let us now restrict to even n's. The inclusion .~eP(I-I) C .LPq(H) for p g q (ef. Appendix i) shows that an n summable Fredholm module (H, F) is also n + 2k summable for any k = I, 2, ... We shall prove (cf. section 4) that the (n + 2k)-dimensional character % + ~k of (H, F) is deter- mined uniquely as an element of H~+2k(d) by the n-dimensional character % of (H, F). More precisely, there exists a linear map S:H~(d) ~ I-I~+~(sr such that "r,+~ = Sk'rn in H~+~(~). The operation S : H~(M) -+ H~+*(d) is easy to describe at the level of cycles. Let Y~ be the 2-dimensional cycle over the algebra B = C with character cr, a(I, I, I) = 2ir~. Then given a cycle over ~ with character % S'r is the character of the tensor product of the original cycle by Z. The reason for the normalization constant 2ir~ appears clearly from the computation of an example (cf. section 2). It corresponds to the following normalization for f co, co e ~", n = 2m, fo) = m!(2i~)" Trace(r162 267 ALAIN CONNES 5 2 Let now H~(d) = (fl~ H~(d). The operation S turns H~(.qr into a module n=0 over the polynomial ring C(a), S being the multiplication by a. Let, n*(0~r ~-- Lim(H~(~r S) = H~(~r | where (3(a) acts on (3 by P(a) z = P(I) z for z ~ (3. The above results yield a map ch*: {finitely summable Fredholm modules over a~r H*(ad). We shall show (section 5) that two finitely summable Fredholm modules over ad which are homotopic (among such modules) yield the same element of H'(ad). When off = C~(V), where V is a smooth compact manifold, one has H~ont(ad) ---- H.(V, (3) where H~ont means that the (n -]- I)=linear functionals ? e C~(0ff) are assumed to be continuous, and H.(V, (3) is the ordinary homology of V with complex coefficients. We can now explain what our construction has to do with the Chern character in K-homology. The latter is (cf. [9]) a natural map, ch, : K,(V) ---> H.(V, (3) where the left side is the K-homology of V ([9])- By [24] the left side is isomorphic to the Kasparov group KK(C(V), (3) of homotopy classes of *Fred.holm modules over the C*-algebra C(V) (1). The link between our construction and the ordinary dual Chern character ch. is contained in the commutativity of the following diagram: homotopy classes of finitely summable / .~ H~o.t(C~(V)) *Fredholm modules over C~ ) KK(C(V), C) oh. > H.(V, (3) For an arbitrary algebra 0d over (3, let Ko(d ) be the algebraic K-theory of ad (cf. [40]). One has (cf. part II, proposition I4) a natural pairing < , > between K0(ad ) and the even part of H*(od). Moreover the following simple index formula holds for any finitely summable Fredholm module (H, F) over d: <[e], ch*(H, F)) -~ Index F + V e ~Proj Mk(d ). Here e is an arbitrary idempotent in the algebra of k � k matrices over d, [e] is the corresponding element of K0(ad), and F + is the Fredholm operator from e(H+| C k) to e(H- | (3 k) given by e(F | I) e. This formula is a direct generalisation of [2o], [34]. It follows that any element .~ of H*(0d) which is the Chern character of a finitely sum- mable Fredholm module has the following integrality property, < Ko(d), -: > C Z. (1) A Fredholm module over a *algebra 0d is a *Fredholm module if and only if < a~, ~] > = < ~, a* B > for aEd, ~,~ ~ H. 268 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY To illustrate the power of this result we shall use it to reprove a remarkable result of M. Pimsner and D. Voiculescu: the reduced C*-algebra of the free group on 2 gene- rators does not contain any non trivial idempotent. Letting -= be the canonical trace on C~(F), and e e Proj(C~(F)), one knows that v(e) 9 [o, i]. Using a suitable Fredholm module (cf. [56], [23], [37]) with character x we shall get -r(e) 9 Z and hence "r(e)~{o, I}, i.e. e=o or e= I. Part I is organized as follows: CONTENTS I. The character of a i-summable Fredholm module ........................................... 53 2. Higher characters for a p-summable Fredholm module ....................................... 56 3- Computation of the index map from any of the characters "r n ................................ 6o 4- The operation S and the relation between higher characters .................................. 6i 5- Homotopy invariance of ch*(H, F) .......................................................... 63 6. Fredholm modules and unbounded operators ................................................ 66 7. The odd dimensional case ................................................................. 69 8. Transversally elliptic operators for foliations ................................................. 77 9- Fredholm modules over the convolution algebra of a Lie group ............................... 8o APPENDICES L Schatten classes ........................................................................... 86 2. Fredholm modules ........................................................................ 88 3- Stability under holomorphic functional calculus .............................................. 92 I. The character of a x-snrnmable Fredholm module Let d be an algebra over G, with the trivial Z/2 grading. Let (H, F) be a i-summable Fredholm module over ~. Lemma 1. -- a) The equality ~(a) = i Trace(,F[F, a]), V a 9 ~, defines a trace on ~. b) The index map, K0(d ) -+ Z, is given by the trace .~: Index F, + = ('~ | Trace) (e) V e 9 Proj Ms(d ). ca=de for all a 9 Proof. -- a) Since d is trivially Z/2 graded, one has As zF = -- Fe one has eF[F, a] ---- eF ~ a -- eFaF = ,F 2 a + FasF = ea + FaeF since F 2 = i. Thus, eF[F, a] = [F, a] cF. Then "r(ab) = ~ Trace(sF[F, ab]) = ~ Trace(eF[F, a] b + cFa[F, hi) 2 2 = - Trace([F, a] ~Fb + [F, b] cFa), Thus (ab) = (ba) for a, b d. which is symmetric in a and b. 269 ALAIN CONNES b) Replacing d by Mq(d), and (H, F) by (H| q, F| I) we may assume that q=I. Let F=[; oW"[ so that PQ=IH_ , QP=I m. With Hx=eH +, H2 = ell- we let P' (resp. Q') be the operator from I-I t to t-I s (resp. H2 to Hx) which is the restriction of eP (resp. eQ.) to t-I x (resp. t-Is). Since [F, e] 9 .o.WX(H) one has P' O' -- In, 9 .ogal(H2) , O~' 1 )' -- IH1 e ,,oCfl(Hx). Thus (proposition 6 of Appendix I) one has Index P' -- Trace(Iri 1 -- O~ 1 )') -- Trace(in, -- P' Q') = Tracem(e -- eOePe ) -- Tracea_(e -- ePeQe) = Trace(~(e -- eFeFe)). But Trace(r -- eFeFe)) = Trace(~(e -- FeFe) e) = Trace(eF(Fe -- eF) e) = I Trace(r e] + ,F[F, e] e) = I Trace(r e]) = -r(e). [] 2 2 Then its character Definition 2. -- Let (H, F) be a I-summable Fredholm module over d. is the trace ~ on d given by lemma I a). Then Corollary 3. -- Let "r be the character of a I-summable Fredholm module over d. < Ko(ag), ~) C Z. Now let A be a C*-algebra with unit and "~ a trace on A such that 1:(x'x) ~o for xeA, I) %" is positive, i.e. 2) v is faithful, i.e. x 4= o :,- x(x* x) > o (cf. [55]). Corollary 4. -- Let A be a C*-algebra with unit and 9 a faithful positive trace on A such that ":(I) = I. Let (H, F) be a Fredholm module over A (cf. Appendix 2) such that a) d ---= {a 9 A, [17, a] 9 .s } /s dense in A, b) -~/d is the character of (H, F). Then A contains no non trivial idempotent. Proof. -- By proposition 3, Appendix 3, the subalgebra d of A is stable under holomorphic functional calculus. Hence (Appendix 3) the injection d ~ A yields an isomorphism, K0(d)~ K0(A ). Thus the image of K0(A) by -r is equal to the image of K0(d ) by the restriction of-r to d so that, by corollary 3, it is contained in Z. If e is a selfadjoint idempotent one has ~(e) 9 [o, I] n Z = {o, i } and hence, since v is faithful, one has e = o or e = I. It follows that A contains no non trivial idem- potent f, f~ =f. [] Before we give an application of this corollary, let us point out that its proof is exactly in the spirit of differential topology. The result is purely topological; it is a state- ment on a C*-algebra, which, for A commutative, means that the spectrum of A is connected. But to prove it one uses an auxiliary " smooth structure " given here by the subalgebra d = {a 9 A, [F, a] 9 s176 270 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY As an application we shall give a new proof of the beautiful result of M. Pimsner and D. Voiculescu that the reduced &-algebra of the free group on two generators does not contain any non trivial idempotent [56]. This solved a long standing conjecture of R. V. Kadison. We shall use a specific Fredholm module (H, F) over the reduced C*-algebra of the free group which already appears in [56] and in the work ofJ. Cuntz [23], and whose geometric meaning in terms of trees was clarified by P. Julg and A. Valette in [37]. Definition 5. -- Let r be a discrete group. Then the reduced C*-algebra A -~ c~(r) of F is the norm closure of the group ring CF in the algebra .Cf(t2(F)) of operators in the left regular representation of F (el. [5i]). Now let I" be an arbitrary free group, and T a tree on which F acts freely and transitively. By definition T is a i-dimensional simplicial complex which is connected and simply connected. For j -~ o, I let T i be the set ofj-simplices in T. Let p e T O and 0 : T~ ~ TI be the bijection which associates to any q ~ T ~ q 4:p, the only i-simplex containing q and belonging to the interval [p, q]. One readily checks that the bijection 0 is almost equivariant in the following sense: for all g ~ F one has o(gq) = go(q) except for finitely many q's (cf. [23], [37]). Next, let H + = 12(T~ H- = I2(T 1) @ C. The action of F on T O and T 1 yields a C~(F)-module structure on /Z(Ti), j ---- o, i, and hence on H + if we put a(~, ),) = (a~, o) V ~ eta(T1), ;~ e C, a e c;(r). Let P be the unitary operator P:H + -> I-I- given by P~v: (o, I), Pgq----r Vq4:p (where for any set X, (r is the natural basis off'(X)). The almost equivariance of 0 shows that Lemma 6. -- The pair (H, F), where H = H + @ H-, F = is a Fredholm [; o module over A and ~/ = { a, IF, a] ~ .5f ~ (H)} is a dense subalgebra of A. Proof. ~ For any g ~ F the operator gP -- Pg is of finite rank, hence the group ring CI" is contained in d = {a ~ C;(F), [F, a] E ~~ }. As CF is dense in C~(F) the conclusion follows. [] Let us compute the character of (H, F). Let a~, then a--P-aaP~~ and _I Trace(~F[F, a]) = Trace(a -- p-1 aP). Let 9 be the unique positive trace on A such that "r(Yag g) = at, where x e F is the unit, for any element a = ag g of CF. Then for any a ~ A = C;(F), a- "~(a) I belongs to the norm closure of the linear span of the elements g ~ F, g 4: I. 27'1 56 ALAIN CONNES Since the action of F on T~ is free, it follows that the diagonal entries in the matrix of a--v(a) I int2(T j) are all equal to o. This shows that for any ae~r one has, Trace(a -- p-1 aP) = x(a) Trace(i -- p-1 IP) = v(a). Thus the character of (H, F) is the restriction of v to d and since -r is faithful and positive (cf. [5I]), corollary 4 shows that Corollary 7. -- (Cf. [56]). Let F be the free group on 2 generators. Then the reduced C*-algebra C*(F) contains no non trivial idempotent. 2. Higher characters for a p-snmmable Fredholm module Let d be a trivially Z[2 graded algebra over C. Let (H, F) be a p-summable Fredholm module over ~. As explained in the introduction we shall associate to (H, F) an n-dimensional cycle over d, where n is an arbitrary even integer such that n >= p. In fact we shall improve this construction so that we only have to assume that n => p -- i, i.e. that (H, F) is (n + I)-summable. Let ffbe obtained from d by adjoining a unit which acts by the identity operator in H. For any T e.Sf(H) let dT = i[F, T] where the commutator is a graded commutator. For each j e N we let ~J be the linear span in .~f(H) of the operators of the form a ~ 1...da t, a ke Lemma 1. -- a) d ~T=o VTe~(H). b) d(T x T2) = (dT1) T z + (-- i)eT, Ta dT, V Tx, T, e .~(H). c) d~ t C ~J +1. d) h j x ~k C ~J + k; in particular each fit is a two-sided ~-module. e) ~k C ~~ Proof. -- a) If T is homogeneous, then F(FT -- (-- I) ~ TF) -- (-- I)eT+I(FT -- (-- I) ~T TF) F = F 2 T -- TF ~ = o. b) The map T ~ [F, T] is a graded derivation of .Lf(H). c) Follows from a), b). d) It is enough to show that for a ~ j, ae~ one has (a ~ da 1. . . da j) a e ~j. This follows from the equality (da j) a = d(a t a) -- d da, by induction. e) Since (H,F) is n+ I summable one has dae.~+l(H) for all aed and I I I e) follows from the inclusion .Lr p � -oq ~q C ~" for r = p + q (cf. Appendix I). [] 272 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Lemma i shows that the direct sum ~ = G ~J of the vector spaces ~J is naturally j=0 endowed with a structure of graded differential algebra, with d2= o. Lemma 2. -- For any T 9 .Lf(H) suck that [F, T] 9 ~I(H) let Tr,(T) = I- Trace(r T])). a) If T is homogeneous with odd degree, then Trs(T ) = o. b) If T 9 s then Trs(T ) = Trace(~T). c) One has IF, ~"] C s and the restriction of Trs to ~n defines a closed graded trace on the differential algebra ~. Proof. -- a) Since F[F, T] is homogeneous with odd degree one has ~F[F, T] = -- F[F, T] and Trace(eF[F, T]) = Trace(F[F, T] s) = -- Trace(eF[F, T]) thus Trace(r T]) = o. b) i_ Trace(eF[F, T]) = i Trace r -- FTF) for all T with 0T = o (mod 2). 2 2 If T 9 .~al(H) then Trace(r = -- Trace(FcTF) = -- Trace(eT), so that - Trace(r T]) = Trace(r c) One has [F,~n]Cf~+tC.LPl(H) by lemma x. Since d z=o one has Tr,(dco) = o V co 9 f/n-~. It remains to show that for coa 9 f~", co~ 9 ~', nl + n~ = n one has Tr,(coa co2) = (-- I)"'"' Tr,(co, %), or equivalently, that Trace(r d(co 1 co,)) = (-- x)" Trace(r d(coz col)). Since cF commutes with dcoj and dco,, one has Trace(r d(cox co2)) = Trace(r co2) + (-- I) n' Trace(r dco2) = Trace(r 2 dcox) + (-- i)"' Trace((r dco,) col) = (-- I) ~' Trace(r d(m~ cot) ). [] We can now associate an n-dimensional cycle over ~r to any n + I summable Fredholm module (H, F). Definition 3. -- Let n = 2m be an even integer, and (H, F) an (n + i )-summable Fredholm module over d. Then the associated cycle over ~' is given by the graded differential algebra (fl, d), integral the f co = (2iz:)" m ! Trs(co) V co 9 ~2" and the homomorphism r: : ~r -+ G 0 C .W(H) of definition I. 8 ALAIN CONNES The normalization constant (2irc)"mI is introduced to conform with the usual integration of differential forms on a smooth manifold. To be more precise let us treat the following simple example. We let F C 13 be a lattice, and V = 13[F. Then V is a smooth manifold and the 0 operator yields a natural Fredholm module over C~~ We consider 0 as a bounded operator from the Sobolev space H + = { ~ ~ L~(V), 0~ e L2(V) } to H- = Lz(V). The algebra C~ acts in H + by multiplication operators, and the operator F is given by , where ~ ~ 13, i, r F  the orthogonal of ~+~ the lattice P (to ensure that 0 + e is invertible). We let (eg-)ger" be the natural orthonormal basis of Lz(V) = H-, s~-(z) = 113/r -" exp i(g, z> for z e 13[F, and (%+) be the corresponding basis of H +, Cg+ = (0 + e)-x ~-. Thus ,+(z) = (ig + ,)-~ r for z e 13]F, and we may as well assume that the r form an orthonormal basis of H +. For each gsP let UgeC*(V) be given by Uo(z ) =expi(g,z), then Uoa+o = = UglUg= for gl, gz e F" and the algebra G| is naturally isomor- phic to the convolution algebra 5e(F') of sequences of rapid decay on F', C~ ={ZagU0, a e S'~ One has Ugr = %~ and _ i(g + k) + r %++~, U~ ,~ + = U~(ik + r e~-= (ik + ~)-x so+~ -- ik + for any g,k~F t . We are now ready to prove Lemma d. ~ With the above notations, (H, F) is a 3.summable G~ and I .I 0 1 Try(f~ i[F'fl] i[F'f2]) = ~i~ f df ^ df 2 V f~ 2 e C~ where V is oriented by its complex structure. Proof. -- For g ~ P" one has (!(g + k) + (FU 0-UgF)~k + =F~ ik+s _ o+k--%+k _ /g _ _ {i(g+k)_+e I %+k -- ik + r ~g+k -- ig ~++~. Since 5~(F  C tl(P it follows that and similarly (FUg -- U s F) ~- -- ih + e (H, F) is p-summable for any p > 2. To prove the equality of the lemma we may assume that f~ = Ug i with go, gi, g~ e F I. From the above computation we get N, uj [r, Uj IF, UgJ I + g~ + k -- i g~ + k -- i ~go+gl+g~+k 274 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY and IF, U.o] [F, [F, Uj +g2+k" = -- - Trace(r Ug0] [F, Ugx] [F, Ug,]) is equal Thus Trs(Ug ~ i[F, Ugl] i[F, Uj) 2 to o if go + gl + g*, # o and otherwise to: ker  1-k g2 -b k -- i ~ --k k -- i This sum can be computed as an Eisenstein series ([70]). More precisely let u, v be generators of r -L with Ira(v/u) > o and El(z ) the function N M El(z ) = lim ~ (Lira ~ (z-4-k)-l) where k=~u-bvv. Then the above expression coincides with g1(E1(-- is) -- E1(g ~ -- is)) -- g~(E1(g~ -- i~) -- E~(gl + g~ -- is)) = =i=(n, m, - nl where g~=niu+m~v (cf. [7o], p. I7). Let (~,[~) be the basis of C over R dual to (u,v). Then P=27r(Za A-Z~), Ug(xa-]-y[~) =e~"*ei"Y for all x,y~R, g=nu +mveF  For go+gl+g, 4~ one has (v Ug0 dUg, dUo, = o and otherwise fV Ug0 dUg1 dUg2 = f2rc f~ ( (inx) (im2) -- (in~) (iml) ) dx dy = (2i.) 2 > (n 1 m 2 -- n 2 ml). [] A similar computation yields the factor (2in)"m! for n = 2m. Proposition 8. -- Let n = 2m, (H, F) be an (n A- i)-summable Fredholm module over ~r and 9 be the character of the cycle associated to (H, F), -r(a ~ ..., a ~) = (2i7r)" m! Tr~(a ~ da ~ ... da"). Then a) x(a 1,...,a",a ~ ='~(a ~ for a jed; b) ~ (-- I) j'c(a ~ ..., a j a j+l, ..., a "+1) + (-- I) "+lx(a "+1 a ~ ..., a") = o. Proof. -- Follows from proposition i of part II. [] With the notation of part II, corollary 4, one has -~ e C[(~1) by a) and bv = o by b), i.e. v ~ Z[(zr Remark 6. -- All the results of this section extend to the general case, when],~' is not trivially Z/2 graded. The following important points should be stressed, ~) Since a ~ ~ .~r can have non zero degree rood =, it is not true in general that f to=o for c0ef~", n odd. 276 60 ALAIN CONNES ~) Since the symbol d has degree i, the n-dimensional character -r, of an (n q- i)- summable Fredholm module (H, F) over ~r is now given by the equality, x,,(a ~ ..., a") = c,(-- x) e~+e~"+ "'" + e~*+~+ ... Tr,(a 0 da ~ ... da"). n+2 Here c, is a normalization constant such that c,+~ = 2ir~--c,,, we take Y) In general, the conditions a), b) of proposition 5 become n Oaj) t 0 a') "r(a', . . ., an, a ~ = (-- i)n( - I) ~176 "tea,a' , . . ., a") V aJ e.~ r (-- I) j'l~(a 0, ..., a j a t§ ' ..., a n+l ) b') .i=o + (_ 0.+,(_ "r(a TM a ~ a 1, . .., a") = o. The general rule (cf. [49]) is that, when two objects of Z/2 degrees ~ and ~ are permuted, the sign (-- i) ~ is introduced. 3" Computation of the index map from any of the characters ~. Let d be an algebra over C, with trivial Z[~ grading. Let n = 2m be an even integer, (H, F) an (n + I)-summable Fredholm module over d, and -r, the n- dimensiona character of (H, F). Let (x,) be the class of x, in H~(~) = Z~(d)/bC~-X(d). By part II, propo- sition I4, the following defines a bilinear pairing ( , ) between Ko(.~t ) and H~(~): (e, 9) = (2i~)-m(m!)-'(9 # Wr) (e, ..., e) for any idempotent e e Mk(d ) and any 9 ~Z~(d)- Here 9 # Tr ~Z~(Mk(~r is defined by (9 # Tr) (a ~174 ~ ..., a"| m") = 9(a ~ ..., a") Trace(m ~ ... m") for any a jed, m jeMk(C ). When the algebra d is not unital, one first extends 9 e Z~(d) to ~ e Z'~(ff), where ~r is obtained from ~r by adjoining a unit, ~(a~ + X~ I, .. ., a" + X" I) =9(a ~ Va~e~ r hieC. Then one applies the above formula, for e e Mk( ~. Theorem 1. -- (Compare with [25] and [34]). Let n = 2m and (H, F) an (n + I)- summable Fredholm module over zi. Then the index map K0(~ ) ---> Z /s given by the pairing of K0(d ) with the class in H~(d) of the n-dimensional character v, of (H, F): Index F + = ([e], (v,)) for e e Proj Mq(d). 276 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 6i Proof. -- As in the proof of lemma I. I b) we may assume that k = i, that ~1 is unital and that its unit acts in H as the identity. Let F=[; oQ], sothat PQ=Ia_ , Q P= In+. Let H l=eH +, H,=eH-, and P' (resp. O~) be the operator fromH 1 to Hz (resp. H~ to Hx) which is the restriction of eP (resp. eO) to H x (resp. H~). Thus ~HI- Q' P' (resp. Ia,- P' O~) is the restriction to H i (resp. H~) of e--eFeFe. As e--eFeFe=--e[F,e]*e, and [F,e] ~"+I(H), we get (Appendix I, propo- sition 6) Index P' = Trace e(e -- eFeFe) "+l. One has (e, %,) -- (- r)" Trace(r e]~"+~). As IF, e] ---- elF, e] + [F, e] e, one has Trace(r e]) 2"+t) = Trace(eFe[F, e] [F, e] ~"~) + Trace(r e] e[F, e]2"). cF=--Fr F[F,e] m+l=-[F,e] m+lF, so that Now Trace(r e] ~ + 1) = _ Trace(F,e[F, e] ~ +l) = -- Trace(,e[F, e] ~+a F) = Trace(,eF[F, e]m+l). As elF, e] ~= IF, e] ze we get Trace(eF[F, e] z'+l) = 2 Trace(r e] ,[F, e] ~'') = 2(-- I) "~ Trace r -- eFeFe) m+l. [] 4" The operation S and the relation between higher characters In part II, theorem 9, we show that the operation of tensor product of cycles yields a homomorphism (% +) ~ q> # + of Z~(d) � Z~(&) to Z~+m(d| for any algebras d, ~ over C. Taking ~=C and aeZ~(C), ~(X0, Xl, X,) =2inX 0xlkS yields the map S, Sq~ = 9 # ~ from Z~(d) to Z~+2(~1| C) = Z~+2(~1). By part II, corollary Io, one has SB~(~r B~+2(~r Now let n = 2m be even, (H, F) be an (n + i)-summable Fredholm module over d. As .L~+t(H) C .oq~+S(H), the Fredholm module (H, F) is (n + 3)-summable, and hence has characters %, -~,+z of dimensions n and n + 2. Theorem 1. -- One has %+2 = S% in H~+~(~r Proof. -- By construction, -r. is the character of the cycle (~, d, f) associated to (H, F) by definition 3. Thus (part II, corollary Io) S% is given by n+l I S-~,(a ~ ..., a "+z) = 2i Y, (a ~ da 1 ... da j-l) a j aJ+i(da j+9" ... da "+z) 0 9 n+l = (2/~) "+1 mt Z Tr,((a ~ dat.., da j-i) a t aJ+l(da ~+~ ... da"+2)). 277 6~ ALAIN CONNES By definition, % +2 is given by %+2(a ~ ..., a "+2) = (2iTr)"+t(m + I)l Tr,(a ~ da t ... da"+Z). We just have to find % e C~+1(~qr such that b?o = S~r, -- %+2. We shall construct ? E C~+t(,~r such that n+l b~(aO, ..., an+2 ) = -2 y~ TL((aO da t ... daJ_t) aj aj+t(daJ+2 ... da,+2) ) n+l We take ? -- ~o (- I)j qfl' where q~i(aO ' .-., a . +1) = Trace(eFa ~ da j + t ... daJ-1). One has a~ da j+l.., da j-t ~f~"+t C S~ so that the trace makes sense; moreover by construction one has ? ~ C~+t(.~r To end the proof we shall show that (-- i) j-t b?J(a ~ ..., a n+2) Tr,(a ~ da 1 ... da" + z) + - (-- I) j Tr,((a ~ dal.., da j-t) a j aJ+t(daJ +2 ... dan+9~)). Using the equality d(ab) = (da) b+adb, with a,b~d, we get b~(a ~ ..., a" + 2) ---_ Trace(eF(a i+t da j+2 ... da "+z) a~ 1 ... daJ)) + (-- i)i -1 Trace(eFaJ+l(dai+Z... da~ da i-1) a j) + Trace(eFaJ(da ~+t ... da "+z) a~ t ... dai-t)). ... da "+2) a~ t ... da j-t) E f~". Using the equality Let [3 = (da ~+2 Trace(~ d[3) = TL(~ d[3) = TL(i[F, 0c] [3) V 0c ~ SO(H), ~0~ = -- 0re, we get (-- i)J -t Trace(r j+2 ... da ~ ... da j-t) a j) = TL(i[F , a j Fa j+t] [3). Thus, bgJ(a ~ ..., a ~+z) = Trace(da j cFa j+l [3) + TL(i[F, a j Fa j+l] [3) + Trace(r jda j+l [3) = Tr,((Fd(a j a ~+1) + i[F, a i FaJ+l]) [3). One has Fd(a ja j+l) +i[F,a iFa j+t] :--i(da jda j+t-2a ~a ~+t) and the above equality follows easily. [] This theorem leads one to introduce the group HeY(d) which is the inductive limit of the groups I-~fl(d) with the maps, 278 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY With the notation of part II, corollary io, one has, n~ : H[~(a/) | C where C(a) acts on C by P(~) -+ P(t) (cf. part II, definition t6). Definition 2. -- Let (H, F) be a finitely summabte Fredholm module over ~. We let ch*(H, F) be the element 0f H*(d) given by any of the characters "~2,,, m large enough. By part II, corollary t 7, one has a canonical pairing ( , ) between Hev(~/) and Ko(~) and theorem 3. t implies the following corollary. Corollary 3. -- Let (H, F) be a finitely summable Fredholm module over d. Then the index map K0(~r ) -+ Z is given by Index F + -~ (ch.(e), ch*(H, F)) V e ~ Proj Mk( ~. For such a formula to be interesting one needs to solve two problems: t) compute H*(~); ~) compute ch*(H, F). In part II we shall develop general tools to handle problem t. 5" Homotopy invariance of ch'(H, F) Let d be an algebra over C. In this section we shall show that the character ch*(H, F) ~ Hov(M) of a finitely summable Fredholm module only depends upon the homotopy class of (H, F). Let H 0 be a Hi]bert space and H the Z/2 graded Hi]bert spacewith H+=Ho, H-=H o. Let F~L~~ F=[~ Io], Lemma 1. -- Let p = 2m be an even integer. For each t ~ [o, I] let ~t be a graded homomorphism of d in 5e(H) such that I) t -+ IF, nt(a)] is a continuous map from [o, i] to .~q~P(H)for any a ~ d, 2) t -+ rq(a) ~ is a C 1 map from [o, I] to H for any a ~ ~, ~ ~ H. Let (Ht, F) be the corresponding p-summable Fredholm modules over d. Then the class in H~ + 2( ~t) of the (p + 2).dimensional character of (Ht, F) is independent of t ~ [o, i]. Proof. ~ Replacing d by ffwe can assume that ~ is unital and that nt(i) = i, V t ~ [o, i]. By the Banach Steinhaus theorem, the derivative 8,(a) of the map t-+ nt(a) is a strongly continuous map from [o, I] to Lf(H). Moreover, ~t(ab) = ~t(a) 8,(6) + ~,(a) ~t(b) for a, b ~ d, t ~ [o, I]. For t ~ [o, 1] let ~, be the (p + 2)-linear functional on d given by p+l ~t(a ~ ..., a p+t) = Y~ (-- t) k-lTrace(~t(a ~ [F, nt(al)] ... k=l [r, ~,(ak-1)] ~,(a k) [F, 7r,(ak+l)] ... [F, 7r,(a'+l)]). 279 64 ALAIN CONNES Using the equality 8t(ab) = rq(a) ~t(b) + ~t(a) ~t(b), V a, b e ~r one checks that q~t is a Hochschild cocycle, i.e. bgt----o, where p+l b~pt(a ~ . . ., a 7'+~) ---- Z (-- r)q q~t(a ~ .... , aq a q+l, . . ., a p+~) q=0 ..~ (__ i)P+2 q)t(alo+2 a O, ~1, "'', aV+l), V a j e ~r Let 9 be the (p + 2)-linear functional on ~r given by q~(a ~ ..., a p+I) = %(a ~ a p+I) dt. fo "., (Since II~,(a)II and IIn,(~)II are bounded, the integral makes sense.) One has One has b~=o and s ~ v+l) =o if a J= I for some j4=o. dt Y, (-- i)kTrace(~[F, ~t(a~ ... q~(l,a ~ t , ...,a v) =f~ k=0 [r, ~,(~- 1)] 8,(~) [F, ':t(~ + ~)] --. [r, ~,(~')]). Let .r,(a ~ ..., a p) = Trace(r ~ [F, ~,(al)] ... [F, ~xt(aV)]). One has /- (.rt+~(a ~ ..., a v) -- -rt(a ~ ..., aV)) ---- Trace (r ~(nt+ s(a ~ -- nt(a~ ~,+ s(at)]... IF, 7~,+ s(al~)]) + Trace (r ~ IF, rq(at)]... [F, ~(rq+,(a v) -- rq(aV))]). When s --+ o one has, using t) and 2), Trace (~7:t(a ~ [F, rq(at)]... IF, rq(a k- 1)] iF, ~(rq +s(a ~) -- rq(a~))]... [F, rq +8(aV)]) = (-- I) kTrace (e[F, ~t(a~ ... [r, ~,(a~-l)] I-s (~'+8(ak) - ~'(a~)) [r, ~,(~+t)] ... [r, ~,+s(~')]) -+ (-- ~)~ Trace(r rq(a~ ... [F, rc,(ak-1)] ~t(a k) [F, r:,+~(a~+~)] ... [F, ~,+,(aV)]). Thus ~(I, a 0, ..., a p) = ~'~ dt -= "q(a ~ ..., a v) - "ro(a ~ ..., a p) and the result fol- lows from Part II, lemma 34, since bq~=o and B oq~='r 1-%. [] 280 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 65 Theorem 2. -- Let d be an algebra over 13, H a Z/2 graded Hilbert space. Let (Ht, Ft) be a family of Fredholm modules over d with the same underlying Z/2 graded Hilbert space H. [o Let O~ be the corresponding homomorphisms of ~ in ~,~ and F t = . Assume that for some p < oo and any a e d, P~ I) t ~ p+(a) -- Qt p~-(a) Pt is a continuous map from [o, I] t0 ~'(H), 2) t ~ p+(a) and t ~ Qt p~(a) Pt are piecewise strongly C 1. Then ch*(Ht, Ft) e H~'(~ r is independent of t ~ [o, I]. [:o] [: :] Proof.- Let T t = , then TtFtT~-I = and Q, T,p,(a) TT,=[p+~a) o ]. Q, O-[(a) P, Then the result follows from lemma i and the invariance of the trace under similarity. [] Corollary 3. -- Let (H, Ft) be a family of p.summable Fredholm modules over d with the same underlying d-module H and such that t ~ F t is norm continuous. Then ch*(H, Ft) is independent of t e [o, r]. Proof. ~ Since the set of invertible operators in ..q'(H +, H-) is open, one can replace the homotopy F t by one such that t ~ Pt is piecewise linear and hence pieeewise norm differentiable. [] Let now A be a C*-algebra and .~r C A a dense *subalgebra which is stable under holomorphie functional calculus (cf. Appendix 3). By theorem 2, the value ofch*(H, F) only depends upon the homotopy class of (H, F). We thus get the following commu- tative diagram, Homotopy classes offinitelysummable} oh. HeV(d ) *Fredholm modules over d KK(A, 13) , Hom(Ko(A), Z) C I-Iom(K,(A), 13) where a) the left vertical arrow is given by proposition 4 of Appendix 3, b) the right vertical arrow is given by the pairing of Ko(~ ) with HeY(d) of part II, corollary 17 together with the isomorphism Ko(d ) ~ K0(A ) (Appendix 3, pro- position 2), c) the lower horizontal arrow is given by the pairing between KK(A, 13) and KK(13, A) = K0(A ). 9 66 ALAIN CONNES 6. Fredholm modules and unbounded operators Let d be an algebra over C. In this section we shall show how to construct p-summable Fredholm modules over ~ from unbounded operators D between d-modules. We shall then apply the construction to the Dirac operator on a manifold. We let H be a Z/2 graded Hilbert space which is an ~-module and D a densely defined closed operator in H such that x) cD= --De, 2) D is invertible with D -1 9 ~r 3) for any a e%t the closure of a--D-laD belongs to .LPV(H) (where p e [I, oo[ is fixed). [o ~9.[. Let H1 be the Z[~ graded ~t.module Proposition 1. -- a) Write D = D1 given by H + = H +, H~ = The Hilbert space H + with a~ = D~ -1 a Dl~, for ~ 9 DomD 1 a 9 Let Fi=[~ Io]. Then (H1, F~) isap"summableFredholmmodule over ~'. b) The following equality defines an element .c e Z~(~), n = 2m, n > p -- I z(a ~ ..., a n) : (2~i)"*m! Trace(D-l[D, a ~ ... D-'[D, an]), V a s ~'. c) Let (H~, F,) be constructed as in a) from H- and D,. Then .~ : x i-n: s where v~ is the character of (H~, F j). Proof. ~ a) For a 9 ~, let ~(a) be the operator in H + defined as the closure of D~ -lad 1. Since a--D -laD 9 pp, we see that ~(a)--a is bounded and belongs to ACP(H+). Since D1D~ -1 : I one has n(ab) = n(a) n(b) for a, b 9 Thus the module H~- is well defined and one has IF1, a] ~ .~~ , V a 9 d. b) Follows from c). [a-- D~-laD1 o ] c) One has D-I[D, a] ---- so that, for any a s 9 ~r o a -- D~-I a D2 x(a ~ ..., a") = TraceK+((a0 -- D~ -1 ao D1) ... (a, -- Di -1 a, D1) ) -- TraceH-((ao -- D~ -1 a0 D2) ... (a, -- D~ -1 a, D2)). Now the character v i of (Hi, F1) is given by = (2~i)" m! TraceH§ ~ -- Di -1 a ~ D1) ... (an -- Di -1 a ~ D1)). Similarly one has %(a ~ ..., a") = (~xi)"m! TraceE-((a 0 -- D~ -1 a 0 D2) ... (a, -- D~ -1 a~ D~)) [] 282 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Let us now assume that off is a *algebra and H a *module (i.e. (a*~, n) = (~, an>, V~,~H, aeoff). For any ~0~C~(off), let ?* be defined by o*(a ~ . . ., a n ) : 9(a~,...,a;) Va~eoff. One checks that ~*~ C~ and that (bq~)* : (-- r)" bq~*. Corollary 2. -- If D is selfadjoint, then .r = ((2ni)-m(mt) -1 -rl) + ((2ni)-"(m!) -a ~:1)*. Proof. -- One has D 2 = D~, thus TraceH-((a o -- D~ -t ao D~) ... (a, -- D~ -~ a, D2)) = TracerI-((a0 -- D~ -1 a0 D~) ... (a, -- D~ -i a, DE) ) = (Tracea-((a~ -- D1 a~, D~ -1) ... (4 -- I)1 a~ D~-~))) - = (Tracea,((Di -1 a~ D1 -- a~) ... (a~ -- D~ -1 a; D1)))- = -- ((2rci)-"(ml) -~ ~)* (a0, ..., a,). [] We shall define the character of a pair (H, D) satisfying i) 2) 3) as -r(a ~ ..., a") -- (2hi)" m! I Trace(r D-lID, a ~ ... D-~[D, a']). When D = F with F2= I we get the same formula as in section 2. Since = ~ ('rl -- ~:z) where vj is the character of a Fredholm module determined by (H, D), = S k the results of section 4 still hold for the character % i.e. "r,+z~ v, in H~+~(d) for any k -- i, 2 ... We let ch*(H, D) be the element of HeV(off) determined by any of the -r,. Corollary 3. -- Let d be a 9 algebra, H a Z/2 graded Hilbert space which is a 9 module over off, and D a (possibly unbounded) selfadjoint operator in H such that, ~) ~ D = -- De, ~) the domain of D is invariant by any a ~ off and [D, a] is bounded, 2') D -~ e .oq 'p. Then D satisfies conditions I) 2) 3) above and for any selfadjoint idempotent e ~ Mk(off), the operator D, = e(D | i) e /s selfadjoint in e(H | Ck). Its kernel is finite dimensional and invariant under ~, with Signature ,/Ker D, : ( [el, ch*(H, D) ). Proof. -- Since D -1 ~ Lap, one has D-lID, a] ~ .WP for all a s off, so that D satisfies I) 2) 3). For the rest of the proof we may assume that k = I. By ~), D, is densely defined in ell. It is selfadjoint by [57], since D -- (e De + (I -- e) D(I -- e)) is a bounded operator. Let f be the closure of D-le D, then f is a bounded operator with f-- e ~ Ae~ and f2 = f Let us show that the kernel ofj~ in eI-I is the same as the kernel of D,. Clearly ~ ~ Ker D, implies ~ s Kerfe. Conversely, let ~ ~ Kerfe. Let us show that e~ e Dora e De. Let ~, ~ Dora D, ~, ~ e~. Let ~, =f~, = D -x e D~,. One has fe~=o, hence ~,~o. Thus ~,--~,~DomD, ~,--~%~e~ and eD(~,~--~q,) ----eD~,--eD~,=o. This shows that e~DomeD and that eDe~=o. 283 68 ALAIN CONNES Now, as f--e e Lot, fe defines a Fredholm operator from eH to fH, and its kernel is finite dimensional. The operator e commutes with fe and one has, Signature (~/Ker D,) = dim Ker(fe), m -- dim Ker(fe),H-. Let us show that the codimension of the range offe infH + is equal to dim Ker(fe),a-. In fact both are equal to the codimension of the range of e De D -1 H- in ell-. Thus, Signature (~/Ker D,) = Index J? :eH + ->fH +. With the notation of proposition I the right side of the above equality is the index of (F~+), so that, by theorem 3. i, it is equal to .-., The conclusion follows from corollary 5 combined with the equality e = e*. [] In corollary 3 the condition " D is invertible " is still unnatural, we shall now show how to replace it by 7') (I + D~) -1 e S ~ Let H be a Z[2 graded Hilbert space which is a module over the algebra .~r Let D be a (possibly unbounded) selfadjoint operator in H verifying 0t) and [~) of Corollary 3- To make D invertible we shall form its cup product (cf. [6]) with the following simple Fredholm module (He, F0) over the algebra C. We let H e be the Z/2 graded Hilbert space H~=G, we let G act on the left in Heby X-+[~ :]eoL~a(H0), andwe let Proposition 4. -- a) Let H ---- H ~3 H c be the graded tensor product of H by H c viewed as an d | E = sr left module. For any m 4= o, m e R, the operator D,~ = D ~ i + mi ~ F c is an invertible selfadjoint operator in H which satisfies ~) ~) y) if D satisfies o~) ~) 7'). b) Corollary 3 still holds under this weaker hypothesis. c) ch*(H, Din)= [zm] e HOV(d) is independent of m (where % is the character of (H, D~,)). Proof. -- a) One has D~ = (D ~ + m 2) ~ i, so that D,~ is invertible. Moreover [D~tl = (D2+ m2)-m~ i e.Sf p. Since conditions ~) ~) are obviously satisfied by D,, we get a). b) Let ec=[: :]e~oqa(HG). Forany e=eZ=e*e~r onehas (e (~ %) D,~(e @ ec) = e De Q ec, thus Signature (e/Ker e De) = Signature (r | ec/Ker(e (~ ee) D,,(e G e~)) = <[e], ch*(H, D,,)> by corollary 3. c) Follows from Corollary 9 and Lemma 5. I. [] 284 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY The above construction of the operator D,, from the operator D associates to the Dirac operator in R 3 the Dirac Hamiltonian with mass m. Let now V be a compact even dimensional Spin manifold. Let g be a Riemannian metric on V, S the bundle of complex spinors and D the Dirac operator in L*(V, S) = H. By construction H is a Z/2 graded Hilbert space, with H  ---- L2(V, S +) and is a module over ~r C~176 One has: ~) ~D------D~; 9) the domain of D is invariant under any f e d and [D,f] is bounded; y') (I 4-D2) -les for any p>dimV. Thus proposition 4 applies and combined with proposition I b) it yields for each m e R, m # o an element %, of z~mv(c~(v)), the character of (H, Din). Theorem 5. -- a) With the above notation, vm(f ~ ...,f") is convergent, when m -+ 0% for any fo, . . .,f" e C~r b) The limit "c(f ~ ...,f") is given by .r ( fO, . . .,f") = f fO dfl ^ ... ^ df" 4- (S 2~1) (fo, ...,f") 4- (S co~) (fo, ...,f") 4- ... 4- S,/2 cos(,/4)(f o, ...,J") where S is the canonical operation Z~-+ Z~ +2 (el. Part 11I), o~j is the differential form Aj(px, ..., P~) describing the component of degree 4J of the .~ genus of V in terms of the curvature matrix of the metric g, and is considered as an element of Z~-4J(d) by the formula ,~(fo, fl, ...,f"-4J) = fvfO dfl A ... h df "-4j ^ co s . Here the manifold V is oriented by its Spin structure. This theorem will be proven in part III using the technique introduced by E. Getzler in [28]. 7- The odd dimensional case For nuclear C*-algebras A, there are two equivalent descriptions of the K-homology KI(A). The first, due to Brown, Douglas and Fillmore ([ii]) classifies extensions of A by the algebra o~f of compact operators, i.e. exact sequences, of C*-algebras and homomorphisms o -+ ~f -+ 8 -+ A -+ o. The second, due to Kasparov classifies Fredholm modules over the Z/2 graded C*-algebra A | C1 where C 1 is the following Z/2 graded Clifford algebra over C, C + ={7,1,),eC}, i the unit of C 1 C~ =(X~,XeC}, ~=I. 285 ALAIN CONNES 7o In the work of Helton and Howe on operators with trace class commutators and in the further work [23] [I2] [20], differential geometric invariants on V are assigned to an exact sequence of the form, o ---> -oq~ ---> a -+ C~(V) ~ o. In this section we shall clarify the link of these invariants with our Chern character. We show that, given a trivially Z]2 graded algebra d over C, :) ap-summable Fredholm module (H, F) over d | C1 yields an exact sequence, o ~.oq ~p/2 -+8 -+d ~o; 2) the cohomology class I-r] eH~"-l(d), m GN, m>p/2 of the character of the above Fredholm module only depends upon the associated exact sequence, and can be defined directly (without (H, F)); 3) when .~ = C=(V), the fundamental trace form ? of Helton and Howe ([3:]) is obtained from the character -r by complete antisymmetrisation: ? = Zr -r ~ Hence, using the results of part II (lemma 45 a) and theorem 46) we see that ? is the image of-r under the canonical map :: H~-I(C~(V)) -. H'-I(C~(V), C~(V)*). Since the kernel of I is the direct sum of the de Rham homology groups H2,,_a(V , C) @ H,,,_5(V, C) | @ Hi(V, C), we see that some information is lost in the process when the latter group is not trivial. This fits with the results of [3:] and [25] where the fundamental trace form is used either in low dimensions or for spheres. Our formalism thus gives an explicit formula for the lower homology classes of Helton and Howe ([3:]). Let us begin with :). We letH tbe theZ/2 gradedHilbert space H + =C, H~" =C. We let C 1 act inHlby, Lemma 1. -- Let d be a trivially Z/~ graded algebra, (K, P) a pair, where K is a Hilbert space in which d acts (by bounded operators), while P e .o~(K) satisfies the conditions a) [P,b] e.L~'P(K), Vbed, b) P~= i. (H, F) is a p-summable Fredholm module over the Z]2 graded algebra d | C 1. Proof. -- By construction H is a Z/2 graded d | C1 module. The operator F satisfies eF=--Fe, F 2= :. Finally for any x=aQI -q-b| x the graded commutator [F, x] is given by [--[P,b] --[P,a]] iEF, x] = [P, a] [ P, b] e .~P(H). [] 286 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 7I Lemma 2. -- Let % be the n-dimensional character of (H, F) for n >_ p -- I. Then, a) if n is even one has %=o; b) if n is odd, one has % = ~ | Y, where Y is the graded trace on Gt, Y(Z + ~t~) = ~t V X + ~t~ e Gx, and where n--1 , 0 %(a,...,a") = (--I) z c, Trace(P[P,a ~ [P,a x] ... [P,a"]), Va ie~r c) one has x~ e Z~.(..~r Proof. -- One has by definition (cf. remark i .6) %(x ~ ..., x") = (-- I) ~ c, Tr,(x ~ dx t ... dx"), dx j = i[F, x~], for x ~ ...,x"e~r174 x ~ homogeneous, q=~deg(x~+l). Replacing .~r by ~qr we may assume that ~1 is unital and that its unit acts as the identity in K. We shorten the notation and replace x| by 0tin ~r174 It acts in H by the matrix i- -i o~= [: :1. One has x~=ax for xe~r174 t and (r162 this shows i. J that when n is even, any to ~ f~" satisfies IXto ~ tool,, As e~ = -- ~r this shows that for n even, n~p-- I, one has Tr,(to) =o Vt0e~". By remark I .6, %(x ~ . .., x") = o for x i ~ d| Gx, x i homogeneous, Let n be odd. Z Ox ~ = o (mod ~). for a i ~ ,~, ei e { o, I }, Since Foc=--~F, one has d~=o, and hence, '~r = I (mod 2), %(a ~ 0r ..., a" o~',) = %(0ca ~ a a, ..., a"). Now ,or oo. = al = t ."l [ o thus, with --I O ~ to = o~a ~ da 1 ... da n, _x F(Feto -- toFr = _I iF~e da ~ ... da" 2 2 2 (- I) ~ (PIP, a ~ ... [P, a"]) | This shows that .. = ~,(a, ..., a") V(~'0 ~" ... ~). Tn(a 0 0~o, ., an o~n) , o It follows that Finally, the character % satisfies conditions a') b') of remark i. 6 y). < ~ z~(d). [] 287 ALAIN CONNES 7~ To any pair (K, P) verifying the conditions a) b) of lemma r, we have thus associated, for any odd n>p--i, the (n + 1)-linear functional -r~ e Z~(~r We shall now show that the class of z~ in H~(~r only depends on the extension of ~1 by .oq a~/z associated to (K, P) as follows, Iq-P Proposition 3. -- Let d, K, P be as above andput E 0 -- -- 9 .~e(K), E = Range of E0, p(b) =E 0bE 0 9 for any b 9 a) One has p(ab) -- p(a) o(b) 9 .LaP/2(E) for all a, b 9 d. b) Let g = p(zt)+ LaP/Z(E) C-qa(E), and d~" be the quotient of ~t by the ideal d" = {a 9 ~1, p(a) 9 .~~ then one has a natural exact sequence, o + sePn(E) -+ ~" -+ d' -+ o. Proof. -- a) Since p2= I, one has F_~=E 0. Hence E o abE o -- E o aE 0 bE o = -- E0[Eo, a] [Eo, b] 9 Na'/2(K). b) By a), g is a subalgebra of .la(E). One has s C ~' and p yields an isomorphism p' of ,~r with g'/.lap~. [] Let J=.i~ 'p/zC g. Then for any integer m>p/2 we have jmc.9,1, so that the trace defines a linear functional -~ on J" such that x(ab) = x(ba) for a 9 b 9 k -k- q_> m. Moreover p : ~1 -+ g is muhiplicative modulo J. We shall show in this generality how to get an element qh,,-1 of Z~'-x(~r and relate it to -r" in the above situation. Proposition 4. -- Let Y~ be an algebra, J c Z a two-sided ideal, m e N, and .r a linear functional on J'~ such that "r(ab) = v(ba) for a e J k, b 9 kq-q=m. Let p : d ~ Y, be a linear map which is multiplicative modulo J. a) Let ~ be the 2m-linear functional on d given by ~(a ~ ..., a ~"-1) = ~(~o ~ ... ~m-~) - ~(~ ~3 . . . ~-0, where ,j = p(a ~ a s+l) -- p(a ~) p(a~+l), j ---- o, I, ..., ~m -- I. Then ~ 9 Z~'~-l(d). b) Let p' : d --+ Z satisfy the same conditions as p, with p(a) -- p'(a) 9 for a 9 ~r then, with obvious notation, one has ~" -- r 9 B~m-~(~r c) Let (K, P) satisfy conditions a) b) of [emma I and "~" be given by lemma 2, for n=2m--I, m_>p. Let 2~---- g, J=LfP/2(E), and p be as in proposition 3. Then the corresponding ~ 9 z~m-l(d) satisfies v = - (2 -("+~) cZ ~) ~'. 288 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Proof. -- a) One has, by construction, ~(2,...,a 2m-l,a ~ =--@(a ~ Va jEd. With the notations of a), let ~+ = X(r ... r One has O(aJaJ-blaJ+2) __ ~(aJa j§ ~(a j+2) -- (o(aJa j+l a j+2) _ p(a ~) o(aJ+laJ+2)) = o(a j) p(a j+l a j+2) _ o(a j a j+l) o(a j+~) _-- o(a j) ej+l -- r p(aJ+2). Using this equality, we get b?+(a ~ a TM ) ?+(a "+1 a ~ 2, 9 9 *, -- o 9 a t~) = ~(p(a ~ sl ~... ~. - So... s._l p(.+l)). Similarly, with @- -- ?+ -- ~, we have b?-(a ~ . . ., a "+l) - ?-(a ~ a 1, a s, . . ., a "+1) = - ~(p(a 1) ~ ~... s.+~ - s~ ~.... s. pCa~ Thus bg(a ~ ..., a "+1) = cp+(a "+l a ~ ..., a") -- "r(~ 0 ... s,_ l p(a"+l)) 9 .., a n+l) ...... -- (?-(a ~ 2, + "r(p(2) ~ s.+l) = x(A~ s._l) where A : p(a "+1 a ~ a 1) -- p(a "+1 a ~ p(a 1) -- p(a "+1) r (P( a"+l a~ - p(a "+l) p(a ~ a~)) + s.+, p(a ~) = o. Therefore ~p e Z~(.~). b) Let L=p'--O; then Lis alinear map from a~ to J. With Pt----o+tL it is enough to show that the cocycle cpt associated to Pt, satisfies ~ ~p~ = bd h for a continuous family +t r C~-1(~) 9 Clearly it is enough to do it for t----o. Letting ---- ?t we have (J,) t=0' ,'(a ~ ..., a") =-r(A-- B), where A=g~...s,_l+SoV~S 4...s._l+ ... +~o~... r i t t B -- sl ~s 9 9 9 s, + s i s s ... s, + ... + S I S8 . 9 . ~n and cj = L(a ~ a j +1) _ 0(a j) L(a j +1) _ L(a ~) 9(a j +1). Let +o(a ~ "-1) =x(L(a ~ l~3... ~,-2) and let %(a ~ ..., a n-1 ) = d/o(aJ , a j +l, ..., aJ-'). Using the same equality as in a) we obtain b+~(a ~ ..., a") = "r((p(a ~ s 1 sa-.. L(a2u+l) ." s,-1) -- (~0 ... ~z~-2 P(a~) L(a2~+I) ..- s,-1) + (So... ~-~ L(a2~a2~+a) -.. r -- (r a~-2L(a~) P(a2~+1) "" s.-1) + (So ... ~.~_, L(a ~) ~+~ ... s._~ ~(a")) -- (p(a" a ~ 2) -- p(a" a ~ 0(2)) ~... s,~_2 L(a ~) ~a~+1 ..- s._,.)). 10 74 ALAIN CONNES The last two terms cancel the first two in bhbzk_l(a ~ ..., a") = "r((p(a" a ~ a 1) -- p(a") p(a ~ at)) ~... ~k-2 L(a~) -.. *,,-z) - (pCa ~) ~... L(a ~) ... ~,) - (~,... ~_~ ~_~ 9 .. ~,) -- (r L(a2k-1) ... r P(a~ 9 Thus we get, for k= 1, 2, . . ., m -- I, b(q,k-~ + +,k) ( a~ ..., a") = x((o(a ~ ,1 ... ~k-t L(a2*+t) ... r -- (O(a~ r ~k-3L(a*~-l) .-. *,-~) + (~o... *~k-24k''' e,-t) - (~ ... ~_~ ~,_~ ... ~,)). As bd~o(a ~ ..., a") = "r((o(a ~ L(a 1) ~... *,-1) -- (P(a~ ~1 ... L(a")) + (~ ~... ~.-1) - (,, ... ~.-~ ~')), one obtains n--1 y. %=r $=0 c) Let o(a)= EoaE o ~.oq'(E), where Eo---. One has o(a ~ aa) -- o(a ~ p(a 1) = -- Eo[Eo, a ~ [Eo, a 1] = -- I Eo[p ' aO ] [p, at]. Therefore, since E o commutes with [P, a ~ [P, al], m--1 rl (p(a~ a ~k+l) - p(a ~) p(a~+l)) = (_ 4)-'~ ~ [1', ~q. k=o S=o Thus we obtain q~( a~ -.-, a~-l) = Trace(eo e2 ... ~m--2) -- Trace(c1% .. 9 ~m-t) = (-- 4) -m Trace(Eo( h [P, a s] -- fi [P, a s + t])). j=o j=o Similarly, if we let Eo= t--Eo, E'=Range of Eo, p'(a) :EoaEo~.o~a(E ') for a e ~r we have, with obvious notation, 9'(a ~ ..., a ~-1) = (-- 4) -m Trace(E~( fi [P, a j] -- II [P, aS+l])). S =o S=o One has P=2E o-I =E o-Eo, thus ,P - ,P' = (-4)-" (- ~)"-~ (G.)-~ "~..' Since F_~+E 6= I, one has (,p + ,p') (a~ a-) = (- 4)-" Trace( n [1', aq -- I] [P,,~s+~]) = o. S=o S=o Thus ~=--2- ~-tc~-1%. [] 290 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY The construction of the character of an extension of d by -~P/~ can be summarized as follows: Theorem 5. -- a) Let E be a Hilbert space, p a linear map of ~t in .o~~ which is multi- plicative modulo .oq~ then the following functional %, n = 2m- I, re>p[2 belongs to %(a ~ ..., a") : -- 2 "+2 c, Trace((~ 0 ~... ~,-1) -- (s1r "" r where ~j = p(a ~ a j+t) -- p(a j) p(aJ+l). b) The class of % in H~(~r depends only on the quotient homomorphism c) The class of % in H~(~r is unaffected by a homotopy Ot such that i) [1 p,(ab) -- p,(a) p,(b)[[p~ is bounded on [o, I] for any a, b ~ ~; 2) for ae~r ~ eE, the map t-+p~(a) ~ is C 1. d) The index map Kl(d ) ---> Z is given by Index~(u) = ([u], %) V u e OL(d). e) One has SIr,] = [%+~] in H[+~(d). Proof. -- a) and b) follow from proposition 4. c) Follows from the proof of proposition 4 b). d) Follows from the equality Index ~(u) = Trace(i -- ~'(u- 1) ~(u))" -- Trace(I -- ~(u) ~'(u- 1)),, (of. Appendix I) and the definition of the pairing between Kl(d ) and I-IX(d ) (Part II, proposition I5). e) Follows from the following algebraic lemma, whose proof is left as an exercise to the reader. Lemma 6. -- With the notation of proposition 4 one has (2z-~ S)(~,.-1)= 4 (m + ~) ~2,.+1 We shall thus define the Chern character of the given extension as the element of H~d(~) = lim(H~"-l(d), S) given by any of the characters %, n odd. Let us now clarify the relation between v, and the fundamental trace form of Helton and Howe ([3I]). We assume now that ~r is commutative. The fundamental trace form is defined, under the hypothesis of theorem 5, by the equality, T(a ~ ..., a") : Trace(Zr p(a ~176 ... p(a~ where ~ runs through the group ~,+1 of all permutations of {o, i, ..., n} and c(e) denotes its signature. 291 76 ALAIN CONNES Proposition 7. -- Let d be a commutative algebra, p and E be as in theorem 5, p: ~r -+ ~e(~.)/~e,~(v.). a) FOr p = I trig fundamental trace form T(a", a x) is equal to ~-~ q(a ~ al). b) For p> I, one has (-- I) m (n -q- I) ]~((~) Tn(a0 aO(1 ) .... , aO(,))" T(a ~ ..., a") = 9 "+sc,, Proof. -- a) One has, by definition, "rl(a~ , a l) = (Trace(p(a ~ a 1) -- p(a ~ p(al)) -- Trace(p(d a ~ -- p(a 1) p(a~ As a la ~ ~ x one gets the result. b) For any n + i linear functional d/ on d, let Oqb be given by 0+(a ~ ..., a") = Z ~(~) +(a n(~ ..., an(")). ~n+l Since v, satisfies ~,(a 1, ..., a", a ~ = -- v,(a ~ ..., a"), one has Z r "r,(a ~ a "(1), ..., a a(")) _ ~ ~ ~(~) ~,(r ..., a-C-)) _ ~ K. n -~- I n~@n+X n + I Let us write T n = T: -- "7n, where, with the notation of theorem 5 a), + 0 2n+2 x, (a, ..., a") = -- c, Trace(% e~... r One has x[(a ~ .. .,a") = x, + (a, 1 ..., a~ and hence 0r, = 0~ + -- 0r; = 20~ +. As in the proof of a) one has p(a 2k a 2~+I) _ p(a ~) p(a 2k+I) _ (p(a 2k+I a ~) -- o(a 9"k+I) p(a~)) = [v(a~+~), v(a~)]. Let ~k be the transposition between 2k and 2k + I ; then m--1 + ).~ ( 1-I (I -- 0~2,k) ) T n = (-- I (-- 2n+2Cn) k=0 � Trace([o(a~ o(a~)] ... [p(a"-~), p(a")]). Since 0(I -- ~-a~) = 20, we get 0r + = ~-~ 0q~, where +(a ~ .... , a") = (-- I) ~ (-- e+~c,) � Trace([p(a~ o(al)] ... [p(a"-1), p(a")]). The result now follows easily. [] 292 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 8. Transversally elliptic operators for foHations Let (V, F) be a compact manifold with a smooth foliation F, given as an integrable subbundle F of TV. We shall show that any differential or pseudodifferential ope- rator D on V, which is transversally elliptic with respect to F yields a finitely summable Fredholm module over the convolution algebra d = C~(Graph(V, F)) ([i5] , [i6]). We deal here with the obvious notion of transversally elliptic operator; a more general notion will be handled in Part VI. Let E q- be complex vector bundles on V which are equivariant for the action of Graph(V, F) on V. This means that for any 7 ~ Graph(V, F), s(7) = x, r(7 ) =y, one is given a linear map 4 -+ 74 of F_~ to E~ with the obvious smoothness and compa- tibility conditions. Definition 1. -- Let D be a pseudodifferential operator of order n from E + to E-. Then D is transversalty elliptic with respect to F if and only if its principal symbol is a) invariant under holonomy, and b) invertible for 4  F, 4 9 o. More explicitly, a) means that for any 7 e Graph(V, F), 7:x~y, one has a((dT) t 4) = Yg(4) 7 -1, V 4 e F~, where d7 is the differential of the holonomy, a linear map from TflF, to Ty[F u. Let E = E + 9 E-, and let us show that each of the usual Sobolev spaces W'(V, E) of sections of E is a module over ~r = C~(Graph(V, F)). Let G = Graph(V, F). Lemma 2. -- For any s e R, the equality (k (x) fo~k(7)f(y), k e CT(G), fe W"(V, E), y = s(y), *f) defines a representation of C~(G) in W'(V, E). Before we prove it, we have to explain the notation. Elements of C~~ are not quite functions but sections of the line bundle s*(G), where s : G -+ V is the source map and ~ the line bundle (trivial on V) of i-densities in the leaf direction. This gives meaning to the integral fo x k(7)f(y ) for scalar functions f. For sections of E one has to replace f(y) e E u by 7f(Y) e E, and then the integral is performed in E,. When dealing with Sobolev spaces which are not spaces of functions (i.e. s < o) the statement means that k. extends by continuity to W'. Proof. -- The definition of the Sobolev spaces Ws(V, E) is invariant under diffeo- morphism. More precisely given open sets V1, V 2 C V and functions ~i e C~~ with (support ~) C V~, any partial diffeomorphism tF : V 1 -~ V, covered by a bundle map defines by the formula T4 = q~x tF*(~, 4) a bounded operator in each of the W s. Hence (as in [I5] ) to show that k, is bounded in W ~ one may assume that k ~ C~(Gw) C C~(G) where W is a small open set in V (i.e. the foliation F restricted 293 ALAIN CONNES to W is trivial) and G w is the graph of the restriction of F to W. ]'hen one can write k 9 as an integral of operators of translation along the plaques of W and the statement follows (say by taking the local Sobolev norms to be translation invariant). [] Note that unless F= = T, for all x, the operators k. are not smoothing; they are only smoothing in the leaf direction. Lemma 3. -- Let D be a transversaUy elliptic pseudo,differential operator from E + to E- (both bundles are holonomy equivariant and the transverse symbol of D is holonomy invariant). Let Q.be any (x) pseudo-differential operator on V from E- to E + with order -- q (with q = order D) and transverse symbol aD 1. Let H + = W~ E+), H- -~ W~ E-) and F = [ ~ ~o ] " Then ( for any s 9 It) the pair (H, F) is a pre-Fredholm module over .~ = C~~ It is p-summable for any p > Codim F = dim V -- dim F. Proof. -- Let us first show that k(F 2 -- I) and (F ~ -- I) k belong to .~v(W') for anys, andp>na=CodimF. (We take n=dimV, n x -dimF, n2=-CodimF). Both DO..-- x and QD -- I are pseudo-differential operators of order o on V with vanishing transversal symbol, and we shall show that if S is such an operator, then kS and Sk are in .~v(W ~ for any k e C~~ It is enough, as in the proof of lemma 2, to prove it for k e C~~ where W is a small open set in V. This shows that the problem is local, and hence we may as well take for (V, F) the torus T" = T "2 � T"' (T = R/Z) with the foliation whose leaves are the T"' � {x}, x 9 T"'. Let a be the total symbol of S; then S is of the form (2=)-" f e '<~ ~' 6(x, ~) f(x -- s) Z(s) ds d~, (sf) = where s varies in R" (which acts by translations on T"), ~ in R. = (R")*, and X e C~~ ") is identically i near o. One has G=T"'� '~� and k 9 acts on functions by (,,) fk(xDyx, x,)f(yx, x~) dy x where x = (xx, x,) 9 T". (kf) To show that Sk 9 .oq~v(W~ it is enough to show that, given s, the .~V-norm of (I + A)~ ~- A) -'/2, k~,(x) = expi2~(~,x), does not grow faster than a poly- nomial in 0r = (0it, ~t, ~z) e Z"' +"' 4-,, Also since any k= as an operator is the product of a multiplication operator by a k=,, ~' of the form (-- ~, ~, o), it is enough to estimate [1(i + A)mSk=,(I + A)-'/2IIv, and as k=, commutes with A one is reduced to the case s = o. Finally it is enough to estimate Z ]l S~ k~, ]]~, where S~ has total symbol ~ independent of x: ~(~) = f e ~<~'=> a(x, ~) dx, ~ 9 Z". (x) For instance take Q, with symbol o(~) = (i --Z) ~ where Z ~ C~~ is equal to t on V C F "l" and p : T ~ --+ F is a linear projection. 294 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Now both Ss and k s, are diagonal in the basis e~,,,, (e~,,,(x) = exp i2~<0t", x>, 0~" ~ Z"). The operator Ss multiplies e,,, by (%. ~)(2~,t"), where ~ is the Fourier transform of X), andks, (~'= (--[~,~,o)) multiplies %, by o if at' 1' #[~ and by x if ~'1' = [~" Thus (ll lip) = * p. This is finite for p > n,, since by hypothesis one has for ~ (and hence a s 9 ~) an inequality I (x, <c(, + l[ lll)(i + [l xll + l[ ,ll) --a Since the same inequality holds for the partial derivatives with respect to x, one gets that the Cs's (for the o's) are of rapid decay in a, thus the conclusion follows. Let us check that iF, k] z .~P, p > n~, for any k e C2(G ). If P : W' -,,- W '-~ is a pseudo-differential operator of order k and its principal symbol vanishes on F x, we have kP and Pk in .oq~ ', W '-k) for any p > nz. This shows that to prove that if the principal symbol of P is holonomy invariant one has [P, k] e -LP v, one can assume that k ~ C2(Gw), W a small open set. One is then back to the above case where V = T"' x T"'. Applying again the above result one can now assume that P is exactly invariant under the action of the compact group T'. Now the action of k s C 2(G) in W' is of the form kf = f~, k, Ut(f) dt, where U t is the translation by t e T"' and k t is the multiplication by a smooth function of xeV (and teT"'). Thus [P,k] =f[P, ktlU tdt =fPtUtdt where P, is pseudodifferential with order --x. Using Fourier expansion one checks that any keC~(G) is of the form k=k 1,k2, thus [P,k] =[P,k dk~+k1[P ,k2] and both terms are in .L~ 'v by the above result, rq Remark 4. -- In the special case when the foliation (V, F) comes from a locally free action of a Lie group H (not necessarily compact), the graph of (V, F) is equal to V � H. The convolution algebra C~(H) becomes a subalgebra of C~~ (by composing f z G~(H) with the proper projection V � H -+ H). Thus given a transversally elliptic operator D for (V, F) one can restrict its n-dimensional character (n > dim V -- dim F) to G~~ If both D and a parametrix Q. are exactly H-invariant, then one can compute r~ from the distribution character Z of D. The easy computation gives this restriction "~, 9 = s'z, (. = 2m). The central distribution X is defined as in [2] by the equality x(f) --= Trace(action of fin Ker D) -- Trace(action of fin Ker D t) (eft [2], Remark, p. 17). In the simplest examples with H non compact, the distribution character Z of D is not invariant under homotopy. However, by the above results, its class in H'(C~'(H)) is stable. 295 8o ALAIN CONNES 9. Fredholm modules over the convolution algebra of a Lie group Given a Lie group G, we let ~ = C~(G) be the convolution algebra of smooth functions with compact support. The Mi~enko extension ([5o]) gives a natural construction of Fredholm modules over ~r = C~(G) parametrized by a representation 7: of the maximal compact sub- group K of G. In this section we shall show in the two examples G = 113 and G = SL(2, 11) that the corresponding modules are p-summable and go a good way in the computation of their Chern characters. For SL(2, 11) we shall find a precise link with the surface of triangles in hyperbolic geometry which is a standard 2-cocycle in the group cohomology with coefficients in C. This link appears very natural if one has the example of G = 113 in mind. The method that we use goes over to semi-simple real Lie groups of real rank one. For such groups, in the corresponding symmetric spaces G/K the angle under which one sees a given compact set B C G/K from a distance d tends to o as e-~d when d-+ oo. Thus the p.summability follows as in lemma I below using Russo's theorem ([63], p. 57). For groups of higher rank the problem of constructing natural p-summable Fredholm modules is open. The case G = 11s Let G ---- 112. We define a Fredholm module over the convolution algebra d ---- C~(G) as follows: H + ---- L2(R2), H- ---- L2(R 2) (with the action of d by left [ o o0 translation), and F--= where the operator D:H + -+ H- is the multipli- cation by the complex valued function ~(z) = z/Izl v z ~11, = c, z, o. but this is unimportant since only its class The function ~ is not defined at z = o in Lc~ 2) matters to define D. Lemma 1. -- The pair (H +, F) is a Fredholm module over d = C~~ It is p-summable for any p > 2. Proof. -- One has F 2 = I by construction. For fe C~(R 2) and ~ e H +, one has ~) (s) ~(s) ff(t) ~(s - t) dt -- ff(t) ~(s -- t) ~(s -- t) dt ([D,f] ds'. = ff(s -- s')(r -- ~(s')) ~(s') Thus it is the integral operator with kernel k(s, s') =f(s -- s') (~(s) -- ~(s')). Since fhas compact support one has k(s,s') =o if d(s,s')>C for some C<o% where d is the Euclidean distance. Also for [ s] large and d(s, s') ~ C, the term ~(s) -- ~(s') 296 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 8i -- P one has is of the order of I[] s ]. This shows that for any p > 2, with q p~, -'-'j(J[k(s,s')lqds)mds'<oo (and similarly for the kernel k* = k(s', s) ). So Russo's theorem ([63] , p. 57) gives the conclusion. [] We shall now compute the character x 2 of (H, F). By a straightforward compu- tation, as in section I, one gets "r2(f~ f2 ) = ~i~r f~o + ~" + ~'=o f~176 fi(si) f~(s~) c(s~ sl' s~) dsi ds~" where the function c(s ~ S 1, SO'), s~eR 2 is given by ~ s) s - s ~ s - sO - sq as c( o, sl, = f with ~(s ~ s) = I -- ~(s) -~ q~(s --s~ To get this, one just has to write the trace of ds. an integral operator as the integral fk(s, s) We shall compute c(s ~ s i, s 2) ; we try to prove the next lemma in such a way that the proof goes over to the case of hyperbolic geometry. Lemma 2. -- One has c(s ~ s l, s 2) = 2i=(s i A s2). Proof. -- Let us first simplify the integrand [~(s ~ s) [~(s i, s -- s ~ [3(s ~', s -- s o -- si). For that we consider the Euclidean triangle with vertices o, s ~ s o + s 1 (remember that s o + s i + s 2 -- o). Then $(s) -1 ~0(s -- s ~ = e ~ where 0~ = -~ (o, s, s ~ is the angle (between --~ and =) obtained by looking at the edge (o,s o ) from s. Thus ~(s ~ = i --e ~. Similarly ~(s 1,s-s o ) = i--e ~ where [~=-~(o,s--s o ,s 1) =~(s o ,s,s ~ 1) and [3(s 2,s-s ~ x) = I --e iv where y= <~ (o,s--s ~ 1,-s ~ 1) =<)~(s ~ 1,s;o). Since ~+} +y=o we get (I -- e i~) (I -- e ~) (I -- e iv) ---- -- e i~- e i~- e ~v + e -i~ + e -i~ + e -iv = -- 2i(sin o~ + sin [5 + sin 7)" Define ds. S(A, B, C) = f (sin <~ AsB + sin <~ BsC + sin <~ CsA) Then c(s ~ s 1, s ~) = -- 2iS(A, B, C), where A=o, B=s o , C=s~ i form the triangle A, B, C. The integrand is 0(Is] -3) for large s so that the integral is well defined. To prove that S(A,B, C) is proportional to the Euclidean area of the triangle (A, B, C), the main point is to show that S is additive for triangles Ti, T 2 such that T 1 u T, is again a triangle. Let us prove this. Let , be the symmetry around the straight line which contains three vertices, say B1, C~ ---- B e and C, and let A1 ---- As be the only vertex outside this line. Writing the integral defining S as a limit of integrals 11 8~ ALAIN CONNES over a-invariant subsets eliminates the terms of the form sin-~ B 1 sC1, sin -~ B s sC s and sin <~c B 1 sCs. Moreover one has sin -~ C1 sA1 = -- sin %t A s sB2. The equality S(T1 w Ts) = S(T1) A- S(Ts) is now clear. A 1 ~ A s ~t The next point is that S(A, B, C) ~ o if the triangle ABC is positively oriented (i.e. if the orientation ABC fits with the natural orientation of R s ---- C). To see that, consider the disk I) R with center A and radius R. Then D R is invariant under the symmetries around both sides AC and AB so that the integral expressing S reduces to f sin -~g BsC. Let cr be the symmetry around BC, then the complement of the line BC in DR has (for R large) two components D' and D" such that a(D') C D". As on D"\~(D') one has sin <~c BsC > o, one gets the answer. DR 298 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 83 Now we can define the functional S on all subsets C of R ~ which are finite unions of closed triangles. The collection ~ of such subsets is a compact class in the sense of probability theory and thus we see that S defines a translation invariant Radon measure on R 2. Thus S is proportional to the area, and the constant of proportionality is easy to check. Corollary 3. -- The 2-dimensional character ~:2 of (H, F) /s v~(fo, fl, ff) = fro dfl ^ dj~= where fo,fl,fz ~ C~(R ~) have Fourier transforms fi. The ease G = SL(2, R) Let G = SL(2, R). In fact we shall use the realization of G as SU(I, I) i.e. 2 by 2 complex matrices g= [~ ~], 10c12--1~]~=I. Asmaximal compact subgroup K, we choose K = e_~0 , 0 e R/2~Z , and we identify G/K with the unit disk U ~z+~ for zeU, geG. Tothe in the complex plane C, on which G acts by gz -- ~z + ~ [[e~~ ~ ])=e~~ corresponds an induced line bundle En character X, of K given by Z~[ o e -~~ on U whose sections correspond canonically to functions ~ on G such that ~(gk) = x~(k) -l~(g) for k eK, g eG. The tangent bundle of U considered as a complex curve corresponds to Z~ where x2(k) = e i2~ which is the isotropy representation of K. At each point p e U, p + o, there is a unit tangent vector ~(p) e Tp(U), i.e. the one-dimensional complex tangent space T = E2, such that o belongs to the half-line starting at p in the direction of q~(p). (We use the unique G-invariant metric of cur- vature -- I: 2(1 --]Z[2) -I Idz] as a Riemannian metric on U.) 299 84 ALAIN CONNES Being a section of E~ (on U/{o}), q~ can be considered as a function on G; given geG, g=[~ ~], gCK one gets q)(g) -- I I It has a simple interpretation in terms of the KA + K decomposition. One has ?' / teR and k,k'eK, aeA +. ~(kak') = Z_2(k'), where A = sh t ch ' ' For each n ~ Z we let D. be the operator of multiplication by ~0 from L2(U, E.) to L~(U, E.+~) (1). Lemma 4. -- a) For each n e Z, the pair (H., F.) is a Fredhotm module over d = C~ ~ (G), where H + = L~(U, E.), H~- = L~(U, E.+~), [; b) The direct sum (| H,, | Fn) is also a FredhoIm module and is p-summable for any p > i, as well as all (H,, F,). Proof. -- The algebra ~ acts by left convolution in H,. One has by construction F~ = i, F2= I (where F = | It is clearly enough to prove b). Now H + = | H + = L2(G) where ~ acts by the left regular representation; also H- = LZ(G), and the operator D = | D, is given simply by the multiplication by the function q~(g). As in the case of R z we get 4) (g) ([D,f] f f(gg'-~) ~(g') dg' -- f f(gg'-~) 9(g') ~(g') dg' = fk(g, g') ~(g') dg' where k(g, g') = (~(g) -- ~(g') ) f(gg'-~). We want to show that (H, F) is 2-summable, i.e. that f l k(g, g') lg dg dg' < or. Sincefhas compact support, it is enough to show (with d a left invariant metric on G) that f l v(g-1) _ ~(g,-1)12 dg < ov where (g, g ) < C < oo d ' (a) In this special case G = SL(2, R) we rely on the natural conformal structure of U = G]K, but the true nature of the construction is to take the Clifford multiplication by c? (cf. [5o]) for which one just needs an invariant spin c structure on G]K. 300 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 85 (more precisely f M(g) 2 dg < oo where M(g) = sup {[ q0(g -1) -- ~(g,-1)[, d(g, g') < C}). But by construction, if we let p = gK, p' = g'K e U, then [q~(g -1) -- q~(g,-1) [ is of the order of the angle .,~.pOp'. The basic formula in hyperbolic geometry ch c ~ ch a ch b -- sh ash b cos-~ C implies that i q~(g)-i _ q~(g,-1) [ is of the order of exp(-- d(o,p)). Since the area of the disk of center o and radius d = d(o, p) is of the order of exp d, one easily gets f M(g)2 dg < oo. [] As in the case ofR 2 we shall now compute the 2-dimensional character v~ of (H, F). (The computation of "r~ (for (H,, F,)) and its relation with characters of discrete series is postponed until part VII.) Note that obviously z2 = Y.z~. By a straightforward computation we get .~(fo, f~,f,) = 2i7: ~,~,g, =1 fO(gO) fX(g~) f2(g2) c(gO, g~, g,) dgl dg2 where the function c(g ~ gl, g~), gi e G, gO gl g2= I, is given by c(gO, gl, g~) = f ~(gO, g) ~(gl, (gO)-l g) ~(g2, (gO g,)-l g) dg with ~3(g ~ g) = i -- ~(g)-i ~((g0)-i g). We now relate c(g ~ ga, g2) to the 2-cocycle A(g 1, g~) which is given by the (oriented) area of the hyperbolic triangle (in the PoincarE disk U) with vertices o, (gl)-~(o), g~(o). Note the relations A(kg ~,g~) =A(g ~k,kg ~) =A(g ~,g~k) =A(g a,g2) VkeK and A(gO, gl) = A(g~, g~) __ A(g~, gO) for gO g~ gg. = I. Lemma 5. -- One has c(g~ 2) = 4i~A(g ~, g~) (where gOgXg~ = I). Proof. -- Let A = o, B = g~ C = g~ and let us consider the hyperbolic triangle T = (A, B, C) in the Poincar6 disk U. For g E SL(2, R) let p = g(o) 9 U. The value of q~(g)-i ~((gO)-lg) only depends on the three points A, p, B. Since ~((gO)-lg), considered as a function of g, is the section of the tangent bundle T(U) which to p e U assigns the unit tangent vector at p looking at g~ = B, we get ~(g)-~ ~((gO)-~g) = exp i~):ApB. Thus ~(g0, g) = i -- exp i-4:ApB. Also ~(gl, (gO)-lg) = I -- exp i~3, where ~ = -<~ (o, (g0)-i g(o), g'(o)) = <): (g~ g(o), gO gl(o)) = .~ BpC, and similarly one has ~3 (g2, (gO gl)-1 g) = I -- exp/'F, 7 = <): CpA. As in the Euclidean case one has ~r q- ~ + y = o so that the same computation as in lemma 9. ~ gives c(g ~ gX, gZ) = _ ~iS(A, B, C), where S(A, B, C) = fu (sin -<): ApB -k- sin <;t BpC q- sin <~ CpA) dp. 301 86 ALAIN GONNES Now the proof of lemma 9.2 is written in such a way that it goes over without changes to the hyperbolic case. For instance it is still true that the disk D R with center A is invariant under the symmetries around AC and AB while the complement of the line BC in D R has two components D', D" with aBc(D') C D". Thus as in lemma 9.2 one gets a G-invariant Radon measure on U so that S is proportional to the hyper- bolic area. Appendix x: Schatten classes In this appendix we have gathered for the convenience of the reader the properties of the Schatten classes SOP needed in the text. Let H be a separable Hilbert space, SO(H) the algebra of bounded operators in H and SO~176 the ideal of compact operators. For T ~ SO| we let ~,(T) be the n-th singular value of T, i.e. the n-th eigenvalue of [ T[ = (T* T) 1/~ (cf. [63] ). By definition, the Schatten class SOP(H) is, for p e [i, oo[, SOP(H) ---- {T ~ SO(H), Z~.(T)P < oo}. Proposition 1. -- a) SOP(H) is a two sided ideal in SO(H). b) SOP(H) is a Banach space for the norm IIT = (z~t,(T)P) lip. c) SOP(H) c SO (H) for p < r I I I d) Let p, q, r ~ [I, OO] with -=- +-. For any S ESOP(H), T~SOq(H), one r p q has ST eso'(H) and IISTll,< IlSll. llYllr Proof. -- See [63]. One could equivalently define SOP(H) starting from the trace on S~ which we consider as a weight, i.e. a map: SO(H) + -+ [o, oo] defined by Trace(T) = Z (T~,, ~,) for any orthonormal basis (~,) of H and any T e SO(H) +. (See [51] theorem 2.14. ) Proposition 2. -- a) SOP(H) = {T e SO(H), Trace IT] p < oo}. b) For T ~SOP(H) one has [ITllp = (Trace [TIP) 1/p. c) Trace(A* A) -- Trace(AA*) for all A e SO(H). d) The trace extends by linearity to a linear functional on SOX(H) and Trace(T) = Z <T~., ~,> for T E SOl(H) and any orthonormal basis (~,) of H. e) ]Trace(T)] ~ 117111 for T esol(H). 302 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Proof.- See [5 I] and [63]. The next theorem, due to Lidskii, expresses Trace T from the eigenvalues of T. Since LalC Laoo the eigenvalues of T e Lax form a sequence (~,), with ~, ---> o when n~oo (see [63]). Theorem 3. -- Let T 9 .El(H), then Z I X.(T) [ < oo and Trace T = ZZ,(T). Corollary 4. ~ Let A, B e La(H) be such that AB and BA belong to Lal(H). Then Trace(AB) = Trace(BA) (el. [63] , p. 5o). We shall now prove two results needed in part I. They are an easy modification of lemma 3.2, p. i58 in [30]. Proposition 5. -- Let p e [I, or[, S, T 9 La(H) and assume that [S, T] e LaP(H). Then: cr if f is an analytic function in a neighborhood of the spectrum of S, one has [f(S), T] e LaP(H); f~) if S is selfadjoint and if f is a C ~~ function on the spectrum of S, one has [f(S), T] e LaP(H). Proof. -- ~z) Let y be a simple closed curve containing the spectrum of S, with f analytic on y. Then, f(s) = (i/2i=) f.ff(x)(~, - s) -1 dx and hence, [f(S), T] ---- (I[2i~) fvf(k ) [(), -- S) -1, T] dL Now [(k -- S) -~, T] = (), -- S) -~ IS, T] (), -- S) -1, which implies that the map k ~f(),) [(k -- S) -1, T] is a continuous function from y to LaP(H). Thus the integral converges in LaP(H) and [f(S), T] e LaP(H). ~) Let us show that II[e"S,W]llp is O(Itl) when t~oo. For any U~La(H), with [U, T] e Lap one has, II[U",T]llp<nll[W,T]llpllWll o-1 Vn ~N. This shows that IIEe"S,W]llp is bounded on any bounded interval. Since, with U ---- e us, one gets, IIEe "'s, W]llp < n II[e "s, W]llp v n EN, it follows that II[e"S,T]llp<C(i + Itl) for all t. Then take f to be compactly supported, so that f=g with (1 + Itl) g eL~(R). This yields, lilY(S), W]llp < f lg(t)l II[e ''s, T]llpdt < C f lg(t)l(I + Itl) dt< oo [] 303 88 ALAIN CONNES Proposition 6. (Cf. [34]-) -- Let p e [I, oo] and P, Q, e s be such that I -- PO ~ .SfP(H), I -- QP e .~P(H). Then P is a Fredholm operator and for any integer n > p, one has, Index P = Trace(I -- Qp)n __ Trace(i -- PQ)". Proof. -- Since I -- QP and i -- PQ are compact operators, P is a Fredholm operator. Moreover I is an isolated point in K = { I } t3 Spectrum(i -- PQ) t3 Spectrum(I -- QP). Let y be the boundary of a small closed disk D with center I such that D n K = { i }. Set e=~ ifv dX -- xfv dX . Then e = e ~, f =fs; Ex = Range of e, F x = Range off are finite dimensional, and admit respectively E 2 = Kere, F 2 = Kerr as supplements in H. For any ~z ~C one has, Q(~ - PQ) = (~ - QP) Q. Thus, for any XCK, one has (X -- (I -- Qp))-I Q= Q(x- (I - pQ))-l. This shows that Qf = eQ and similarly that Pe =fP. Thus, P(Ex) C F1, P(Es) C V,, Q(F1) C E,, Q(F2) C E,. Let Pj (resp. Q~) be the restriction of P (resp. Q) to Ej (resp. Fj), j = I, 2. By cons- truction the restrictions of QP to E~ and of PQ. to F~ are invertible operators, and hence, a) Index P ----- dim E x -- dim Fx, b) Trace(IE, -- Q2 P2)" : Trace(IF. -- P~ Q.2)" v n > p. The spectrum of IE, -- Q1 P1 and of IF, -- P1 Qa contains only { I }, thus, c) Trace(IE, -- Q1 P1)" -- Trace(IF, -- Pt Q1) n = dim E 1 -- dim F 1. Combining a), b), c), one gets the conclusion. [] Appendix 2: Fredholm modules The notion of Fredholm module is due to Atiyah [3] in the even case, and to Brown, Douglas, Fillmore [I i] and Kasparov [42] in the odd case. Their definitions are slightly different from the definition below and our aim is to clarify this point. Let X be a compact space and A-= C(X). An element of Atiyah's Ell(X) is given by two representations, a+ : A -+ s176 a- : a -+ .W(H-) 304 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY of A in Hilbert spaces H +, H- and a Fredholm operator P : H + -+ H- with para- metrix Q, which intertwines a + and a- modulo compact operators, Po +(a) Q-a-(a) cot ~ Va 9 The typical example is obtained when X = V is a smooth compact manifold, H + = L2(~ +) are Hilbert spaces of square integrable sections of bundles ~ over V, a~: are the obvious actions of C(V) by multiplication, and P is an elliptic operator of order o from ~+ to ~-. With Q a parametrix of P, let F e ~(H + | H-) be given by .[; One has, ~) H is a Z/2 graded *module over A= C(X), ~) [F,a] 9 Va 9 y) F 2- I 9 Note that in general F 2 4= I since P is not invertible. Definition 1. -- Let ~1 be a Z/2 graded algebra over C. Then a pre-Fredholm module over d (resp. Fredholm module over d) is a pair (H, F) where I) H is a Z[2 graded Itilbert space and a graded left d-module, 2) Feff(H), Fs=--r [F,a] E~ Va 9 3) a( F~ -- I) 9 ~f V a 9 ~ (resp. F ~ = I). We shall now show how to associate canonically to each pre-Fredholm module a Fredholm module. Let Hr be the Z[2 graded Hilbert space Hg =C and let H=HQHr be the graded tensor product ofHby H 0. One has H+ =H +| H- =H-| +. We turn H into a graded left d-module by a(~ (~ ~) = a~ Q e~ Va~d, ~eH, "~eH c, ext, w,th .__[; we define where e c .~e(Hc) , e['o :1 P '-PQ] 0.= -Qy (Qy - 2) o_..1' -vQ One checks that = I, QP = I, so that ~2= i. P~7). ~~ Proposition 2. -- Let (H, F) be a pre-Fredholm module over d. a) (H, F) is a Fredholm module over d. b) Let H 0 be the Hilbert space H with opposite Z[2 grading and o-module structure over ~1, then (H0, o) is a pre,Fredholm module over dt. 12 ALAIN CONNES 9 ~ c) One has ~ = H r H 0 as a Z/2 graded sg-module, and a(ff--F| e~ Vae~r Proof. -- a) Since F ~ = I, conditions I) and 3) are verified. Let us check that IF, a] e.~e', Va e~r One has by hypothesis [F,a] eOff, a(F ~-- I) e.~ and hence (1 ~- I) a~r Let us first assume that a is even. One has Pa -- aP a(i -- PQ,)] e .Y{'. Ptz -- aP = (I -- Q P) a Let now a be odd and Since 0. = ~-' one gets 0." -- ,,~ ~,X", hence IF, a] e .X'. LP(H) be the corresponding operator in H. By hypothesis one has [:. 1 (QP - i) al~ e ~, a~t(QP -- I) e gr (PQ-- x) a,t e ~", at2(PO _- I) e ~. The action of a in ~=H+| is given by the matrix T = T' where T'= t T"= [o [,: :1 ?: :1 One has to check that T" ~ + 0.T' e ~ and T' i~ + fiT" e or. This follows easily. from our hypothesis. b) Obvious. [o ~ [: o] [o :] Then F'= with P'= , Q' = c) Let F'=F| P' o With a even one has, a(~ P') = o a(Q-Q') =[a(2-QP) Q-aQo a(I-QP)] e~'o The odd case is treated similarly. [] Let p e [ I, oo[. We shall say that a pre-Fredholm module (H, F) over d is p-summable when, a) IF, a] e.~'(H) for aed, 13) a(F ~- I) e.LaP(H) for aed. Proposition 3. -- Let (H, F) be a p,summable pre-Fredhotm module over d. Then (ffI, F) is a p-summable Fredholm module. Proof. -- In the proof of proposition 2 one can replace 3g" by any two-sided ideal. [] We shall now discuss the index map associated to a Fredholm module. 306 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Lemma 4. -- Let s/ be a Z/2 graded algebra, (H, F) a Fredholm module over ~/. a) Let ff = s/ | C be obtained from ~t by adjoining a unit. Let ~ act in H by (a+),I)~=at +?~ for ar ),eC. Then (H,F) is a Fredholm module over ~. b) Let H,~=H| F.----F| and M.(.~r =~r174 act in H,, in the obvious way. Then (H,, F,) is a Fredholm module over M,(~). The proof is obvious. Let us now assume that ~ is trivially graded. Proposition 5. -- Let (H, F) be a Fredholm module over d. There exists a unique additive map ~ : K0(~' ) -+ Z such that for any idempotent e e M,(d), q~([e]) is the index of the Fredholm operator from ell+, to eI-'I-; given by T~=eF,~ V~eeH +. Proof. -- One checks that T is a Fredholm operator with parametrix T' where eF, ~ for ~q e eHn, and that the index of T is an additive function of the class T' ~q of e in K0(~ ). Finally we shall relate the above notion of Fredholm module with the Kasparov A _ B bimodules, we recall (cf. [42]). Definition ft. -- Let A and B be C*-algebras. A Kasparov A -- B bimodule is given by l) a Z/2 graded C*-module 8 = ~+ | o ~- over B, 2) a 9 homomorphism 7: of A in .L~'B(8), 3) an element F of .oq~B( e) such that Va cA, a(V - r') VaeA, Va~A. v) a( i) (Cf. [42] for the notions of endomorphism (-9~B(8)) of 8 and of compact endomor- phism (~g'n(r Let us take B = C. Then a Kasparov A--C bimodule is in particular a pre-Fredholm module over A. Conversely, Proposition 7. -- Let A be a C*-aIgebra and (H, F) a * Fredholm module over A. Let F' = F [F[-1. Then (H,F') is a Kasparov A-- C bimodule. Moreover, for each t, F l = F 1F [- t defines a * Fredholm module (H, Fj). Proof. -- One has [F*F,a] e~Y', VaeA, thus [IFl',a] cog" for seR, aeA. Let F = J A be the polar decomposition of F, with A ---- I F ]. One has F- 1 _-- A- 1 j-1 which gives the right polar decomposition of F ---- F -1, thus J -j-1 and J -j* -- F', so that (H, F') is a Kasparov A -- C bimodule. Finally jAj -1 = A -1 and hence j As j-1 _~ A- s for any s e R, so that J A s is an involution for any s. It follows that (H, F,) is a * Fredholm module. [] 307 9o ALAIN CONNES Appendix 3: Stability under holomorphic functional calculus Let A be a Banach algebra over C and d a subalgebra of A, A and ,~ be obtained by adjoining a unit. Definition 1. -- d is stable under holomorphic functional calculus if and only if for any hen and aeM, CM,(A) one has, f(a) M.(d) for any function f holomorphic in a neighborhood of the spectrum of a in M,(A). In particular one has GL,(d ~) = GL,(.~) n M,(~7), hence if we endow GL,(~ ~) with the induced topology we get a topological group which is locally contractible as a topological space. We recall the density theorem (cf. [4], [4o]). Proposition 2. -- Let d be a dense subalgebra of A, stable under holomorphic functional calculus. a) The inclusion i : d ~ A is an isomorphism of Ko-groups i,: Ko(d ) ~ Ko(A ). b) Let GL~(~ be the inductive limit of the topological groups GLn(~7). Then i. yields an isomorphism, ~k(GL~(~7)) ---> =k(GL=(A)) --- Kk+I(A ). Let now (H, F) be a Fredholm module over the Banach algebra A and assume that the corresponding homomorphism of A in .'Ae(H) is continuous. Proposition 3. -- Let pe[i, oo[ and d={aeA,[F,a]e.s Then ~ is a subalgebra of A stable under holomorphic functional calculus. Proof. -- One has IF, ab] = IF, a] b + a[F, b] for a, b e A. Thus as .s is a two-sided ideal in .oq~ ~r is a subalgebra of A. Let n ~N, (H,, F,) be the Fredholm module over M,(A) given by lemma 4 b) of Appendix 2 and ~, the corres- ponding homomorphism: M,(A) ---> .ge(H,). One has, M,(d) = {a e M,(~,), IF,, 7:,(a)] e .o'AeP(H,)}. Moreover Sp(n,(a)) C Sp(a), and since =, is continuous, ~:,(f(a)) =f(~,(a)) for anyf holomorphic on Sp(a). The conclusion follows from proposition 5 of Appendix I. [] Let now A be a C*-algebra and ~r a dense 9 subalgebra of A stable under holo- morphic functional calculus. Proposition 4. -- Let (H, F) be a 9 Fredholm module over .~. Then the corresponding 9 homomorphism ~ of~in .s is continuous and extends to a 9 homomorphism ~ of A in .Lf(H) yielding a * Fredholm module over A. 308 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Proof. -- We can assume that d and A are unital and ~(i) = i. Let a e d, then the Spectrum of a* a in .~ is the same as its Spectrum in A. Thus the norm of a in A is ]] a ]] = pl/~ where 0 is the radius of the Spectrum of a* a in ~f. One has Spectrum (=(a* a)) C Spectrum(a* a), thus [] =(a)][* = Spectral radius of~(a* a) < p = ][ a][ 2. This shows that rc is continuous. Let ~ be the corresponding 9 homomorphism of A in .o~~ For a e A, I-F, ~(a)] is a norm limit of I-F, =(a,)], a, e d, which are compact operators by hypothesis, thus [l v, ~(a)] e 3/" for all a e A. [] 309 II. -- DE R.HAM HOMOLOGY AND NON COMMUTATIVE ALGEBRA In part I the construction of the Chern character of an element of K-homology led to the definition of a purely algebraic cohomology theory H~(d). By construction, given any (possibly non commutative) algebra ~r over C, H~(~r is the cohomology of the complex (C~, b) where C~ is the space of (n + 1)-linear functionals q~ on d such that qo(a 1,...,a",a ~ ~-(-- I)"~(a ~ Va ~ed and where b is the Hochschild coboundary map given by (bqg) (a 0, ..., a n+l ) = ~ (-- i)/r 0, ..., aJa j+l, ..., a n+l ) j=0 + (-- I)"+lT(a"41a ~ ...,a"). Moreover H~(d) turned out to be naturally a module over H~((1) which is a polynomial ring with one generator r of degree 2. In this second part we shall develop this cohomology theory from scratch, using part I only as a motivation. We shall arrive, in section 4, at an exact couple of the form H*(d, d*) H~(d) > H~(~) relating H~(d) to the Hochschild cohomology of ~ with coefficients in the bimodule of linear functionals on d. This will give a powerful tool to compute H~(d) since Hochschild cohomology, defined as a derived functor, is computable via an arbitrary resolution of the bimodule d (cf. [I3] [47]). For instance, if one takes for d the algebra C ~ (V) of smooth functions on a compact manifold V and imposes the obvious continuity to multilinear functionals on d, one arrives quickly at the equality (for arbitrary n) H"(d, ~*) = space of all de Rham currents of dimension n. (This will be dealt with in section 5. The purely algebraic results of sections I to 4 easily adapt to the topological situation.) The operator I o B : Hn(~ r d*) -+ Hn-l(d, d*) coincides with the usual de Rham boundary for currents, and the computation of H~(~) will follow easily (cf. section 5). In particular we shall get H*(~ r = Ordinary de Rham homology of V, 310 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY where H*(d) is defined as in part I by H'(d) = H;,(d) As another application we shall compute H*(d) for the following highly non commutative algebra, Fix 0 ~ R/Z, 0 r Q./Z. Then ~10 is defined by d 0 ~- { Y~an,. U" V'; (a., ,~)., m ~ Z sequence of rapid decay }, where VU = (exp i2n0)UV gives the product rule. The algebra d0 corresponds to the " irrational rotation C*-algebra " studied by Rieffel [58] and Pimsner and Voiculescu [55]. It arises in the study of the Kronecker foliation of the 2-torus [16]. In section I we introduce the following notion of cycle over an algebra ~ which is crucial both for the construction of the cup product H~(~) � H~(~) -~ H~ +'(~ | ~) and for the construction of B : Hn+l(d, ~*) ---~H~(~). By definition a cycle of dimension n is a triple (~, d, f) where ~ = 0 ~i is a j-0 algebra, d is a graded derivation of degree I such that d ~ = o, and f : ~" -~ (l graded is a closed graded trace. A cycle over an algebra d is given by a homomorphism p: ~r _+ f~0 where (f2, d, f) is a cycle. In part I we saw that any p-summable Fredholm module over .& yields such a cycle. Here are some other examples. 1) Foliations Let (V, F) be a transversally oriented foliated manifold. Using the canonical integral of operator valued transverse differential forms [i4] of degree q = Codlin F, we shall construct in Part VIa cycle of dimension q over the algebra d= C~~ F)). 2) C* dynamical systems Given a C* dynamical system (A, G, ~) (el. [I9] ) where G is a Lie group, the construction of [I9] associates a cycle on the algebra A ~ of smooth elements of A to any pair of an invariant trace -r on A and a closed element v ~ A (Lie G). 3) Discrete groups In part V we shall associate a cycle on the group algebra C(F) to any group cocycle co e Z"(F, C) and obtain in this way a natural map of the group eohomology Hn(I ", C) to H[(C(r)). Given a cycle of dimension n, ~r ~ ~ over d, its character is the (n + I)-linear functional ~(a~ ..., a") = rp(a ~ dp(a ... dp(a"). 311 9 6 ALAIN CONNES We show that x 9 Z[(d) =- C~(d) r~ Ker b, that any clement of Z~(d) appears in this way and that the elements of B~(d) = bC~-l(d) are those coming from cycles with fl ~ flabby. (See [40] for the definition of a flabby algebra). Then the straightforward notion of tensor product of two cycles gives a cup product HI(d) | H~'(~) -+ H~ +'~(d | ~). We then check that H~(C) is a polynomial ring with a canonical generator a of degree 2 and we define at the level of cochains the map S : HI(d) -+ H[+*(M) given by cup product by a. In section 2 we show that the standard construction of the Chern character by connexion and curvature gives a pairing of H~V(d) with the algebraic K theory group K0(~r ) and of H~d(d) with K x. The invariance of this pairing naturally yields the group H*(d) = H~(~) | C, inductive limit of the groups H~(d) with map S. We then discuss the invariance of H~(d) under Morita equivalence. In section 3 we show that two cycles over ~ are cobordant (cf. 3 for the definition of cobordism) if and only if their characters xl, x, differ by an element of the image of B, where B is a canonical map of the Hochschild cohomology Hn+l(d, d*) to H~(d) defined as follows: = 2; (0 yEF where F is the group of cyclic permutations of {o, ..., n}, ~ ..., a = ..., r is the signature and (B0-r)(a ~ ='r(~,a ~ ~) § (-- ~)"r(a ~ '~,~) for all g a d. Thus defined at the level of cochains, B : C"(~', .~/*) + C"-l(d, ~*) commutes (in the graded sense) with the Hochschild coboundary b, which yields the basic double complex of section 4. The above result yields a new interpretation of H*(d) as H*(~) = (Cobordism group of cycles over a~r | a ~ (I which is completed in section 4 thanks to the exact triangle H*(d, ~*) H~(d) + H~(~ r where I is induced by the inclusion map from the subcomplex C[ to C n. This exact triangle gives in particular the characterization of the image of S which was missing in part I (cf. theorem 16): z 9 S if and only if 9 is a Hochschild coboundary. It also proves that H~(~/) is periodic with period 2 above the Hochschild dimension of ~r 312 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY By comparing the above exact triangle with the derived exact sequence of o ~ C~ -+ C" ~ C"/C~ -~ o we prove that there is a natural isomorphism H"(C/Cx) ~ H~-1(~r We then show that the cohomology of the double complex C"' " = C"-'(d, d*), d~ = b, d~ = B is equal to HeY(d) for n even, and H~162 for n odd. The spectral sequence associated to the first filtration ( 2~ C"") does not converge in general, and in fact has initial n>p term E2 always o. This and the equality S = bB -x are the technical facts allowing to identify the cohomology of this double complex with H*(~r The spectral sequence associated to the second filtration ( ~ C"") is always ra>q convergent. It coincides with the spectral sequence associated to the above exact couple and i) its initial term E 1 is the complex (H"(d, d*), I o B) of Hochschild cohomology groups with the differential given by the map I o B; 2) its limit is the graded group associated to the filtration F"(H*(d)) by dimensions of cycles. Finally we note that in a purely algebraic context the homology theory (which is dual to the cohomology theory we describe here) is more natural. All the results of our paper are easily transposed to the homological side. However, from the point of view of analysis, the cohomology appeared more naturally and, for technical reasons (non Hausdorff quotient spaces), it is not in general the dual of the homology theory. This motivates our choice. Part II is organized as follows: CONTENTS x. Definition of H~(~ r and cup product ...................................................... 97 2. Pairing of H~(~r with Kl(~r i = o, ~ ................................................... xo7 3- Cobordism of cycles and the operator B .................................................... x 14 4. The exact couple relating H~(~r to Hochschild cohomology ................................. tx 9 5- Locally convex algebras ................................................................... 125 6. Examples ................................................................................ x27 I. Definition of Hi(d ) and cup product By a cycle of dimension n, we shall mean a triple (f~, d, f) where is a graded algebra, d is a graded derivation of degree i with d z = o and : ~" --> C is a closed graded trace. 13 98 ALAIN CONNES Thus one has: I) f~� ~+~ Vi, j 9 iWjs 2) dta' c '+I, a(toto') = (ato) to' + (-- to #to,, d" = o; Given two cycles fl, fl' of dimension n, their sum f~| is defined by (~")~ = f~ @ fU, (tox, to~) (to~, to~) = (to~ to,, to~ to~), d(to, to') = (dco, dto') and Given cycles f~, f~' of dimensions n and n', their tensor product f~" = ~ | f~' is the cycle of dimension n + n' which as a differential graded algebra is the tensor product of (f~, d) by (~', d'), and where f(to| fto' V~ 9 co'eft'"'. For example, let V be a smooth compact manifold, and let C be a closed current of dimension q (<dimV) on V. Let f~i, i 9 ...,q} be the space &~ AIT'V) of smooth differential forms of degree i. With the usual product structure and dif- ferentiation f~ = O f~i is a differential algebra, on which the equality f to = (C, to ), i~0 for to 9 f~q, defines a closed graded trace. In this example f~ was graded commutative but this is not required in general. Now let ~1 be an algebra, and f~(~r be the universal graded differential algebra associated to aar ([I] [39]). Proposition 1. ~ Let .r be an (n + I )-linear functional on d. Then the following conditions are equivalent: I) There exists an n-dimensional cycle (~, d, f) and a homomorphism p: ~ -+ ~o such that v(a ~ a') p(a ~ d(p(aX)) .. d(pCa")) V a ~ ., e d. 9 . .~ -~- f " . . a n 2) There exists a closed graded trace T of dimension n on f~(~) such that 9 .,~ 9 ., a n "r(a ~ a") =T(a ~ t .. da n ) V ~ a ., 9 d. 9 "*, ~ "''9 "~ an 3) One has r(a 1, a", a ~ (-- x)" .r(a ~ a") for a ~ , 9 ~t and (-- I) i'~(a0, ..., a iai+l ..., a n+l ) + (-- I) "+l"r(a "+1 a ~ ..., a") = o for a ~ ..., a "+1 9 d. Proof. ~ Let us first recall the construction of the universal algebra f~ (d) (It] [39])- Even if d is already unital, let ff be the algebra obtained from d by adjoining a unit: ff={a+)~x;aed,), 9 For each heN, n~ i, let fP(d) be the linear space -+ o-(d) d. 314 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY 99 The differential d:~n-+ ~n+1 is given by d((a ~ + X ~174174 | = ~|176174 | 9 By construction one has d 2 ---- o. Let us now define the product ~i x E~ i -* f~ i+i. One first defines a right d-module structure on El" by the equality (~~174174 | n) a-- ~ (-- I)n-i~~174174174 j=O Let us check that (toa) b = to(ab) 'v' to 9 fV', a, b 9 ~r One has n+l ('d'O| ... |174 aJ+l| ... |174 n+l) a n+2 = ~_a Cj, k O~j,k k~O where e~, k = o ifj = k, e~,~ = (-- i) "+k-1 if j< k and cj, k : (-- I) n-k if k <j = = a~174 |174 |174 | 2. while % k ak, j ......... Thus if one expands ((,~o| ... | a "+1) a "+2, one gets twice the term ~j,k for j, k 9 { o, I, ..., n} and with opposite signs: (-- I) "-j (-- r) "+l-k and (-- I) "-k (-- I) ~-~. Thus ((~o|174 ~)a.+l)a.+2= ~ (_ i),-~r j=O _--_ (~o| ... | a,,+~). This right action of ~r on f~" extends to a unital action of ~ One then defines the product: f~ x ~ -4 f~i+j by to(~~174174 | = to~o|174 | v co 9 ~. It is then immediate that the product is associative. With to=~~174 l|174 . 9 one has, for a 9162 d(toa) = ~ (-- I)n-iI|176174 |174 | j=O n+l (do))a= ~ (-- I)"+l-iI|176174174174174 j=O = (-- I) "-1 to da + d(toa). Thus (~, d) is a differential graded algebra, and the equality ~O da 1. . . da ~ = ~o | a l | . . . | a,, shows that it is generated by d. One checks that any homomorphism ~r -~ f~,o of ~r in a differential graded algebra (f2', d'), d'2= o, extends to a homomorphism ~ of (f~(d), d) to (ta', d') with -p(.~o dat ... da") = p(a ~ a'(p(al)) d'(o(a2)) ... d'(p(a")) + X ~ d'(p(a')) ... d'(p(a")) for a i 9162 ~o 9 ~-o = (a o,xo). 315 loo ALAIN CONNES Thus I) and 2) arc obviously equivalent. Let us show that 3) =~ 2). Given any (n + i)-linear functional q~ on d, define ~ as a linear functional on ~"(d) by ~((a ~ + x ~ ,) | (1' | | a") = ~0((1 ~ (1', ..., (1"). By construction one has r = o for all co e s Now, with ": "satisfying 3) let us show that "~ is a graded trace. We have to show that ~(((1o d(11 ... d(1~)((1,,+, d(1~+2 ... a(1,+,)) = (-- ,)k("--k)'~((ak+' dak+2.., dan+')(a~ dak)). Using the definition of the product in ~(~/) the first term gives Z (- ,)~-~ ~((10, ..., (1j (1j+,, ..-, ~,+,), and the second one gives n--k Z (-- I) k(n-k)+n-k-j Z((1 TM, ..., (1k+l+j (1k+t+j+t, ..-, ak). j=0 The cyclic permutation ),, )~(l) ---- k + x + t, has a signature equal to (-- i) "(k+t) so that, as .:x = r x by hypothesis, the second term gives n+l - Z (- ,)~-j~((1o, ..., (1j(1j+,, ...,(1,+~). k+t Hence the equality follows from the second hypothesis on ":. Let us show that I) :~3). We can assume that ~----s ~ One has ,,r O, (ll, ..., an) = f ((1O d(11)(d(l?. . " d(1n) = (__ i)n-l f (d(12 . . . da")(a~ da 1) = (- ,)"f(a(1~... da"a(1 ~ a' = (-- ,)".:((1', ..., (1", (1o). To prove the second property we shall only use the equality fat~ = f,oa for o) egl", a e.~. From the equality d(ab) = (da) b + a db it follows that (da a . .. da '~) a'~+ ' = ~Z (_ i),,-J da ~ . . . d(ai aj+a) ... da,~+l J-1 + (-- i)"aada~.., da"+', thus the second property follows from (Note that the cohomology of the complex (El(d), d) is o in all dimensions, including o since s176 ---- d.) Let us now recall the definition of the Hochschild cohomology groups H"(d, Jr') of ~r with coefficients in a bimodule ,/4 ([i3]). Let ~ ---- d | ~0 be the tensor 316 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY I01 product of ~r by the opposite algebra. Then any bimodule ~' over ~r becomes a left ~r module and by definition: H"(~r162 = Ext~,(~g,../t'), where ~qr is viewed as a bimodule over ~r via a(b) c =abc, V a, b, c ~ ~/. As in [I3] , one can reformulate the definition of H"(~, ~r using the standard resolution of the bimodule ~r One forms the complex (C"(~r ..it'), b), where a) C"(~qr ..r is the space of n-linear maps from ~r to .,r b) for T e C"(,~r162 bT is given by (bT) (a 1, ..., a "+l) = a 1T(a 2, ..., a "+l) +,~(-- ,)'T(al,...,a'a '+l, ...,a "+l) + (-- i) "+1T(a', ..., a n ) a "+1 Definition 2. -- The Hochschild cohomology of a~/ with coefficients in ~r is the cohomo- lo~y H"(d, ..Of) of the complex (C"(d, ..r b). (Note the close relation of the ~'~(~r with the standard resolution and the use of the bimodules ~r in the process of reduction of dimensions: see for instance [36], p. 8). The space d* of all linear functionals on d is a bimodule over ~ by the equality (a~b) (c) = ~(bca), for a, b, c ~ ~. We consider any T E C"(.~, .~r as an (n + 1)- linear functional -r on ~r by the equality x(a ~ 1,...,a") =T(a 1,...,a")(a ~ Va ~d. To the boundary bT corresponds the (n q- 2)-linear functional by: (b'~) (a ~ ..., a "+l) = v(a ~ a l, a ~, ..., a "+l) 9 . "'', --., a ~ ...,a n ). + ~l(-- I)i'r(a O, aia i+l, a n+l) + (-- I)n+l,r(a n+l Thus, with this notation, the condition 3) of proposition r becomes a) "r v = r "r for any cyclic permutation "I" of {o, I, ..., n}; b) b'r = o. Now, though the Hochschild coboundary b does not commute with cyclic permutations, it maps cochains satisfying a) to cochains satisfying a). More precisely, let A be the linear map of C"(d, d*) to C"(d, d*) defined by (Ao) = Z r ~-e, where I' is the group of cyclic permutations of { o, I, ..., n }. Obviously the range of A is the subspace C~(d) of C"(d, d*) of cochains which satisfy a). One has Lemma 3. -- boA=Aob' where b':C"(d,d*) ~C"+1(~r r is defined by the equality (b'o)(x ~ ..., x "+l) = ~ (-- I) iq~(x ~ ..., xJx j+l, ..., x"+l). j=0 317 I02 ALAIN CONNES Proof. -- One has ((Ab') ~)(x~ ..., x "+1) = Y,(-- i) ~+I"+1)* q~(x~,..., x~+~x~+~+l, ..., x ~-1) where o<i<n, o<k<n+ i. Also ((ba) @)(x ~ ..., x "+1) = ~ (-- I) j (A?)(x ~ ..., x j x j+l, ..., x "+1) j=0 + (-- I)"+l(A?)(x"+lx~ ..., Xn). For j~{o,...,n} one has (A~)(x ~ ..., x jx j +1, ..., x,+l) = ~ (__ I)"k 9(x k, ..., x~x ~+1, ..-,xk-1) k=0 n+l + Z (- I)"(k, ~1 ~(x k, ..., x "+1, x ~ ..., x~ x ~+~, ..., x~-~). k=j+2 (Aq~)(x "+1 x ~ ..., x") = ~0(x n+l x O, ..., X n) Also, + ~: (- ilJ ~(~J, ..., x", ~"+~x ~ ..., xi-1). In all these terms, the xJ's remain in cyclic order, with only two consecutive x~'s replaced by their product. There are (n + I) (n + 2) such terms, which all appear in both bA~ and Ab' ~. Thus we just have to check the signs in front of Tkj (k +j + i) where Tk.j=~(x u, ...,x~x ~+1, ...,xk-1). For Ab' we get (--I) ~+l"+l)k where i=j--k(modn+~) and o<i<n. For bA we get (-- I) j+"~ if j~k and (-- I) ~+"ln-1) if j< k. When j~ k one has i =j--k thus the two signs agree. When j<k one has i~-n+2--k+j. Then as n + 2 -- k +j + (n + I) k =j + n(k -- I) modulo 2 the two signs still agree. [] Corollary 4. -- (C~(off), b) is a subcomplex of the Itochschild complex. We let H~(~) be the n-th cohomology group of the complex (C~, b) and call it the cyclic cohomolog~ of the algebra off. For n = o, H~(d) = Z~ is exactly the linear space of traces on d. For off=C one has H~=o for n odd but H~=C for any even n. This example shows that the subcomplex C~ is not a retraction of the complex C", which for off = C has a trivial cohomology for all n > o. To each homomorphism p : off -+ ~ corresponds a morphism of complexes: p* : G~(~) -+ C~(off) defined by (p* ~p)(a ~ ..., a") = ~p(p(a~ ..., p(a")) and hence an induced map p*:H~(~) -+H~(off). 318 NON-COMMIYrATIVE DIFFERENTIAL GEOMETRY ~o3 Proposition 6. -- I) Any inner automorphism of ~r defines the identity morphism in H[(~). 2) Assume that there exists a homomorphism 0 : ~-+ a~r and an invertible element X [:o] [:o] of M~(ad) (here we suppose ~r unitaI) such that X X -t = for a 9 sJ. Then H[(~) is o for all n. p(a) p(a) Proof. -- i) Let a ~ ,~/ and let 8 be the corresponding inner derivation of d given by 8(x) =ax--xa. Given ~eZ~(d) let us check that +, 4(a ~ . a") - ~: ~(a ~ ~(a'), ... a"), is a coboundary, i.e. that 4 9 B~(d). Let 4o(a ~ ..., a "-t) = ~(a ~ ..., a "-1, a) with a as above. Let us compute bA4o = Ab' 4o. One has: tl--1 (b'4o)(a ~ ...,a n ) = Y, (-- I)~r ~ ...,a~a i+x, ...,a n ,a) i=0 = (b~?)(a ~ ..., a n , a) -- (-- I) n q~(a ~ ..., a n-l, aria) - (- x) "+1 ~(aa ~ ..., a "-1, a"). Since b~ = o by hypothesis, only the last two terms remain and one gets Ab' 40 -- (-- i)" 4. Thus 4 = (-- I) n Ab' 40 = b((-- i) n A4o ) E B~(~). Now let u be an invertible element of d, let ~ e Z[(~) and define 0(x) -- uxu -1 for x 9 d. To prove that q~ and q~ o 0 are in the same cohomology class, one can [: o] replace d by M~(d), u by v= and q~ by q% where, for a~ed and b ~ 9 M~(s u-1 92(a ~174 b ~ a t| b t, ..., a n| b") = q~(a ~ ..., a n) Trace(b ~ ... b"). Now v=v lv~ with vl= , v2 = . One has [: :][: :][:, o I [ o :1 I --I 7~ v~ = exp a~, a~ =~ v~, thus the result follows from the above discussion. 2) Let T 9 Z[(~r and ~0~. be the cocycle on M~(~r defined in the proof of i). For a e,~, let e(a)= p(a) o(a) . By hypothesis e and ~ are homomorphisms of ,~ in M~(,ff) and, by i), ~. ~ and ~. ~ are in the same cohomology class. From the definition of ~ one has q~2(~(a~ ..., ~(a")) = 9(a ~ ..., a") + e?(p(a~ ..., p(a")), ~(~(a~ ..., ~(a")) = ,?(~(a~ ..., ~(an)). [] Following Karoubi-Villamayor [4x], let (3 be the algebra of infinite matrices (a~i)~,~e~ with a o e s such that 0~) the set of complex numbers {%) is finite, [~) the number of non zero a~'s per line or column is bounded. 319 xo 4 ALAIN CONNES Then C satisfies condition ~) of proposition 5, taking ~ of the form p(a) = Diag(a, o, a, o, ...). The same condition is satisfied by ~1 | C for any ~1, thus: Corollary 6. -- For any ~ one has H[(C~r = o where C~1 = C | s/. We are now ready to characterize the coboundaries B~ C Z~ from the corresponding cycles, as in proposition i. For convenience we shall also restate the characterization of Z~. Definition 7. -- We shall say that a cycle is vanishing when the algebra f~o satisfies the condition ~) of proposition 5 ([41]) 9 f~o, Given an n-dimensional cycle (f~, d, f) and a homomorphism ~: ,~r we shall define its character by ,~(a ~ ..., a") = f p(a ~ d(o(aX)) ... d(o(a")). ,I Proposition 8. -- Let -c be an (n + i)-linear functional on ~(; then x) v e Z~,(~r if and only if .r is a character; ~) x e B](.~r if and only if x is the character of a vanishing cycle. Proof. -- ~r is just a restatement of proposition I. p) For (fl, d, f) a vanishing cycle, one has H~(f~ ~ = o, thus the character is a coboundary. Conversely if -r ~B~(d), "r = b~ for some d/e C~-l(d), one can extend + to Cd = C | d in an n-linear functional d~ such that ~(I | a ~ ..., I | a n-x) = d?(a ~ ..., a n-l) for all a ~ e ~r and such that ~x = r ~ for any cyclic permutation ~, of {o, ..., n --x }. (Take for instance ~(b ~ ..., b "-1) = +(o~(b~ ..., a(b"-l)) where ~(b) = bl~ ~ d for any b= (b~) eccl.) Let p:d~Cd be the obvious homomorphism o(a) = l| Then "r' = b~ is an n-cocycle on Cd and 9 = p* "r' so that the implication 3) ~ 2) of proposition 1 gives the desired result. [] Let us now pass to the definition of the cup product H~(~) | H~'(~) -~ H~ +'(d | ~). In general one does not have f~(d | ~) = ~(~/) | f~(~) (where the right hand sid is the graded tensor product of differential graded algebras) but, from the universale property of ~(.~/| we get a natural homomorphism 7: : f~(~/| -+ f~(~/) | f~(~). Thus, for arbitrary cochains (p ~ Cn(~/, Ja/*) and ~b ~ Cm(~, ~*), one can define the cup product 9 # ~ by the equality (v +)^ = | ;)o 320 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY xo 5 To become familiar with this notion, let us compute 9 # %5 where 9 ~ C"(~r d*) is an arbitrary cochain, and where %5 ~ CI(C, C) (so that ~ = C) is given by d~(i, I) = r. Here ~r174162 so that 9# %5~C"+1(~1, a~r One has (9 # +)(~0, ..-, ~,+,) = (~| ~)(~(~o| ~)d(~'| ~)... d(~"+'| ~)). One has rcd(al| =dal| +aa| As r"= I one gets I(dI) ~ ~-o thus the only component of bidegree (n, r) of (rc(a~174 ~) d(a*| ~) ... d(a"+~| t)) is (a~ da ~ ... da") a"+~| ~ d~. Hence we get # %5 = ~ (- ~?+" 9(~~ ~'~+', ..., a "+') --(-~)" b' 9 with the notation of lemma 3- Theorem 9. -- I ) The cup product 9, q~ ~ ? # ~ defines a homomorphism H~(d) | H~'(8) ~ H~ + "(.~ | 8). 2) The character of the tensor product of two cycles is the cup product of their characters. Proof. -- First, let 9 e Z~(d), q~ e Z~'(8); then ~ (and similarly ~?) is a closed graded trace on f~(d), thus 7~ | ~ is a closed graded trace on f~(~r174 f~(8) and 9 # %5 e Z~+m(d | 8) by proposition x. Next, given cycles f~, f2' and homomorphisms p : d -+ f~, O' : 8 ~ f~', one has a commutative triangle a(~r | a(8) ~| Thus 2) follows. It remains to show that if 9 e B~,(~r then 9 # %5 is a coboundary: 9 # + ~ B~ +'(~ e 8). This follows from 2), proposition 8 and the trivial fact that the tensor product of any cycle with a vanishing cycle is vanishing. [] Corollary 10. -- i) H~(C) is a polynomial ring with one generator ~ of degree z. 2) Each Hi(d ) is a module over the ring H*x(C ). Proof. -- I) It is obvious that H~(C) = o for n odd and H~(C) -- C for n even. Let e be the unit of 13; then any 9 e Z~(13) is characterized by 9(e, ..., e). Let us 14 I06 ALAIN CONNES compute 9 # q~ where q~ 9 hb 9 Z~"'(C). Since e is an idempotent one has in ~(C) the equalities de = ede + (de) e, e(de) e = o, e(de)' = (de)' e. Similar identities hold for e| e and ~(e @ e) 9 f~(C)| f~(C) and one has ~((e| d(e| d(e| = edede| + e| (m + m')! Thus one gets (9 # ~b) (e, ..., e) -- m! m'! 9(e, ..., e) +(e, ..., e). We shall choose as generator of HI(C ) the 2-cocycle 6 6(I, I, I) = 2i~. ~) Let 9~Z~(~r Let us check that 6#9=9#6 and at the same time write an explicit formula for the corresponding map S :H~(a~r ~ H~+a(d). With the notations of I) one has 2~---~ (9 e 6) (a ~ "" "' a"+") = ~| 2z--~x ~ (aO| | ... d(a"+'| = ~(~0 a~ a, ~... ~.+,) + ~(~o ~(~ a~) da'.., da "+') + ... + ~(a" d~'.., d~'-~(~' a '§ d~+~.., da "§ +... + ~(a ~ I ... da"(a "+x a"+2)). The computation of 6 # 9 gives the same result. For 9 9162 let S?=6#~=q~#6 9 By theorem 9 we know that SB~(a~r C B~+'(off) but we do not have a definition of S as a morphism of cochain complexes. We shall now explicitly construct such a morphism. Recall that ~ # ~b is already defined at the cochain level by (9 # +) ^ = (~ | ~) o re. Lemma 11. -- For any cockain 9 9162 let $9 9 be defined by $9---- A(6#~); then n+3 a) -- A(6 4. ~) = 6 4* 9 for 9 9 Z~(~), so S extends the previously defined map. n+3 n+I b) bS9 =--Sb9 for 9 9 n+3 Proof. -- a) If 9 9 Z~,(~') then (6 4. 9) x = ~(X) 6 4. 9 for any cyclic permu- tation X of {o, I, ..., n + 2 }. b) We shall leave to the reader the tedious check in the special case + =- 6 of the equality (bAg) ~ hb = hA(9 4. hb) for 9 9 G"(a~r a~r It is based on the fol- lowing explicit formula for A(? 4# 6). For any subset with two elements s = {i,j}, i<j, of {o,~,...,n+2}=Z/(n+3) one defines ~(s) : q~(a ~ ..., ~'-~, ~ d +~, ..., ~J ~+~, ..., ~"+~). 322 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY xo7 In the special case j=n+~ one takes ~(s) ---- ~(a "+~ a ~ ..., a'd +~ .... , a "+t) if i < n + x, ~(s) = q~(a '*+x a "+~ a ~ ..., a") if i -~ n -k- x. Then one gets A(~#~) = Y, (-- ~)~+~(n+3--~i)+~ where, forn even, one has i=l +~ = ~,({o, i)) + ~({ ~, i + x }) + ... + ~({n + ~, i - ~ }), and for n odd +, = ~({o, i}) + ... + ~,({n + 2 - i, n + 2}) --~({n+2--i+I,o})...--~{n+2, i--x}. [] We shall end this section with the following proposition. One can show in general that, ff ~ e Z"(~, ~*) and ~b e Z'(~, ~?') are Hochschild cocycles, then r # + is still a Hochschild cocycle ~ # + e Z" + m(~ | M, ~* | M*) and that the corresponding product of cohomology classes is related to the product v of [I3], p. ~x6, by (n -t- m)! [r # +] -- n! m! [~] v [+]. Since a e Z~(C, C) is a Hochschild boundary one has: Proposition 12. -- For any cocycle ~ ~ Z[(~), S~ is a Hochschild cobounda~y: S~ ~ b+ where +(a ~ .. a "+1) = 2i~ Z (- ~)J~(a~ . daJ -~) ~(a~J +~ . aa")). "~ $--1 .... Proof. -- One checks that the coboundary of thej-th term in the sum defining ~ gives ~(a~ 1 ... da ~-1) a j a j ~l(daJ+~ ... da"+2)). [] 2. Pairing of H~(~/) with I~.(.~r i ---- o, Let d be a unital (non commutative) algebra and K0(.~r Kl(d ) its algebraic K-theory groups (of. [i6]). By definition K0(M ) is the group associated to the semi- group of stable isomorphism classes of finite projective modules over d. Also Kx(d) is the quotient of the group GL~(~r by its commutator subgroup, where GL~(~r is the inductive limit of the groups GL,(d) of invertible elements of M,(~'), under In this section we shall define by straightforward formulae a pairing between H[~(~) and Ko(~ ) and between H~"(d) and K~(d). The pairing satisfies (S~,e)= (~,e), for ~eH~(~), eeK(d) and hence is in fact defined on H*(d) ~- H[(~) | C. As a computational device we shall 323 xo8 ALAIN CONNES also formulate the pairing in terms of connexions and curvature as one does for the usual Chern character for smooth manifolds. This will show the Morita invariance of H~(d) and will give in the case d abelian, an action of the ring K0(d ) on H~(,~). Lemma l& -- Let 9eZ~(d) and p, qeProjM~(d) be two idempotents of the form p = uv, q = vu for some u, v e Mk(d ). Then the following cocycles on = {x e Mk(a~r xp ~- px ~- x} differ by a coboundary +l(a ~ ..., a") = (~ # Tr)(a ~ ..., a"), +~(a ~ ..., a") ---- (? # Tr) (va ~ ..., va"u). Proof. -- First, replacing d by Mk(d ) one may assume that k = I. Then one can replace p, q, u, v by , , , and hence assume the existence ['o o] [: o][: :l [: :1 o q of an invertible element U such that UpU- 1 = q, u ---- pU- 1 = U- a q, v = qU ----- Up [' ' "]) ta e U = . Then the result follows from proposition 5-I. [] v i--q Recall that an equivalent description of Ko(d) is as the abelian group associated to the semi-group of stable equivalence classes of idempotents e ~ Proj Mk(a~ ). Proposition 14. -- a) The following equality defines a bilinear pairing between K0(~t ) and H~V(,~): ([el, [~]) = (2i~) -m (m!) -x (~? # Tr) (e, ..., e) for e ~ Proj Mk(d ) an b) One has ([e], IS,] ) = ([e], [9] ). Proof. -- First if ~ r B~m(d), ~ ~ Tr is also a coboundary, , # Tr = b+ and 2rn hence (~#Tr)(e,...,e) =bt~(e,...,e)----- ~3 (-- I) ~+(6 ...,e) = +(e, ...,e) = o, i=0 since +x = _ ~b. This together with lemma 13 shows that (~ # Tr) (e, ..., e) only the result, one gets the additivity and hence a). 2m b) One has --:-Sq~(e, ..., e) =  ~(e(de) ~-1 e(de) "-j+1) and, since e~= e, one 21~ j=l has e(de) e = o, e(de) ~ = (de) ~ e, so that 2i~ S~(e, ..., 8) ---- (m + I) ~(e, ..., e). [-] We shall now describe the odd case. 324 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY xo9 Proposition 15. -- a) The following equality defines a bilinear pairing between K1(~r ) and Hx~ (,~r : < [~], b] > = (2i~)-" ~-(~'+') (m -- I/2) .., I/2 (~p#Tr)(u -1-I,u-x,u -1-1,...,u-t) where ~ ~ Z~'-l(d) and u ~ GLk(d ). b) One has < [u], [$9] > = < [u], [9] >. Proof. -- a) Let ~ be the algebra obtained from d by adjoining a unit. Since ~r is already unital, ,~r is isomorphic to the product of ~r by 13, by means of the homo- morphism p:(a,Z) ~(a+),I,k) of~to a~r215 Let ~eZ~(~) be defined by the equality '~ 0 ~((a, x~ ..., (a", ~")) = ~(a ~ ..., a"), V (a', X') ~ Let us check that b~' = o. For (a ~176 ..., (a,+1,),,+1) e ff one has ~ q~((a, o k~ ..., (a i, ),) (ai+t, Xi+ 1), --., (a,+l,)`,+ 1)) = 9(a ~ ia i+1, ...,a n+l ) +X ig(a ~ ...,a i-l,ai+ 1, ...,a "+1) 'd+z, a-+l) +)~+19( a~ ..., . ~ 0 . (a,+l,)`,+l)) Thus bq~((a , X~ .. , _ )`0 q~(a l, a,+l) + (_ i),-i )`0 q~(a,+l, .... -- . .. ~ a 1, a n) o. Now for u e GLi(.~r ) one has q~(U -1 -- 1, U -- I, ..., /Z -I -- I, U -- 1) = (7~ p--l)(~--I if, "'', ~--I, ~) where ~ = (u, I) e ~qr X 13. Thus to show that this function Z(u) satisfies )~(uv) = )~(u) + Z(v) for u, v e GLx(~), 9 a n- 1) a i one can assume that q~(I, a ~ = o for E ~r and replace Z by z(u) = ~(u -1, u, .., u -~, u). Now one has with U= [: o] , V= [; ;1 Z(uv) = (cp # Tr) (U -a, U, ..., U -1, U), Z(u) +Z(v) = (9#Tr)(V -1,v,...,v -l,v). Since U is connected to V by the smooth path lCOS t sin -- cos t sin tJ 325 IIO ALAIN CONNES it is enough to check that (? # Tr) (U~ -a, Ut, ..., Ut) = o. Using (U~-I) ' = -- U~ -1 U~ U~ -1 the desired equality follows easily. We have shown that the right hand side of 15 a) defines a homomorphism ofGLk(d ) to C. The compa- tibility with the inclusion GL k C GL k, is obvious. To show that the result is o if ? is a coboundary, one may assume that k = i, and, using the above argument, that ~ =b~ where 4/eC~ -1, d~(i,a ~ ...,a ~-~) =o for a ~. (One has b~= (b+) ~ for +eC~-l.) Then one gets b+(u -~, ...,u -1,u) =o. b) The proof is left to the reader. [] Definition 16. -- Let H'(~ 0 = H~(~r | Here H~(C), which by corollary IO I) is identified with a polynomial ring C[a], acts on C by P(a) ~ P(I). This homomorphism of H~(C) to C is the pairing given by proposition 14 with the generator of K0(C ) = Z. By construction H*(a~r is the inductive limit of the groups H~(d) under the map S : H~(.~ 0 -+ H~+~(~r or equivalently the quotient of H~(.~) by the equivalence relation ~0 ~ S% As such, it inherits a natural Z/2 grading and a filtration: F" H*(~g) = Im H~(d). We shall come back to this filtration in section 4. Corollary 17. -- One has a canonical pairing between H~ and Ko(~r and between H~d(d) and K~(d). The following notion will be important both in explicit computations of the above pairing (this is already clear in the case ~ = C~~ V a smooth manifold) and in the discussion of Morita equivalences. Definition 18. -- Let d ~ fl be a cycle over ~, and 8 a finite projective module over ~. Then a connexion V on @ is a linear map V : ~' --> g | ~1 such that V(~.x) = (V~) x q- ~ Q dp(x), V~e#, xed. Here o ~ is a right module over d and ~x is considered as a bimodule over d using the homomorphism ~ : z~ -> fl0 and the ring structure of ~*. Let us list a number of obvious properties: Proposition 19. ~ a) Let e e End~r be an idempotent and V a connexion on ~; then -> (e| i) V~ is a connexion on eg. b) Any finite projective module g admits a connexion. c) The space of connexions is an affine space over the vector space Hom~(g, g| ~1). 326 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY i i i d) Any connexion V extends uniquely to a linear map of ~ 8 | ~ into itself such that V(~| =(V~)o)+~| V~eg, o) e~. Proof. -- a) One multiplies the equality I8 by e | I (on the left). b) By a) one can assume that 8 = Ca| for some k. Then, with (~')~=1 ..... the canonical basis of g, put Note that, if k = i (for instance), then ~r| = O(i) ~1 and Va = p(I) dp(a) for any a e d since .~r is unital. This differs in general from d, even when p(I) is the unit of ~0. c) Immediate. d) By construction ~is the finite projective module over ~ induced by the homo- morphism p. The uniqueness statement is obvious since V~ is already defined for ~ e 8. The existence follows from the equality V(~a) o) + ~a | do) := (V~) ao) + ~ | d(ao)) for any ~eg, ae~r and o) e~'l. [] We shall now construct a cycle over End,c(g). We start with the graded algebra Enda( ~ (where T is of degree k if Tg ~ C g"j+k for all j). For any T e Enda( ~ of degree k we let ~(T) --~VT-- (-- I) kTV. By the equality d) one gets V(~O)) = (V~) O) -~- (-- I) deg ~ do) for ~ e ~, o) e ~, and hence that 8(T) e Enda(~, and is of degree k + I. By construction 8 is a graded derivation of Endo( ~. Next, since o~is a finite projective module, the graded trace ,[: ~"--)-C defines a trace, which we shall still denote by f, on the graded algebra Enda(~. nmma zo. - one f = o for T F.nd.( Y) of degree n- ,. Proof. -- First, if we replace the connexion V by V'= V + F, where F e Hom~,(@, g| ~), the corresponding extension to ~ is V' : V + ~, where eEnda(g ~ and is of degree i. Thus it is enough to prove the lemma for some connexion on g. Hence we can assume that g = e.~ k for some e eProj Mk(.~ ) and that V is given by 19 a) from a connexion V 0 on ~r Then using the equality 8(T) ---=e 80(T) e for T e End ,~C End ~0 (go = .~tk), as well as ~o(T) = 8o(eTe) = ~o(e) T + 8(T) + (-- i) ~ T bo(e), one is reduced to the case g = ~r with V given by 19 b). Let us end the computation say with k = I. Let e ---- 9(I). One has ~= eft, Endn(~) := e~2e, 8(a) = e(da) e. Thus f S(a) = f (a(e e) - (de) a--(-- I)~176 ade) =o. [] 327 II2 ALAIN CONNES Now we do not yet have a cycle over End~,(g) by taking the obvious homomorphism of Endd(d ~ in Endn(~), the differential ~ and the integral . In fact the crucial - f property 82= o is not satisfied: Proposition 21. -- a) The map 0 = V ~ of ~ to ~ is an endomorphism: 0 ~ Endn(~'~ and 8~(T) =0T--T0 for all T~Endn(g ). b) One has ( [d~], [v] ) = ~.~ If (0/~) "m , when n is even, n = era, where [~] e K0(.~ ) is the class of ~', and .r is the character of f~. Proof. -- a) One uses the rules V(~) = (V'~) co + (-- I) "~ ~ d~ and d' = o to check that V~(~) = V~(~) ~. b) Let us show that f0 '~ is independent of the choice of the connexion V. The result is then easily checked by taking on d ~ = e~ ~ the connexion of proposition ~ 9- Thus let V' = V + P where r is an endomorphism of degree ~ of ~. It is enough check that the derivative of f 0~ is o where 0 t corresponds to V~ = V + tP. Also to it is enough to do it for t=o. We get: dldt 07= 0, 0, k=0 f -'f As (dot) :FV+VF=~(I')one has \a~ l 1=o 0 m mfS(o "-1 = o. [] f = ") Thus, while M * o, there exists 0 ~ ~' = Endn(~ ) such that M(T)-----0T--T0, VTE~)'. We shall now construct a cycle from the quadruple (~', 8, 0, f). Lemma 22.- Let (~1', 8, O, f)bea quadruple such that ~' is a graded algebra, ~ a graded derh~ation of degree I of ~' and 0 ~ ~'~ satisfies 8(0) = o and 82(~) = 0o~ -- ~o0 for o~ ~ ~'. Then one constructs canonically a cycle by adjoining to ~' an element X of degree I with dX = o, such that X ~ = 0, ~l Xc~ = ~ V ~ ~ f~'. Proof. -- Let ~1" be the graded algebra obtained by adjoining X. Any element of ~" has the form co = on + wl, X + Xo21 + Xc0,, X, co~j ~ 9V. Thus, as a vector space, t)" coincides with M2(~'), the product is such that 328 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY If3 and the grading is obtained by considering X as an element of degree i; thus [o)ij] is of degree k when con is of degree k, oh2 and co~1 of degree k -- i and co** of degree k -- 2. One checks easily that ~" is a graded algebra containing ~'. The differential d is given by the c6nditions do) : 8(r + Xo) --(-- I) deg~ o)X for ca e~'C ~", and dX=o. One gets to) l -- -- J + O)21 r J _ o0 'o]. One checks that the two terms on the right define graded derivations of f~" and that d2=o. Finally one checks that the equality defines a closed graded trace. [] Putting together proposition 2I a) and lemma 22 we get: Corollary 23. -- Let d ~ ~ be a cycle over d, 8 a finite projective module over M' and ~' = Endue(g). To each connexion V on ~ corresponds canonically a cycle dt' P-~ ~' over d/'. One can show that the character 1:'~ Z~(d') of this new cycle has a class [x'] ~ H~(d') independent of the choice of the connexion 7, which coincides with the class given by lemma 13. One can then easily check a reciprocity formula which takes care of the Morita equivalence. Corollary 24. -- Let d, ~ be unital algebras and ~ an d, ~ bimodule, finite projective on both sides, with d = EndB(g), 5~ = End~(g). Then H~(d) is canonically isomorphic to H~(~). Finally when d is abelian, and one is given a finite projective module 8 over ~, then one has an obvious homomorphism of ~r to d' = End~,(8). Thus in this case, by restriction to d of the cycle of corollary 23 one gets: Corollary 25. -- When d is abelian, H~(d) is in a natural manner a module over the ring Ko(~ ). To give some meaning to this statement we shall compute an example. We let V be a compact oriented smooth manifold. Let ,~f = C~176 and ~ be the cycle over given by the ordinary de Rham complex and integration of forms of degree n. Let E be a complex vector bundle over V and 8 = Coo(V, E) the corresponding finite 15 Ix 4 ALAIN CONNES projective module over ~r = C~176 Then the notion of connexion given by deft- nition I8 coincides with the usual notion. Thus corollary 25 yields a new cocycle 1: e Z[(d), ~r = C~~ canonically associated to V. We shall leave as an exercise the following proposition. Proposition 26. -- Let co k be the differential form of degree ~k on V which gives the component of degree 2k of the Chern character of the bundle E with connexion V : ~k ---- i/h! Trace ~ , where 0 is the curvature form ([x 7]). Then one has the equality where "~ ~ Z[-~(,ff) is given by ~ o f ^.. ^ v/ed=C (V), ok(f, " = fO dfl . df.-~ and where .c is the restriction to d = Coo(V) of the character of the cycle associated to the bundle E, the connexion V, and the de Rham cycle of ~t by corollary 23. 3. Cobordism of cycles and the operator B By achain of dimension n +I we shall mean a triple (fl, 0h, f) where ~ and 9~2 are differential graded algebras of dimensions n + i and n with a given surjective morphism r : ~ ~ 0~ of degree o, and where f : L2" + 1 ___> C is a graded trace such that fdr Vcoe~" such that r(r =o. By the boundary of such a chain we mean the cycle 0~, where for o~' e (0~)" one (f') takes co'= dr for any r e~n with r(o~) -----o~. One easily checks, using the ; f surjectivity of r, that f' is a graded trace on O~ which is closed by construction. Definition 27. -- Let d be an algebra, and let d o_~ ~, ~ p~ ~, be two cycles over d (el. proposition I). We shall say that these cycles are cobordant (over d) if there exists a chain ~" with boundary ~ @ ' (where ~' is obtained from ~' by changing the sign of f) and a homo- morphism p" : ~ --> ~" such that r o p" ~ (p, 0'). Using a fibered product of algebras one checks that the relation of cobordism is transitive. It is obviously symmetric. Let us check that any cycle over ~r is cobordant to itself. Let ~0 _-- Coo([o, i]), ~1 be the space of C ~ x-forms on [o, I], and d be the usual differential. Set 0~ = C | C and take f to be the usual integral. Then taking for r the restriction of functions to the boundary, one gets a chain of dimension i with boundary (C| r ~(a, b) = a-- b. Tensoring a given cycle over ~ by the above chain gives the desired cobordism. 330 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY ~5 Thus cobordism is an equivalence relation. The main result of this section is a precise description of its meaning for the characters of the cycles. We shall assume throughout that the algebra .~r is unital. Lemma 28. ~ Let xt, % be the characters of two cobordant cycles over a~t. Then there exists a Hochschild cocycle 9 e Z"+a(d, ..r such that v 1 -- % ----- B o % where (Bo ~)(a ~ ..., a") = ~(i, a ~ ..., a") -- (-- ~)"+~ ~(ao, ..., a", I). Proof. -- With the notation of definition 27, let q~(a ~ ..., a "+1) -= fp"(a ~ dp"(a 1) ... dp"(a"+l), V a i e d. Let r = p"(a ~ dp"(a t) ... dp"(a n) e n"n. Then by hypothesis one has (~t - ~)(~0, ~, ..., ~,) = f d,o. Since p"(a ~ ~-- p"(I) p"(a ~ one has do.) = (d{9"(I)) [~"(a O) d~"(G 1) ,,, d[~"(a n) + ~"(I) d[:}H(a O) ,,, d~}H(an), Using the tracial property of f one gets fdo~ = (-- I) n q)(a 0, a x, ..., a", I) + r a ~ ..., a"). the tracial property of f one checks that q~ is a Hoschchild cocycle. [] Using again Lemma 29. ~ Let -cl, v 2ez~(d) and assume that Vl--V2~--Boq~ for some e Z"+I(~ r ~'). Then any two cycles over ~ with characters "~x, "~ are cobordant. Proof. ~ Let ~r ~ ~ be a cycle over d with character -~. Let us first show that it is eobordant with (~(d), ?). In the above cobordism of ~ with itself, with restriction maps ro, q, we can consider the subalgebra defined by rx(r ) e ~', where ~' is the graded differential subalgebra of ~ generated by p(d). This defines a cobor- dism of ~ with ~'. Now the homomorphism "~ : ~2(d) -+ ~' is surjective, and satisfies ~'*f =-~. Thus one can modify the restriction map in the canonical cobordism of (~(d), -2) with itself to get a cobordism of (~2(a~), "~) with ~'. Let us show that (~(,.qg), "~a) and (~(~), ~) are cobordant. Let ~ be the linear functional on ~"+l(d) defined by I) ~(G 0 da 1 ... da "+1) ---- ~(a ~ ..., an+l), 2) ~.(da 1 ... da "+i) -- (Bo ~) (a 1, ..., a"+l). Let us check that ~ is a graded trace on ~(d). We already know by the Hoch- schild cocycle property of q~ that ~(a(bo~)) = ~((br a), V a, b e d, r e ~,+l. 331 xx6 ALAIN CONNES Let us check that ~t(am) = ~t(o~a) for o~ = da 1 ... da TM. The right side gives n+l ~z( Z (-- I)"+l-Jda ~ ... d(a~a ~+1) ... da'~+lda) + (-- I) "+' ~(a 1 daZ.., da) n+l = Y. (-- l)n+a-~(Bo ~)(a 1, ..., a~a ~+I, ..., a "+1, a) =1 + (-- 1) TM q~(a 1, a z, ..., a "+1, a) -- a n + ~', a). -- (-- I)" ((b' B 0 r -- ~0) (a 1, a s, ..., Now one checks that for an arbitrary cochain $ 9 C"+1(~r ~qr one has Bob~ + b, Bo 9 = 9 _ (_ ~),+1 9x, where X is the cyclic permutation X(i) = i- i. Here q~ is a cocycle, bq~ = o and b' B 0 q~ -- q~ ----- (-- I) n q~x so that v@oa) ---= q~(a, a l, ..., a "+1) = ~(ao~). It remains to check that for any a 9 d and co ~ ~" one has ~((da) ~) = (-- I) n ~(co da). For ~ 9 df~"-1 this follows from the fact that B o ~ 9 C~ (recall that B 0 q~ = x 1 -- -@. For co = a ~ da 1 ... da" it is a consequence of the cocycle property of B o ~p. Indeed one has bB 0 ~ =- o, hence b' B o 9(a ~ a 1, ..., a", a) = (-- ~)" B o r ~ a t, ..., a") and since b'B oq~=q~-(-I) "+lq~x we get ~(a ~ ..., a", a) -- (-- I) "+1 q~(a, a ~ ..., a") = (-- I)"+l(Boc?)(aa ~ a 1, ..., a"), i.e. that ~((da) a ~ da 1 ... da") = (-- I)" vt(a ~ da 1 ... da n da). To end the proofoflemma 29 one modifies the natural cobordism between (~(d), ~1) and itself, given by the tensor product of~(d) by the algebra of differential forms on [o, I], by adding to the integral tile term ~ o rx, where r 1 is the restriction map to { I } C [o, I]. [] Putting together lemmas 28 and 29 we see that two cocycles vx, x2 9 Z~(d) correspond to cobordant cycles if and only if ~1- "~ belongs to the subspace Z~(d) n B0(Z"+l(d, ~')). We shall now work out a better description of this subspace. Since Az ---- (n + i) x for any x ~ C[(~), where A is the operator of cyclic antisymmetrisation, the above subspace is clearly contained in the subspace Z"(d) c~ B(Z"+I(~r ~r where B = ABo: C "+1 ~ C". Lemma 30. -- a) One has bB=--Bb. b) One has Z[(d) c~ Bo(Z"+l(d, d*)) = BZ"+I(d, a"). Proof. -- a) For any cochain 9 e C"+l(d, M*), one has B 0 b~ + b' B 0 q~ = ~ -- (-- I) n+l q~X, 332 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY xI7 where ~ is the cyclic permutation X(i) = i- I. Applying A to both sides gives AB o br + Ab' B o ~ = o. Thus the answer follows from lemma 3 of section I. b) By a) one has BZ"+I(~ r d*.)C Z~,(d). Let us show that BZn+l(,r d*) C Bo Zn+l(d, ~r Let ~eBZn+l(d,d*), so that ~3=Bq~, r We shall construct in a canonical way a cochain + e Cn(d, d*) such that I I --~-~Bo(~o--b+). Let 0-=B or --~. By hypothesis A0=o. Thus there n+I n+i exists a canonical ~b such that ~b -- r +x = O, where X is the generator of the group of cyclic permutations of {o, I, ...,n}, X(i) = i--I. We just have to check the equality B0 b~b = 0. Using the equality B o b+ + b' B o ~b = + -- r qx, we just have to show that b' B o + = o. One has ..... , a n-l) ..., a n-l, x) Bo +(a ~ ., a "-l) ___= +(,, a ~ -- (-- i) n +(a ~ = (__ i)n--1 (+__ ~(~.) +X) (aO, ...,an-l, I) = (-- I) n-10(a 0, ...,a n-1 , I) = (_ ~)n-1 (~(~, a o, ..., an-l, ~) _ (_ ~)n+l ~(a o, ..., an-l, ~, ~)) -3 l- ~ (-- I) n ~(a 0, ..., a n-1 , I). The contribution of the first two terms to b' B o +(a ~ ..., a n) is n--1 (-- I) n-1 2~ (-- I) ~ (r a ~ ..., aJa j+l, ..., a n, i) j=o + (_ ~)n ~(a o, ..., aJ a ~+l, ..., a n, i, i)) = (- I) n (b~0, a ~ ..., a n, ~) - ~(a ~ ..., a", ~)) -- (be~(a O, ..., a n , I, I) -- (-- I) n r O, ..., a n , I)) = 0 since b~? = o. The contribution of the second term is proportional to n--1 2~ (-- I)J~(a ~ ..., aJa j+t, ..., a n, I) ---- b~(a ~ ..., a n, I) = O. [] j=0 Corollary 31. -- x) The image of B:Cn+l-+ C" is exactly C~. 2) B~(d) c Bo zn+l(~r ~r Proof. -- I) =~ 2) since, assuming i), any b% ~ e C[ +l is of the form bB+ ---- -- Bbd/ and hence belongs to BZn+l(~r d*) so that the conclusion follows from b). To prove i) let q~ e C[. Choose a linear functional ~0 0 on d with ~0(I) = i, and then let +(a ~ ..., a "+~) = ~o(a ~ ~(a l, ..., a n+l) + (- ~)" ~((a ~ -- ~o(a ~ ~), a l, ..., a n) ~o(a" +l). 333 ~,8 ALAIN CONNES One has +(~,a ~ =r ~ ...,a") and q,(a ~ ..., a", ~) = v0(~ ~ ~(a ~, ..., a", ~) + (-- ~)" v(a ~ ..., a") + (-- ~)"+~ ~0(a ~ ~(~, a ~, ..., a") = (-- ~)" ~(a ~ ..., a"). Thus B 0+--2r and (p~ImB. [] We are now ready to state the main result of this section. By lemma 4 a) one has a well-defined map B from the Hochschild cohomology group H"+~(~r ', ~r to H[(~). Theorem 32. -- Two cycles over .~t are cobordant if and only if their characters za, z~ e H~(d) differ by an element of the image of B, where B: H"+X(~ r ..~'*) -+ H[(~r It is clear that the direct sum of two cycles over ~/is still a cycle over ~r and that cobordism classes of cycles over d form a group M*(~/). The tensor product of cycles gives a natural map: M*(~q/) � M*(~)-->M*(~/| Since M*(C) is equal to H~(C) : C[~] as a ring, each of the groups M*(~') is a C[(~] module and in particular a vector space. By theorem 32 this vector space is H~,(~r B. The same group M*(~) has a closely related interpretation in terms of graded traces on the differential algebra ~(d) of proposition i. Recall that, by proposition i, the map ": ~ ? is an isomorphism of Z~(d) with the space of closed graded traces of degree n on ~(d). Theorem 33. -- The map v ~ ? gives an isomorphism of H~(~)/Im B with the quotient of the space of closed graded traces of degree n on Q(~r by those of the form d t tL, ~ a graded trace on ~(~t) (of degree n + I). Proof. -- We have to show that, given ": e Z[(d), one has ? = d ~ V for some graded trace ~ if and only if .reImBDB~,. Assume first that -~=d t~. Then as in lemma 28, one gets -~----B 0 cp where (p e Z"+1(d, d*) is the Hochschild cocycle ~(a ~ 1, ...,a "+1) : ~(a ~ da 1... da"+l), V a ~ e ~r Thus v---- AB 0~EImB. n-j-I Conversely, if n:eImB, then by lemma 30 b) one has ~:B 0T for some ~0 e Z"+l(d, d*). Defining the linear functional ~z on ~"+l(d) as in lemma 29 we get a graded trace such that ~(da ~ da 1... da") ----'~(a ~ ...,a"), Vaie i.e. ~(a~) = ~(~), v ~ ~ ~"(d). [] Thus M*(d) is the homology of the complex of graded traces on ~)(d) with the diffe- rential d t. This theory is dual to the theory obtained as the cohomology of the quotient of the complex (~(d), d) by the subcomplex of commutators. The latter appears 433 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY zI 9 independently in the work of M. Karoubi [39] as a natural range for the higher (]hem character defined on all the O uillen algebraic K-theory groups K~.(~r Thus theorem 33 (and the analogous dual statement) allows: I) to apply Karoubi's results [39] to extend the pairing of section 2 to all K~(~r 2) to apply the results of section 4 (below) to compute the cohomology of the complex (~(~')/[ , ], d). 4- The exact couple relat/ng H~(d) to Hochschild cohomology By construction the complex (C~(d), b) is a subcomplex of the Hochschild complex (C"(d, d*), b), i.e. the identity map I is a morphism of complexes and gives an exact sequence: o -+ c~ ~ c- -+ c"/c~ -~ o. To this exact sequence corresponds a long exact sequence of cohomology groups. We shall prove in this section that the cohomology of the complex C/C~ is H"(C/C 0 = H"-I(C~). Thus the long exact sequence of the above triple will take the form o ~ H~ I , -> H~ d') -> H;1(~r --> Hk(d) -> S (d, d') -+ H~ -+ n~(d) -+ ... H"(d) i H"(d, ~g*) -+ H~-'(zr -+ H~+'(~r ~ H"+~(g, d') -+ ... On the other hand we have already constructed morphisms of cochain complexes S and B which have precisely the right degrees: S : H~-~(d) -+ H~+'(M), B: H"(d, d*) -+ H~-I(~). We shall prove that these are exactly the maps involved in the above long exact sequence, which now takes the form B I H~(d) ~ H"(d, M*) -+ H~-I(.~g) L H~+I(,~) ---> ... Finally to the pair b, B corresponds a double complex as follows: C n,m = C"-'(d, d*) (i.e. (]"'' is o above the main diagonal) where the first diffe- rential dl:C~'m-+ (],+l.m is given by the Hochschild coboundary b and the second differential d2: (]~"~-+ C n'm+l is given by the operator B. By lemma 3 ~ of section 3 one has the graded commutation of dl, d~. Also one checks that B2= o so that d~ = o. By construction the cohomology of this double complex depends only upon the parity of n and we shall prove that the sum of the even and odd groups is canonically isomorphic with H~(d) (~) C =- H'(,~') H~lC) (where H:(C) acts on C by evaluation at : := i). 335 I oO ALAIN CONNES The second filtration of this double complex (F q = Y, C"") yields the same m>q filtration of H*(d) as the filtration by dimensions of cycles. The associated spectral sequence is convergent and coincides with the spectral sequence coming from the above exact couple. All these results are based on the next two lemmas. Lemma 34. -- Let + eC"(d, ~*) be such that b+ eC[+l(~). Then B~b eZ~-l(~) and SB+ = 2i~zn(n + I) b~b in H~,+l(d). Proof.- One has B+ e C~ -1 by construction, and bB+-------Bb~b-----o since b+eC~ +1. Thus B+eZ~ -1. In the same way b+eZ[ +1. Let 0----B+, by proposition x2 of section I one has SO = b+' where ~b'(a~ "'" an) = j-1 ...... (- i) j-1 ~(a~ d~J-') aJ(d~ +1 d*)). It remains to show that +' -- e(X) +'x = n(n + i)(+" -- ~(X) +"x) where X(i) =i-- i for ie{o,x,...,n+ i} and ~b"--~beB". Let us first check that (+' - ~(x) +'~) (a ~ ..., *) = (- i) "-1 (n + ~) o(a" a ~ al, ..., a,-1). One has n-1 ~b'X(a ~ ..., a") =  (-- I) j ~((da ~ ... da ~-l) aJ(da j+l ... da n-1 ) a"). j=0 Let % = a~ 1 ... da i-i) a~(da ~+1 ... da "-1) a". Then d% = (da ~ ... da j-l) aJ(da ~+1 ... da "-1) a" + (-- I) j-la~ I ... da "~ ... da "-1) a" -~ (-- I) n a~ I ... da ~-1) ai(da ~+1 ... da"). Thus for je{i,...,n-- i} one has (-- I) j-1 ~(a~ ... da j-l) a~(da~+l ... da")) -- E(~)(-- I) j ~((da 0 . ,. aa j-l) aJ(da j+l ,.. da n-l) (l n ) = (- I) "-1 o(a" a ~ a 1, ..., a"-l). Taking into account the cases j -- o and j = n gives the desired result. Let us now determine ~b", ~b" -- ~b e B"(~, ~*) such that (+" ~(x) +"~)(a ~ ...,a") (- I)"-1 __ -- 0(a" a ~ ..., a"-l). Let 0----B 0~b and write 0=01+03 with A01=o, 03eC~-l(ad) so that 03=-0. Since A01 = o there exists ~b I e C "-1 such that 01 = D~b 1 where D~b I = +1 -- r ~bl x. 336 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY I2I Parallel to lemma 3 of section ~ one checks that D o b ---- b' o D and hence D(b+l ) = b' 0 i. Let t~"=+--b+i. As D=B 0b+b'B 0 we get Dt~=b'B 0t~=b'0i+b'0, hence I b' Dd/'----b'0~.=n ~" Finally since bq~----o one has b'~ = (-- ~)"-l~?(a"a~ ...,a"-~). [] As an immediate application of this lemma we get: Corollary 86. -- The image of S:H~-~(~) -+H~+t(~) is the kernel of the map : I-I +l(se) This is a really useful criterion for deciding when a given cocycle is a cup product by a e H[(C), a question which arose naturally in part I. In particular it shows that ifV is a compact manifold of dimension m and if we take d =- C~(V), any cocycle -~ in H~(d) (satisfying the obvious continuity requirements (of. Section 5)) is in the image of S for n>m-~dimV. Let us now prove the second important lemma: Lemma 36. -- The obvious map from (ImB n Ker b ) [b (Im B) to (KerB n Ker b ) /b (Ker B) is bijective. Proof. -- Let us show the injectivity. Let ~ e Im B c~ Ker b, say ~ s Z~+l(M), and assume ~ eb(Ker B). Then the above lemma shows that q~ and So = o are in the same class in H~+I(M) and hence 9 ~ b(Im B). Let us show the surjectivity. Let ? e Z"+l(d, d*), Bq~ = o and + e C"(,~/, d,), -- e(X) d/x -= B o q0. As in the proof of lemma 3 ~ of section 3 one gets B 0 b+ = B o ~. This shows that 9' ---- q0--b+eZ~(,~) since D 9'--B 0b 9' +b'B 09' =o. Let us show that B+ebG~ -2. Since t~--~(X)+X=B0b~b one has b'B o+-=o. One checks easily that b '* = o and that the b' cohomology on C"(~, ,~*) is trivial (if b' 91 = o one has b'~l(a ~ ",I) ~--o i.e. ~i=b'~?2 where ~ ..., a "-1) = (-- I) ~ ..., a "-1, Thus B 0+=b'0 for some 0eG "-~ and B~=Ab'0=bA0ebG~ -~. Thus since G~ -2 = ImB one has BO/----= bB01 for some 01 eG "-I i.e. +b0 ieKerB and b+eb(KerB). As ~--bd/eZ~ this ends the proof of the surjectivity. [] Putting together the above lemmas 34, 36 we arrive at an expression of S:H~-I(~) ~H~+l(d) involving b and B: S = ~ir~n(n + I) bB -1. 16 122 ALAIN CONNES More explicitly, given ~ eZ~-l(d) one has ~ elmB, thus ~ = B+ for some +, and this determines uniquely b+ e (Ker b n Ker B)/b(Ker B) = H[+I(~). To cheek I I that b+ is equal to 2ire n(n + i~ S~ one chooses + as in proposition i2: ~(a~ .... ' an) = n(n -~ I~) $~1 ~(a~ "" " daJ-1) a'?(daJ+l " " " dan))" As an immediate corollary we get: Theorem 37. -- The following triangle is exact: H'(d, ~*) HI(~ r > H~(d) Proof. ~ We have already seen that Im S-~ Ker I. By the above description ofS onc has KerS :ImB. Next BoI =o since B is equal to o on C x. Finally if q~eZ'(~r and Bq~GB[ -1, B~:bB{} for some {}~C "-1 so that +b0eKerBnKerbCImI+b(KerB) by lemma 36 . Thus KerB=ImI. [] We shall now identify the long exact sequence given by theorem 37 with the one derived from the exact sequence of complexes o -+Cx -+C -~ C/Cx -+o. Corollary 38. -- The morphism of complexes B:C/Cx ~ C induces an isomorphism of H"(C/Cx) with H[-l(d) and identifies the above triangle with the long exact sequence derived from the exact sequence of complexes o --> Cx -+ C ~ C/Cx -+ o. Proof. ~ This follows from the five lemma applied to H"(Cx) > H"(C) , >, H"(a/az) > H"+I(Cx) > H"+~(C) H[(,~) : s > H"(d,d*) 13> H[-'(~) > H~+~(~r > H"+~(s], d*) [] Together with theorem 32 of section 3 we get: Corollary 39. ~ a) Two cycles with characters vl, v2 are cobordant if and only if Sv 1 = Sv 2 in H~(~r b) One has a canonical isomorphism M*(~r | C = H*(~r (of. definition ~6). Z~*(c) e) Under that isomorphism the canonical filtration F" H*(~r corresponds to the filtration of the left side by the dimension of the cycles. 338 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY xo 3 Proof of b). -- Both sides are identical with the inductive limit of the system (H~(~r S). [] Let us now carefully define the double complex C as follows: a) C"" = C"-'(.~, ~*), V n, m E Z; b) for q~eC"", d x~= (n--m+ I) bq~EC"+"m; c) for oEC"'", d~-- --Bq~C "'"+1 (if n=m, the latter is o). n--m Note that dld,=--d2dl follows from Bb =--bB. Theorem 40. -- a) The initial term E 2 of the spectral sequence asociated to the first filtration F vC =  C "''~ is equal to o. n~p b) Let FqC = ~] C"'' be the second filtration, then HP(FqC) = H~(~r for ra>_q n =p-- 2q. e) The cohomology of the double complex C is given by H"(C) = HeV(.~ r if n is even and H"(C) = H~ r if n is odd. d) The spectral sequence associated to the second filtration is convergent: it converges to the associated graded Y,F q H'(~)/F q+x H*(d) and it coincides with the spectral sequence associated with the exact couple. In particular its initial term E 2 is Ker(I o B)/Im(I o B). Proof. -- a) Let us consider the exact sequence of complexes of cochains o -+ Im B ~ Ker B -+ Ker B/Ira B --~ o where the coboundary is b. By lemma 3 6 the first map: Im B ~ Ker B becomes an isomorphism in cohomology, thus the b cohomo- logy of the complex Ker B/Im B is o. b) Let q~(F qC) v= Y~ C"'', satisfy dq~=o, where d=dl+d2. m>>q,n+m~p By a) it is cohomologous in F qC to an element d/ of C p- q'~. Then d~=o means q~ ~ Ker b c~ Ker B, and ~ ~ Im d means ~ E b(Ker B). Thus using the isomorphism (Ker b n Ker B)/b(Ker B) = H~,-2~(~/) (lemma 3 6) one gets the result. c) By the above computation of S as dl d2 -x we see that the map from HP(F q C) to HP(F q-1 C) is the map S from HxP-2q(~a r to H~-2q-'2(d) ; thus the answer is immediate. d) The convergence of the spectral sequence is obvious, since C'"~= o for m > n. Since the filtration of H"(C) given by H"(F q C) coincides with the natural 339 ALAIN CONNES ~2~t filtration of H*(off) (cf. the proof of c)), the limit of the spectral sequence is the associated graded Z F q H~V(off)/Fq+l HeY(off) for n even, F qH~ q+l H~ for n odd. It is clear that the initial term E 2 is Ker I o B/Im I o B. One then checks that it coincides with the spectral sequence of the exact couple 9 [] We shall end this section with several remarks. Remarks. ~ a) Relative theory. Since the cohomology theory Hi(d ) is defined from the cohomology of a complex (C~, b), it is easy to develop a relative theory HI(Off, ~), for pairs off -~ N of algebras, where r~ is a surjective homomorphism. To the exact sequence of complexes o -~ c~(~) --, ~* c-(d) + c-(off, ~) = c-(off)/c-(~) + o corresponds a long exact sequence of cohomology groups. Using the five lemma, the results of this section on the absolute groups extend easily to the relative groups, provided that one also extends the Hochschild theory H*(off, off*) to the relative case. b) Action 0fH*(off, off). Using the product v of [I3] H"(off,./t'l) | H"(off,./t'2) ~ H"+m(off,~r | one sees that H*(off, d) becomes a graded commutative algebra (using off | off = off, as d bimodules) which acts on H*(Off, d ~ (since off Qd d*= d*). In particular any derivation 8 of d defines an element [8] of Hi(Off, off). The explicit formula of [i 3] for the product v would give, at the cochain level 9 .., a TM) . . (~ v 8) (a ~ a', = ~(~(a "+') a ~ a', ., a"), V ~ ~ Z"(Off, off'). One checks that at the level of cohomology classes it coincides with (~ # 8) (a ~ a', ..., a "+') I n+l - -- Z (- i)~ ~(a~ aa j-~) ~(aJ)(aa~+~... ga"+~)), n+Ij=x v <p ~ Z"(off, off*). With the latter formula one checks the equality 8" ~ ---- (I o B) (8 v G) q- 8 v ((I o B) ~) in H"+I(off, off*,) ft (where ~*v(a ~ =~q~(a ~ for all a ~eoff). This is the natural extension of the basic formula of differential geometry Ox = dix + ix d, expres- sing the Lie derivative with respect to a vector field X on a manifold. 340 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY ~5 c) Homotopy invariance of H*(~). Let d be an algebra (with unit), ~ a locally convex topological algebra and ~ e Z[(~) a continuous cocycle (of. section 5). Let Pt, t e [o, i], be a family of homomorphisms Pt : ~r -+ ~ such that for all ae~, the map te[o,I] ~pt(a) e~ is of class C x. Then the images by S of the cocycles p~ ~ and p~ ~ coincide. To prove this one extends the Hochschild cocycle ~ # ~ on ~ | C~([o, i]) giving the cobordism of ~ with itself (i.e. +(fo, fl)_= f:fo dr1, Vf' eCl([o, x])) to a Hochschild cocycle on the algebra Cl([o, I], ~) of Cl-maps from [o, i] to ~. Then the map p : ~/--->C1([o, I], ~), (p(a))t = p~(a), defines a chain over ~ and is a cobordism of p~ ~ with p~ ~. This shows that if one restricts to continuous cocycles, one has p~ = p; : H*(~) ---> H*(d). 5" Locally convex algebras Before we begin with the examples we shall briefly indicate how sections x to 4 adapt to a topological situation. Thus we shall assume now that the algebra d is endowed with a locally convex topology, for which the product d � d ---> d is conti- nuous. In other words, for any continuous seminorm p on ~ there exists a continuous seminormp' such that p(ab) ~ p'(a) p'(b), V a, b e ~/. Then we replace the algebraic dual d* of d by the topological dual, and the space C"(d, d*) of (n + x)-linear functionals on ~ by the space of continuous (n + 1)-linear functionals: ~ e C" if and only if for some continuous seminorm p on d one has Ir ~ ..., a") [ ~_ p(a ~ ... p(a"), V d e s/. Since the product is continuous one has b~ e C "+1, V c? e C". Since the formulae for the cup product of cochains only involve the product in d they still make sense for continuous multilinear functions and all the results of sections i to 4 apply with no change. There is however an important point which we wish to discuss: the use of reso- lutions in the computation of the Hochschild cohomology. Note first that we may as well assume that ~r is complete, since C" is unaffected if one replaces d by its comple- tion, which is still a locally convex topological algebra. Let ~ be a complete locally convex topological algebra. By a topological module over ~ we mean a locally convex vector space .//g, which is a ~-module, and is such that the map (b, ~) ~ b~ is continuous (from ~ � ~r to ~). We say that .~r is topologically projective if it is a direct summand of a topological module of the form ~' ~- ~ 6,= E, where E is a complete locally convex vector space and ~. means the projective tensor product ([29]). In particular ~ is complete, as a closed subspace of the complete locally convex vector space r 341 ALA[N CONNES It is dear then that if~t' a and .,/g= arc topological N-modules which are complete (as locally convex vector spaces) and p : d( x ~ ~= is a continuous ~-linear map with a continuous (;-Linear cross-section s, one can complete the triangle of continuous N-linear maps T .."" 1 l- for any continuous N-linear map f: .1(4 ~ ..r 2. Definition 42. -- Let ..14 be a topological N-module. By a (topological) projective resolution of .K we mean an exaa sequence of projective N-modules aM N-lincar continuous maps which admits a C-linear ~ontinuous homotopy s~ :~/t~->~+i b,+ts,+s~_lb~=id, Vi. As in [36] the module ~r over ~ = d ~,~ ~ (tensor product of the algebra .~r by the opposite atgebra z~ r given by (a | b ~ c = acb, a, b, c ~ M admits the following canonical projective resolution: i) .At = N~.E. (as a N-module), with E. = zr ... @,,M" (n factors); 2) r162 is given by z(a| ~ =ab, a,b ~t; 3) b.(I|174174 = (a~|174174174 n--1 + Y, (-- I) ":I |174 ... | .-- | j=l + (-,)" | a ~ | (al | | a._ ,). The usual section is obviously continuous: Sn((a~b ~174174174 ) = (I| ~174174174174 ) . Comparing this resolution with an arbitrary topological projective resolution of the module ~r over ~ yields: Lemma 43. -- For any topological projective resolution (~r b.) of the module z~ over N = d @~ ,~r the Itochschild cohomology H"(~r zr coincides with the cohomology of the complex -+ Hom~,(..lt a, .~*) -+ ... Homa(..//~ ..~') ~: (where Hom~ means continuous N linear maps). Of course, this Iemma extend~ to any c~mplete topological b~module over aa/. Let us now pas~ to the examples. 342 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY ~7 6. Examples i) d = Coo(V), V a compact smooth manifold. We endow C~176 with its usual Frechet space topology, defined by the semi- norms sup [O'f[ =p,(f) using local charts in V. I~l_<_, As a locally convex space, Coo(V) is then nuclear ([29]) and one has coo(v) c o (v) = coo (v x v). Thus ~ = ~r ~,, ~10 is canonically isomorphic to C=(V � V) and the module over ~ corresponds to the diagonal A: Vfe Coo(V � V), e(f) ----- A*f. Let us assume for a while that the Euler characteristic of V vanishes. The general case will be treated by crossing V with S 1. Let E k be the complex vector bundle on V � V which is the pull back by the second projection pr, : V � V -+ V of the exterior power A * T;(V) of the complexified cotangent bundle of V. By construction, the dual E~ of E 1 is the pull back by pr, of the complexified tangent bundle. We let X(a, b) be a section of E~ such that: a) for (a, b) close enough to the diagonal, X(a, b) coincides with the real tangent vector expg-l(a) (where expb: Tb(V) -+ V is the exponential map associated to a fixed affine connexion); b) X(a,b) 4=o when a4:b. By hypothesis, the Euler characteristic of V vanishes so that there exists on V a real nowhere vanishing vector field Y, with the help of which one easily extends the germ of X around the diagonal to a section of E~ satisfying b). (Use Y as a purely imaginary component.) Lemma 46. -- The following is a continuous projective resolution of the module Coo(V) over Coo(V � V) (with the diagonal action): A* ix ix Coo(V) ~ Coo(V � V) ~- Coo(V 2, El) 4-- ... <--- Coo(V 2, E,) ~ o (n---= dim V) where i x is the contraction withX. Proof. -- Each of the modules ..r k = Coo(V � V, Ek) is finite projective and hence also topologically projective. Obviously i:~ = o. To show that one has a topological resolution it remains to construct a continuous linear section. Let Z,x'eC~176 be such that : X(a, b) = expg-t(a), V (a, b) ~ Support Z'; Z' = I on the support of Z and X -=-- i is a neighborhood of A. 343 x",8 ALAIN CONNES Let ta' be a section of E lsuchthat <X, co')= i on the support of I --Z. Put ~t(a, b) = exp,(tX(b, a)) for (a, b) close enough to A and let s(ta) = z' 7 + - z) ta. By construction s is C~(V)-hnear in the variable a. Fixing a and taking normal coor- dinates around a = 0 one gets q~t(b) = tb, X(o, b) = -- b, so that one can easily check the equality (cp; di x cox) + i x (~; dta~) -/- = q~;(0x c~ ? = tat for any differential form ta x vanishing off the support of Z and satisfying tat(a, a) = o. Applying this with tax = Zta shows that s/x + i x s = id. [] We are now ready to prove: Lemma 4g. m Let V be a compact smooth manifold, and consider ~ = C~~ as a locally convex topological algebra, then: a) The continuous Hochschild cohomology group H~(d, d*) is canonically isomorphic with the space of de Rham currents of dimension k on V. To the (k + I)-linear functional ~0 is asso- ciated the current C such that < C,f0 dr1 ^ ... ^ dfk> = y, ~(~) ~(fo, fo(1),fo(2), ...,f~ a effik b) Under the isomorphism a) the operator I o B : Hk(d, ~q~*) ~ H~-X(~, d') /s the de Rham boundary for currents and the image orb in H~-X(~q~) is contained in the space of totally antisymmetric cocycle classes. Proof. -- a) One just has to compare the standard projective resolution of ,~1 with the resolution of lemma 44, applying lemma 43. Note that (cf. [33]) given any commu- tative algebra ,~/ and bimodule ~4-/, the map T ~ ~ ~(~) T ~ where T e C*(M, ~r ae~k and T~ ..., a ~) = T(a~ a~ transforms Hochschild cocycles in Hochschild cocycles and its kernel contains the Hochschild coboundaries. Next, if q~eZk(d,d*) and q~~ for ~e(Sk, with ~=C~(V), then (under the obvious continuity hypothesis) there exists a current C on V such that <C,fO dfl ^ ... ^ dfk> = ~(fo, fa, ...,fk), V f' ed. Indeed q~ now satisfies the condition ~?(fo, fx f2 fs, . . .,fk +x) = q~(fo fx,f2,f~, ... ,f~ +1) -k- ~(fo f2,fx, fs, . . .,fk+x) for fie C~~ which shows that, as a distribution on V k+x, its support is contained in the diagonal Ak + x = { (x, x, ..., x) e V k + x, x E V }. Thus the problem of existence 344 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY ~a9 of C is local and easily handled say with V = T" or also using local coordinates. Let O k be the space of currents of dimension k on V. Define ~ : -@k -+ Hk( d, d*) by ~(C) (fo, fl, ...,fk) = (C,fO dr1 ^ ... A dfk), V f' ~ C~(V); then the map ~ has a left inverse e given by e(9) = C, where (C,f~ dfl ^ ... ^ dfk) = Ilk!  ~(~r) q~(fo, fo(~),...,fo(k)). a ~b k To check that ~ o ~r = id we may replace V by V � S ~, since the homomorphism p:&----- C ~(V� 1)-+d = C ~(V) given by evaluation at apoint p~S 1 induces a p* split injection Hk(d, d*) ---> t-Ik(&, &*). Thus we may as well assume that the Euler characteristic of V is o. Let X be a section of E~ as above. Let then (.ACE, b~,) be the projective resolution of C~~ given by lemma 44: ..r -= &~ Ek) , 0 k = i x. By lemma 43 the Hochschild cohomology Hk(C~176 (C~~ *) coincides with the cohomology of the complex Homc~o(v,)(..gt'~, C~(V)*). One has a natural isomorphism c~(v 2, Ek) | C~(V) ~ &~ V, A* Ek) and since A* E k is by construction the exterior power AkT;(V), one has a natural isomorphism of Homc~(v,)(.~, C~(V) ") with the space ~k of k-dimensional currents on V. More explicitly, to T e Homc| *) corresponds the current C given by the equality (C,~)=T(to')(i), V(o'~.~r Mr162 Since the restriction of X to the diagonal A is zero we see that the coboundary operator i~ is zero and hence that Hk(.~r .~1") = ~k. To write down explicitely the isomorphism we just need a chain map F of the resolution ,g' to the standard resolution ("gk = (~1 6,~ ~10) 6,~ ~r 6,~ ... 6,~ ~1) above the identity map "go -+ -'g0. Here ~r &~ VXVXV k) and we take (Fto) (a, b, x 1, ...,x k) = (X(x ~, b) ^ ... ^ X(x k, b), ~(a, b) ), V a, b, x i ~ V and co e.//~ = C~~ 2, Ek). One has 9 .., 9 .., X k-l) (b k Feo) (a, b, x 1, x k-') = (Feo) (a, b, a, x t, k--1 -- 3", (-- i)iF~(a, b, x 1, ..., x j, x j, ..., x ~-1) j=l + (-- i) ~ Fta(a, b, x t, ..., x ~-~, b) = (X(a, b) ^ X(x t, b) ^ ... ^ X(x k-t, b), o(a, b)). This shows that b~ Fco = Fix to, V ~, so that bk F = Fb~, and F is a chain map. 17 ~3 o ALAIN CONNES Let q~ e Zk(d, zr be a Hochschild cocycle, the corresponding element of Homc~olv.)(dlk, ~r is given by the equality ,~( ( f | g) | ft | | fk) ( fo) = r f, fl, . . .,fk), /, g,f, E ~t. Let us compute the k-dimensional current corresponding to ~ o F. One has <C,/0 df 1 ^ ... ^ dfk> = ~0 F(o t) (I), where ~, =f0 ol A ... ^ oh, or b) = drY(b) ~ T](V). One has Fo'(a, b, x 1, ..., x k) = <X(x 1, b) ^ ... A X(x k, b), co'(a, b)> =f~ 1-l<X(x', b), a E~k 1 This shows that to compute ~ o F one may replace q~ by the total antisymme- trization r I y. r ~?o on the last k variables. As the differential of the function k l~ k x~<X(x,b),df(b)> at the point x=b is equal to df(b), we conclude that the k-dimensional current corresponding to ~ o F is C = kI 0t(q~) and hence that ~ is an isomorphism. b) Let C e 9k be a k-dimensional current, and q~ the corresponding Hochschild cocycle: ~(fo, fl, ...,fk) =_ <C,fO dr1 ^ ... ^ dfk>. Then Bo q~(fo, ...,fk-1) = q~(i,fo, ...,fk-x) = <C, df~ ... A dfk-t> = <bC,f~ ... ^ dfk-t>. As an immediate corollary, we get: Theorem 16. -- Let d = C~(V) as a locally convex topological algebra. Then: x) For each k, I-t~(z~') is canonically isomorphic to the direct sum Ker b (C ~k) | C) | , C) | ... (where Hq(V, C) is the usual de Rham homology of V). 2) t-I*(d) is canonically isomorphic to the de Rham homology H,(V, C) (with filtration by dimensions). Proof. -- x) Let us explicitly describe the isomorphism. Let q~ a H~(d). Then the current C = ,(I(q~)) given by I ~ q~(fo, f,Ct)...,lOCk)) < C,f~ djl ^ ... ^ dfk > = o is closed (since B(I(q~)) = o), so that the cochain ~(fo, fl, ...,fk) = < C, fO dfl ^ ... ^ df~> 346 NON-COMMUTATIVE DIFFEREN'IIAL GEOMETRY 13x belongs to Z~(d). The class of ~ -- ~ in H~(d) is well determined, and is by cons- truction in the kernel of I. Thus by theorem 37 there exists + ~H~-2(d) with S+ = ~ -- ~, and + is unique modulo the image of B. Thus the homology class of the closed current a(I(~b)) is well determined. Moreover by lemma 45 b) the class of + -- ~ in H~-~(d) is well determined. Repeating this process one gets the desired sequence of homology classes r ~Hk_2~(V , C). By construction, q~ is in the same class (in H~(~)) as C + ~, Si ~j (where for any closed current mj in the class one takes ~ = ~ "~j(fo, fl, ...,fk-2j) = <(aj,fo dft ^ ... ^ df~-21>). This shows that the map that we just constructed is an injection of H~(~ r to Ker b(C ~fk) | Hk-2(V, C) @... | Hk_2,(V, C) | The surjectivity is obvious. ~,) In x) we see by the construction of the isomorphism, that S : Hxk(~) ~ H~+2(M) is the map which associates to each C e Ker b its homology class. The conclusion follows. [] Remarks 47. -- a) In this example the spectral sequence of theorem 39 d) is dege- nerate and the E 2 term is already the de Rham homology of V (with differential equal to o). b) Let 9 ~ H~(C~ Then theorem 46 shows that q~ is in the same class as + ~ S j ~j where the current C is well defined and the homology classes oj are also j=l well defined. One can prove that, once an affine connection V on V has been choosen, one can associate canonically a sequence c0j of closed currents to any ~ ~ Z~(C~ whose support (in V k+l) is close enough to the diagonal A ={(x, ...,x),x~V}. Moreover if ~ is local, i.e. if its support is contained in A, then the germ of oj around any x ~ V only depends upon the germ of ~ around x and the connexion V. This is proven by explicitly comparing the resolution of lemma 44 and the standard one. It remains valid without the hypothesis z(V) = o. c) Let W C V be a submanifold of V, i* : C~~ --> C~ the restriction map, and o -+ Ker i* ---> C~ ---> C~ ~ o the corresponding exact sequence of algebras. For the ordinary homology groups one has a long exact sequence H~(W) ~ H,(V) -~ Hq(V, W) --. H~_,(W) ~ ... where the connecting map is of degree -- I. Since H~, is defined as a cohomology theory, i.e. from a cochain complex, the long exact sequence --, HI(Ca(W)) ~ HI(C~(V)) -+ HI(O~ C~ -->~ H~+I(C~(W) ) ~... 347 ALAIN CONNES x32 has a connecting map of degree -t- 1. So one may wonder how this is compatible with theorem 46. The point is that the connecting map for the long exact sequence of Hochschild cohomology groups is o (any current on W whose image in V is zero, does vanish), thus Im(0) C SH~-~(C~176 d) Only very trivial cyclic cocycles on C~(V) do extend continuously to the C*-algebra C(V) of continuous functions on a compact manifold. In fact for any compact space X the continuous Hochschild cohomology of d = C(X) with coefficients in the bimodule d* is trivial in dimension n ~ 1 (cf. [35]). Thus by theorem 37 the cyclic cohomology of ~/is given by H~"(~r = H~(d) and I-~x"+l(~) ---- o. This remark extends to arbitrary nuclear C* algebras [51]. Example2.-- d=d0, 0elR/Z. (Cf. [16] [I9] [55] [58] 9 ) Let X= exp2~i0. Denote by St(Z*) the space of sequences (a,, ,,),,,, 9 z, of rapid decay (i.e. (In[ -t- [m[) q [a,,,~l is bounded for any q 9 Let d 0 be the algebra whose generic element is a formal sum Y~a,,,, U~ U~, where (a,,,,) e ~9~ *) and the product is specified by the equality U2 U1 = XUx U,. For 0 9 O this algebra is Morita equivalent, in the sense of corollary 24, to the commutative algebra of smooth functions on the 2-torus. Thus in the case 0 e Q,, the computation of H*(d0) follows from theorem 46. We shall now do the computation for arbitrary 0. The first step is to compute the Hochschild cohomology H(d0, ~r where of course d 0 is considered as a locally convex topological algebra (using the seminorms p,(a) = Sup(1 + In [ -}- [m [)q [ a,,,,[). Let us describe a topological projective resolution of d 0 viewed as a module over = do @,, -~0. Put Mr i = ~ | fii where fl --= fl 0 @ t-I x (9 fi2 is the exterior algebra over the 2-dimensional vector space fi x = C * with canonical basis ex, e,. For j = 1, 2 let bj :.~j--*~+, be the ~-linear map such that bl(I| = I| j= 1,2. b2(I| l^e~)) = (U,|174 (XUx| 1@U ~174 As usual, let r o be given by r =ab for a,b 9162 Lemma 48. -- a) (~/t~, hi) is a projective resolution of the module ~r b) H'(d0, d~) =o for i> 2. Proof. -- For v= (na,n,) 9 let U *--U~'U~' 9 X ~=-U *| 9 and yv = i | IJ * e SY. Then X * and Yr commute for any ,~, v' and any element of ~ is of the form x = Y~a~,~, X ~ Y~', where the sequence (a,,r is an arbitrary element of 5g(Z4). One has X ~x r "'"Ix ~+v, Y~Y~' =X "i'Y~+r 348 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Let us check that Ker ~ = Imbl. The inclusion Im b 1C Ker ~ is clear. For x = ~]a~.r X ~ Y~', ~(x) -- o implies Za~,v X ~ X v = o i.e. x = Y~a~. v X'(Y v -- X"). Using the equality o o (I | V~') (I | U~*) -- (VT'| i)(U~'| I) rt i -- 1 o o = (i | u?)( Z u{| up -~-j) (i | u, - u,| i) nj -- i -J- (U~'@ !) ( Z U2 j~ U~'-I-J) (i ~) U 2 -- U 2e i), we see that the left idealKer r is generated by I| l| and I| 2-U~| and hence is equal to Im b 1. Next, one checks that bib z:o. Given x=x l| 1-x s| seKerbl, one o o has Xi(l|174174174 ). To prove that xeImb 2 it is enough to find y e~ such that x 1 :y(Ug| I -- X| With Z----},Us174 U2 one first proves that xl( Z k) : o, using the relation --oo o __ o ~ o Xi( ~, Zk)(I ~Vl Vl~ I ) : xl(I | 1 Vl| I ) ~ (V21| -oo -oo = x~o(I | s Us| ) ~ (U21| = O. --oo Then writing xl = Y.ak Z k, where (aJ is a sequence of rapid decay of elements of the closed subalgebra of ~ generated by Ux| x, i | U1, Us| i, one gets k--1 x~ = Za~(Z ~ - ~) = Za~( Z ZJ)(z- i) =y~(Z- ~). Finally the injectivity of bs is immediate. [] Using this resolution one easily computes H~(~0, do*). We say (of. [32]) that 0 satisfies a diophantine condition if the sequence Ix --k"[ -1 is O(n ~) for some k. Proposition 49. -- a) Let 0 r Q. One has H~ do* ) = C. b) If 0 r Q satisfies a diophantine condition, then Hi(d0, do*) /s of dimension 2 for j = I, and of dimension i for j = ~. c) If 0 r Q does not satisfy a diophantine condition, then H 1, H s are infinite dimensional non Itausdorff spaces. (Recall that by theorem 46, Hi(.~e, ~r is infinite dimensional for j ~ 2 when 0~Q.) Proof. -- We have to compute the cohomology of the complex (Hom~(.Aci, d~), b~). The map T e Hom~(~, d~) ---> T(I) e ~; allows to identify Hom~(c/t'i, d~) with 349 ALAIN CONNES x34 do*| ~-. Moreover, using the canonical trace 9 on d0, v(Y.a~ U ~) = al0,0 / one can identify do* with the space of formal sums 9 = Za U where (a~)~e z, is a tempered sequence of complex numbers (la.,,.I ~ C(In I + [ml) ~ for some C and ~). The linear functional is given by (% x)= "r(gx ) for x e ~. With these notations, the above complex becomes do* do* 9 do* 2- do* o where at(9) = ((U 1 ~ -- 901), (U s ~o -- ?Us) ) and as(gi, (~S) = Us 91 -- Xgl Us -- (~kUl 9s -- 9s U1). Since )~ r O one easily gets Ker a i = C, which gives a). For (gx, gs) eKcras, onc has Us91--XgtUs=XUi9s--gsUl and the cocfficicnts a v of 9 = ~a, U ~ arc uniqucly determined by the conditions a(0. 0) ~ o, Ui 9 -- ~Ua = 9~, Us 9 -- 9Us = cPs. -- -- a I and (X" -- I) a.,, .,_ t = a~,., For Indeed one has (I ~n,) an t-l'n* n,, nt these conditions to be compatible one needs 1 a s 1 S [-Anx an,, n, +IV~ -- i) -1 a., o = o Vn; 0,. = o Vn; Z')- anll + 1, n.(I -- for n 14=o, n s4=o. From the hypothesis as(gt , 92) = o one gets (x a 1 - (I - x') a 2 v nt, -- n I +t, n s -- nx, nl +I Thus the compatibility conditions are: a t a s 1,0 ~ O, 0,1 ~ O. If 0 satisfies a diophantine condition, the sequence (a~) is automatically tempered, which shows that H t (do, do* ) = (I 2. If 0 does not satisfy a diophantine condition, then by choosing say the pair (~Pt, o) where 91 ----- 2~ Ut U~, one checks that the compatibility conditions are fulfilled but n#0 that (a,) is not tempered. This proves b), c) for Ht; the proofs for H 2 are similar. [] At this point, it might seem hopeless to compute H*(d0) (cf. definition 16) when 0 is an irrational number not satisfying a diophantine condition, since the Hochschild cohomology is already quite complicated. We shall see however that even in that case, where H*(d0, do*) is infinite dimensional non Hausdorff, the homology of the complex (Hn(d0, do*), I o B) is still finite dimensional. The first thing is to translate I o B in the resolution used above. Before we begin the computations we can already state a corollary of proposition 49 and theorem 37: Corollary 80. -- (0 r Q,). One has H~ = C and the map I : Hi(d0) -+ Ht(d0, d;) is an isomorphism. (Thus in particular any I-dimensional current is closed.) 350 NON-COMMUTATIVE DIFFERENTIAL GEOMETRY Proof. -- By proposition 49, a) one has t-~x(~r =H~162 *) = C. By theorem 37 the following sequence is exact: o ~ H~(~r i H~(~r ' ~r ~ t-Px(~r L H~(,aco). Since the image by S of the generator -r of I~x(~r is non zero (it pairs non trivially with I Proj~r one gets B =o. [] Lemma 51. -- Let ~ ~ ~r ~ = I-I2(~r d0") ; then (I o B) (~) ~ Ht(~r ~r = Ker ~/Im ,,, is the class of (~, ~) where (q)l)n,. = -- x--l( 1 -- x[n--1}m) (I -- ~.n--1)--I e~., .+1 and (cp2).,, . = X--I(I -- X n(m-1)) (I -- Xrn--1) -1 q)n+l,m" Proof. -- To do the computation we first have to compare the projective resolution oflemma 4 8 with the standard resolution (.,It' k' ---- ~ ~, ~10 ~k ... ), i.e. to find morphisms h : ~r ~ ,//' and k : ,~r -+ ~ of complexes of ~-modules which are the identity in degree o. Recall that b~(I| 1|174 = (a l| 2|174 n--1 + W, (-- i)Ji| ajaj+t| | | n -31- (-- I)n(I| ~174 t|174 n_,). The module map h 1 is determined by ht(I | which must satisfy Vh,(x eel) = bl(~ *es) = ~ | -- Uj| i; thus we can take hl(I| = I| i. One determines in a similar way (but we do not need it for the lemma) &(i * (e,^ e,)) = ~ * U2| U1 - x~ | Ul| U2. The module map k 1 : & ~,, d o --+ B @ f~t is determined by kl(I | O v) (v ----- (nl, n2)) which must satisfy bl(kl(I @ U~)) = b~(I @ U ~) = U~@ I -- I | (U~) ~ As in the proof of lemma 4 8 we take kl(I | ~) ----- A~Qe I + B~Qe 2 where o o o 1 o o __ nl nt A~ -~ U~(U~ ~ -- U~) (Ut U1)- , B~ = U t (U 2 -- U~') (U9 -- U2) -1 where to simplify o o notation we omit the tensor product signs (i.e. Ui, U i mean Uj | I, i | Ui). Now the module map /~:~2' ~-'gcz is uniquely determined by the equality b zk~=k lb~ since ~r176 A tedious but straightforward computation gives: &(I | U~| U 0 = U~' X~'"~ O~ -- X -"~"~o LI~ O;, -- x", ~r3~" I3~',| (el ^ e~). X" U~ -- X-" U 1 Uz -- XU2 351 I36 ALAIN CONNES In fact we shall only need the special cases a) v---- (i, o), ~ arbitrary, b) v arbitrary, Vt = (o, o), c) ~ arbitrary, Vt= (i,o); one may as well check directly that k s = o in cases a), b) (compute k 1 b~) and that o o k2(IQU~QU1) = U 1 " (Us " --)" U~') (U s XUs)-I | (e 1 ^ e~). We thus have determined the morphisms h and k. They yield the morphisms k~ : Homa,(..g~, ds* ) ~ Hom~(.g/~, ~), h~ : Hom~(..lg~, do) ~ Hom~(..gl, ~), and we want to compute the composition Let q~ e ~q~ and let ~ be the corresponding element of Homar(.Wg2, ~): '~(a | b ~ | el ^ ez) (x) = ~(bxa) V a, b, x e ,.~t o. Let +=k~'----~ok s. One has +(x ~ x~, x~) = ~(k~(~ | x~ | ~')) (xO) v xO, x~, x~ ~ do. Let ~b l= (IoB)+. One has by definition, for x ~ led| +I(X O, X 1) = +(I, X O, X 1) -- +(X O, X 1, I) -- +(I, X 1, X O) --~ +(X 1, X O, I). Using b) one gets that +(x ~ 1,I) =o for x ~ le~r thus +l(X O, X 1) = +(I, X O, X 1) -- +(I, X 1, X O) V X O, X 1 6~0. Let then ,r162 = (*l, q~s). One has ,r162 = h; +t; thus Let us compute oj(U*), ,J ----- (nl, n~), j = x, 2. Using a), we have ~(u ~) = +(i, u ~, u~). Using c) we have ~(o ~) = +(~, u ~, o~) = ~(k,(~ | O~| U,))(,) __{: -- ' - ~kcn' +l)nt) (I - )k(nt +1,)-1 q)(U~' U~.-1) ifn~ 4: o, if n~ = o. The knowledge of ~l(U~), V v e Z 2, determines the coefficients a~ of ~l = Xa~ U" by the equality 1 X"~" ~l(U-~). a,~ -~- Hence we get ~.,~ = x.~(~ _ x(.-~)-) (~ _ x-(.-~) -~ x.I---, ~.,~+~, where q~ -~ Xa~ UL The computation of ~?~ = Za~ U ~ is done in a similar way. [] 352 NON-COMMIYrATIVE DIFFERENTIAL GEOMETRY x37 We are now ready to determine the kernel and the image of I o B. Let e d*/Im 0~ e H2(d0, do*) be such that (I o B) ~ e Im aa. Let thus ~b = Zb, U * ~ ~r with oq(d?) = (I oB) @. Then I) (I -- ~kn,) b.x_l,n, = _ ~--l(i -- ~(., --1) n,) (I -- ~.,--1)--1 ant,n,+1 ' 2) (~ - x",) b~,.._~ = - x-'(~ - x-,c.,-.) (~ - x~-')-' ~..+l,~. So the image (I o B) q~ e Ht(d0, do*) is o if and only if the following sequence is tempered: c.,,. = (i -- ~k rim) (I -- Xn) -1 (I -- X") -1 a.+~,.,+l. One has ~ e Im ~ if and only if one can find tempered sequences (c{,r.), j = I, ~, such that (X" I)r .+l, ~ ra + ( x'~ -- I) ~ V n, m. This is equivalent -- 6n, m+ 1 = an+l,m+l, to a~,~ = o and the temperedness of the sequence (Ix"-- i[ + IX'-- i[) -a a,+t,,,+t. Thus the next lemma shows that in all cases the kernel of I o B is one-dimenslonal. Lemma 82. --For any O ~ Q, and (n, m) e Z ~, (n, m) 4:(o,o), one has (Ix"- ~[ + Ix ~- ~ I)-~<_ Iml + I~ -x"~[ I~ - x"l-~l ~ -x'l -~ X = ~0. w/th Proof. -- For n = o, [ (I -- X'") (i -- X")- a ] is equal to [ m [ > I SO that the ine- quality is obvious. Thus we may assume that n 4: o, m 4: o. If [ I -- X""[ > [ I -- X" [ the inequality is again obvious, thus one can assume [I -- X'"[ < ]I -- X" [. With X"=e ~, xe[--~,~[, onehas [I--e~'~[<[I--ei~[ with m4=o, thus Imo~[>n, I,~" -- ~ l ~_ 21m. [] Let us now look for the image of I o B in Hi(d0, do*) ~--- Ker ~,/Im ~1. Any pair (@1, ~,) eIm(I oB) + Imp1 satisfies a], 0 = o, ~,l = o (using lemma 5i). Conversely, if a~, 0:a0s,l=o, let us find ~ed0* (~?:~a~U ~) and ~be~o* (hb = ~b~ U ~) so that, with the notation of lemma 5 I, one has (r ~2) = ~1(+) "4- (I o B) q~o This means: I) a~, m = (I -- X m) b._l, m- X-I(I -- X ("-l)m) (I -- ~.-1)-l~,m+l, 2) ~,~ = (x" - x) b.,~_~ + x-~(~ - x "c~-~)) (~ - x~-~) -~ a.+~,~. Since ~(~1, ~) ---- o by hypothesis, one has (X" -- I) a~+l. = ---- (I -- X') a~,m+ 1. Thus one can find sequences b, a satisfying the above equalities with Ib.,.I = la.+,,.+,l =(~ + I I -x'l I I - x"l-' I ~ - x'l-')-' ~+"" i--X m 4,.+, where for m=o and e4=o the right term is replaced by I--X" " 18 ~3 8 ALAIN CONNES By lemma 5 2, lb.,~l < (i + Iml)(Ix"--I I- q- I~.m--I l)I a.~--~,~ I [(X -- Xm)-~l -- (~ + Iml)(I a.~+x,ml + 1~,~+~[). Thus a, b are tempered and we have shown that (q~x, q~) belongs to the image of I o B in Ha(~o, a'o*). Theorem 53. -- a) For all values of O, HOb(do) -~ C 2 and H~162 ~ C s. b) The map (q~t, e?2) ~ Ker ~ ~ (qh(U~-~), q~2(U~-~)) ~ s 2 gives an isomorphism of H~162 = Ht(~r a'o*)/Im(I o I3) with C 2. c) One has H~162 = H2(~r it is a vector space of dimension 2 with basis Sv ('r the canonical trace) and the functional ~ given by ~(,o, e, ,s) = ,o(8,(,~) 8,(x') - 8s(x~) 8~(,~)) v x' ~ do. In the last formula, 81, 82 are the basic derivations of do: 81(U *) = 2nin 1 U", ~s(U ~) = 2reins U'. Proof. -- Since H"(~r do*) = o for n > 3, one has by theorem 37 an equality H~ = I-I~(d0) = H~(d0)/Im B. By corollary 5 ~ one gets H~(d0)/Im B ---- H~(do, do*)/Im(I o B). Thus b) follows from the above computations. In the same way, one has H~V(do)----I~x(d0) , and the exact sequence o -+ H~(d0) _~s i_~(~,0 ) L H2(d0, do* ) -+ Hi(d0). With 0 r Q. one has H~(d0) = G with generator v, and using corollary 5 ~ and the computation of Ker(I o B), we see that the image of I in the above sequence is the one-dimensional subspace of H2(d0, d0* ) = do*/Im oc 2 generated by U 1 U s (i.e. the functional x ~ v(xU 1 Us), V x e do). Let us compute the image I(q~) of the c? e I-Px(do) given by 53 c). Let ~ Hom~(.//~, ~O*) be given by ~((a Q b 0) Q X 1 ~ X 2) (x 0) = ~0(bx 0 a, X 1, X 2) V a, b, x i e do, with the notations oflemma 5 I. Under the identification of H2(d0, do* ) with do*/Im aa, I(~) corresponds to the class of ~' o/6. One has ~(h2(I | e~ ^ e2) ) (x ~ = q~(x ~ Us, U1) -- Xq~(x ~ Ux, U2) = - 2x(2~i)" ~(xO u~ u~). This shows that H~(do) is generated by Sz and ~. [] We can now determine in this example the Chern character, viewed (as in section 2) as a pairing between Ko(do) and H~162 With the notations of theorem 53, we take S'r and 9 as a basis for He'(do). From the results of Pimsner and Voiculescu [55] 35d NON-COMMUTATIVE DIFFERENTIAL GEOMETRY I39 and of [I9] lemme i and th6or6me 7 the following finite projective modules over d 0 form a basis of the group K0(J~'0) ----ZZ: i) d 0 as a right d0-module. 2) 5~(R), (the ordinary Schwartz space of the real line), with module structure given by: (4. u1) (s) = ~(s + o), (4 us) (s) = e ~' ~(s), v s ~ R, ~ ~ ~(R). We shall denote the respective classes in Ko(~r by [I] and [5r Lemma 54. -- The pairing of K0(d6) with HeY(d0) is given by: a) ([I], Stag) = I, ([~o], 8%') = 0 ~]o, i] b) ([I], ~p) ---- o, ([5~ ~p) ---- I. Proof. -- a) One has x(i) ---- i. We leave the second equality as an exercice. b) Since 8j(1) = o the first equality is clear. The second follows from [I9] thfior~me 7, noticing that the notion of connexion used there is the same as that of definition i8 above relative to the cycle over d 0 which defines ~p namely: where A 1, A s are the exterior powers of the vector space (]3, dual of the Lie algebra of R s (which acts on d 0 by 81, 82). (Cf. [I9] definition 2.) Corollary 55. -- For 0 r 0 the filtration of Hev(,~0) by dimensions is not compatible with the lattice dual to K0(d0). We shall see in chapter 4 that any element of this dual lattice is the Chern character of a 2 + ~ summable Fredholm module on d 0. Problem 56. -- Extend the result of this section to the " crossed product " of C~176 1) by an arbitrary diffeomorphism of S 1 with rotation number equal to 0 [32]. 355 Terminology (references to part II) Chain, section 3 Character of a r introduction and proposition t Cobordism of cyd*s, section 3 Cup product of cochains, section x Cycle,, introduction and section x Cyclic cohomology, section I, corollary 4 Exact couple, section 4 Filtration by dimension, section 2, definition x6, section 4, corollary 39 Flabby (algebra), introduction, section t, corollary 6 and [13] Hochschild cohomology, section x, definition 2 Hoehechild coboundaty, introduction Homotopy invariame, section 4, remark c. Irrational rotation algebra, section 6 Pairing with K-the0ry, section 2 Relative theory, section 4, remark a. Stabilized cyclic cohomology, section 2, definition 16 Suspension map, section x, lemma It Tensor product of cycles, section i Topological projective module, section 5 Universal differential algebra, section t, proposition I and [I] [I4] Vanishing cycle, section i, definition 7 List of formulae in Part II bA = Ab' b "~ = o~ b r'~ ~ o Db = b" D B o b + b' B o = D bB ----- -- Bb B s= o SB = 2ixn(n + I) b n+3 n+t bS = -- Sb n+3 Z~ ~ B o Z n+a = BZ n§ Im B = C~ bC~, C Bo Z ~+1 [9 @ ] = (n+m)! n.t m ! [~] v [+] e(de) e = o e(de) s = (de) ~ e 356 Notation used in part II d, ,~ algebras over C cn(d, ~*) space of n + I linear forms on r ~ ..... a n) = 9(a "r(~ .... , a "Y(n)) V r ff Cn(d, d*), y permutation of { o, I .... , n } and aJ E d ~, . ~ c.(d, w') ~p(ao ..... an+X) = ~0 (- x)J r 0 ..... aJ aJ +1 .... , a n+l) + (-- x) n+x tp(a n+l a ~ ..... a n) J- zn(d, d*) = Ker b, Bn(d, d*) =Im b, Hn(d, ~r = Zn/B n C~(d) = { 9 E Gn(d, M*)}, q~X = r ~ V), cyclic permutation z~(~,) = c~(d) c~ Ker b B~(~') = bC~-'(d) H~(d) = Z~(d)/B~(d) algebra obtained from mr by adjoining a unit /1(~r universal graded differential algebra 9 ~(a ~ da I ... da n) = x(a ~ a 1 .... , a n) (proposition i) ~r = ~r | ~0, ~r = opposite algebra of ~r A 9 = ~ e(y) q~v, I ~ = group of cyclic permutations -t~F b' ~ = ~ (-- z)i ~(x ~ ..., xJ xJ +x, ..., x n+x) V ~ e C"(~r .~') r~:~(~C| --~ ~(~)| VxJE.~ r # ~ = (~| o~ e Z~(C), a(,, ,, ,) = 2i~ S~ = ~ # ~ V ~ e z~(~r H*(aq) = 1~ (H~(ar S) F n H*(~r = Im H~(~') B 0 r o ..... a n-l) = r a ~ ..... a n-') -- (-- i) n ep(a ~ ..... a n-l, I), V ~p ~ cn(~, .~r M*(~r Cobordism group of cycles over I : morphism of complexes (C~, b) ~ (C n, b) Dcp = 9--r ~), Vq~ ~ Cn(~, ~1"), ), canonical generator of cyclic group I" g~ = (n--m+ ,)~ V,~ ~C"," = c"-m(.~,~ *) d~q~= x i~ e Cn,m+ x V~ ~cn, m n--m ~" ~(a ~ ..., a") = ~ ~(a ~ ..... ~(ai), ..., a n) V a i ~ ~, q~ ~ cn(~, Mat') and 8 derivation of .~. i-1 357 BIBLIOGRAPHY [I] W. 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Coll&ge de France, I I, place Marcelin-Berthelot, 75231 Paris Cedex o 5 et Institut des Hautes l~,tudes scientifiques, 35, route de Chartres, 9144 ~ Bures-sur-Yvette Manuscrits refus le I er flnvier 1983 (Part I) et le 1 er avril 1983 (Part II).

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