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C*-algebras, positive scalar curvature, and the Novikov conjecture

C*-algebras, positive scalar curvature, and the Novikov conjecture C-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE by JONATHAN ROSENBERG (1) This note, an appendix to the foregoing paper [10], is based on the philosophy of [9] that topological obstructions to existence of metrics of positive scalar curvature on non-simply connected manifolds should be closely related to the Novikov conjecture on " higher signatures ". I am very grateful to Mikhael Gromov and to Blaine Lawson for patiently explaining their work to me, for suggesting that I examine its relation to the work of Kasparov and of MiScenko et a/., and for helping with many details of this project. I would also like to thank Alain Connes, Paul Baum, Ron Douglas, Jerry Kaminker, John Miller and Larry Taylor for helpful discussions concerning index theorems and/or the Novikov conjecture, and Alain Connes and Mikhael Gromov for useful suggestions about improvements in the exposition of this paper. We shall show below that certain manifolds do not admit Riemannian metrics of positive scalar curvature. In most situations involving closed manifolds of practical geometric interest, our results of this type will be the same as those of [9] and of Corol- lary A and Theorem J of [10]. Nevertheless, it is not clear that their results always imply ours or vice versa. More significantly, we hope to illustrate another way in which operator algebras can be applied to problems of geometry and topology, and to demons- trate a closer relationship between the positive scalar curvature problem and the Novikov conjecture than may be apparent from the Gromov-Lawson method of attack. Our main results are Theorems 3.3 and 3.5 below. However, their proofs involve technical complications that tend to obscure the main idea, so we have first given the proof of an easier special case, which we have formulated as Theorem 2.11. The auxiliary results 2.4-2.10 on the Novikov conjecture (especially Theorem 2.6) may be of independent interest, i. The main construction As in [9] and [10], our tool will be an analysis of the index of the Dirac operator with coefficients in a bundle, together with the idea of the vanishing theorems of [19] (1) Sloan Foundation Fellow. Partially supported by NSF Grant no. MCS-Sa-ooyoG. 409 198 JONATHAN ROSENBER G and [i i]. But whereas Gromov and Lawson use for coefficients ordinary vector bundles of small curvature, we shall use flat coefficient bundles {i.e. bundles of zero curvature), but only at the expense of making the fibres of the bundles infinite-dimensional. This can be explained by the fact that (by Ghem-Weil theory) ordinary flat bundles have trivial rational characteristic classes, whereas infinite-dimensional flat bundles sometimes do not. Our main technical tool will therefore be not the Atiyah-Singer index theorem but rather the index theorem of MiScenko and Fomenko [23]. Thus we begin with a review of the key definitions and concepts in the Mi§cenko-Fomenko theory. Let A be a (complex) G^-algebra with unit. If X is a compact space, by an A-vector bundle over X we shall mean a locally trivial (Banach) vector bundle E over X, in which the fibres have the structure of finitely generated projective (left) A-modules. Morphisms of such bundles will be required to preserve the A-module structure on the fibres. The Grothendieck group of formal differences of equivalence classes of such bundles (with addition coming from the Whitney sum operation) will be denoted K°(X,A); it coincides with the algebraic K-group Ko(C(X)®A), where ® means the (spatial) CT-tensor product. (Of course, C(X)®A^ C(X,A). For all this see not only [23] but also [13], Exercise 11.6.14, and [21], Ch. i.) By the Kunneth theorem of [27], or rather by a relatively easy special case thereof, for compact metrizable X there is a natural isomorphism K°(X, A) ® Q^ (K°(X) ® Ko(A) 0 QJ C (Ri(X) ® K^A) ® QJ (where the tensor products are taken over Z). Thus using the ordinary Ghern character K^X) -^H*(X, %), we obtain the Chern character of Miscenko-Solov'ev [24]: ch : K°(X, A) -> H^X, %) ® Ko(A) C H^X, %) ® Ki(A), which is an isomorphism modulo torsion. Now suppose X is a closed manifold (for us this will always mean a compact connected G00 manifold without boundary), E and F are smooth A-vector bundles over X, and D is an elliptic pseudodifferential A-operator of order n in the sense of [23], § 3, taking smooth sections of E to smooth sections of F. One can define Sobolev spaces IP(X,E) and H^X, F), which are Hilbert G*-modules (see [23], § i, or [15]) over A. The fundamental theorem of [23] asserts that D defines a bounded operator IP(X, E) -^IP'^X, F), that this operator is an A-module map, with adjoint, which is A-Fredholm {i.e. invertible modulo A-compact operators), and that the " A-index " of D is independent of the choice ofj. One must be careful with the definition of index here; the kernel and cokernel of D are not necessarily finitely generated and A-projective, but this is true of D + K for some A-compact operator K. The A-index of D is then the formal difference (in Ko(A)): [ker(D + K)] - [coker(D + K)], which turns out to be independent of the choice of K ([23], § i, and [21], Gh. i). Finally, the A-index of D depends only on the class of the principal symbol of D in K°(T*X, A) 410 C*-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 199 (where T*X is the cotangent bundle of X, and where for locally compact spaces, K* denotes K-theory with compact supports). The A-index is, modulo torsion, given by a formula formally identical to that of Atiyah-Singer [4], the only difference being that the Ghern character must be interpreted as just explained above. We now apply this machinery to the case of spin manifolds of positive scalar curvature. If n is a discrete group, we denote by ETC a classifying space for TT, i.e. a K(TC, i)-space. For any (connected) manifold M with fundamental group TT, the universal covering of M is associated to a classifying map /: M ->BTT, well-defined up to homotopy and inducing the identity on TT^. THEOREM 1.1 . — Let M be a compact smooth even-dimensional spin manifold without boundary, and suppose M admits a Riemannian metric whose scalar curvature is everywhere non- negative and strictly positive at at least one point. Let A be any G-algebra with unit and i^ a flat (i.e. pulled back from ETC, where TC = n^{M)} A-vector bundle over M. Then <A(M) u chl/H, [M]> = o in Ko(A) ®%. PROOF. — By Proposition 3.8 of [i8], we may assume that M has been given a Riemannian metric whose scalar curvature K is everywhere strictly positive, hence bounded below by a constant KQ > o. Let <^4" and y~ be the two half-spinor bundles of M. We may give V a flat connection and thus define Dirac operators (y : r^^®^) -^r00^-®^) ID- : r°°(^- ® v) -> r^.^ ® -T) which are formal adjoints of one another with respect to the A-valued inner products. These are elliptic A-operators, and the usual calculation gives Th^^M) u chlXD4-)]) == A(M) u ch[^], A. ^ where A(M) is the total A-class of M, a certain polynomial in the rational Pontrjagin classes, and where Th : H*(M, %) ->H^(T*M, QJ is the Thorn isomorphism. Thus by the MiScenko-Fomenko Theorem, we get ind^ D4- = <A(M) u ch^], [M]> e Ko(A) ® %. We claim, however, that D4' has A-index zero. Indeed, with respect to the A-valued L^inner products on I^^®^), we have DTD± = V*V + K, by exactly the same calculation as in [9], § i, and in [10], § i. (There is no ^ term since ^ is flat.) We can conclude from this (together with the fact that K >_ KQ > o) that D'^'D- and D~D'1" are one-to-one with dense ranges and bounded inverses. Here 411 200 JONATHANROSENBERG we must use spectral theory and the fact that if a and b are positive self-adjoint elements of a G*-algebra, then a + b >, b and hence \\a + b\\ ^ \\b\\. To justify this step precisely, one can for instance first embed i^ as a direct summand in a trivial bundle of the form M X A" (using [21], Proposition 4.3). The operator 1C v<tv + - extends in an obvious way to F^^ ^A^, and since the coefficient bundle is trivial, we can use ordinary spectral theory on the first factor. Since VV+^^ i Ko/4, we conclude that D^D' and D~D'1' have bounded inverses with respect to the L^norms. A compact perturbation of D'1" will then be invertible as an operator from H84"1^4") to IP(^-), and tensoring with the identity operator on A", we see that D-^- has A-index zero. Hence <A(M) u ch[V], [M]> = o. a. C'-algebras and the Novikov Conjecture The usefulness of Theorem i. i depends on the possibilities for ch[^], i.e. for the characteristic classes of flat A-vector bundles over M. The best one could hope for is that these might generate all the <c characteristic cohomology 39 of M, i.e. that if /: M ->BTT; is the classifying map for the universal covering of M, one has CONJECTURE a. i (Vanishing of higher A-genera). — If M is a closed spin mamfold admitting a metric of positive scalar curvature, and if TT ^ TCi(M), then for all a e H*(B7r, %), <A(M) u/*(a), [M]> = o. Note that Conjecture 2.1 can be rephrased in the form /,(A(M) n [M]) == o. Modulo torsion, this is the same as the conjecture in [10] that the image of the class of (M,/) must vanish under the " J^-homomorphism " Q^BTT) ->KO,(B7r). We shall see now that Conjecture 2. i follows from the same intermediate step needed by Kasparov and by Migcenko in their proofs of the Novikov Conjecture for certain fundamental groups TC. Thus Conjecture 2. i is valid for many (and conceivably all) finitely presented groups TC. One can deduce from this that spin manifolds of certain homotopy types do not admit metrics of positive scalar curvature, and that within other homotopy types of manifolds, such metrics may exist but only subject to certain cons- traints on the Pontrjagin classes. We shall also see in § gB below that the somewhat annoying spin condition in 2. i can be relaxed to give CONJECTURE 2.2. — If M. is a closed oriented manifold admitting a metric of positive scalar curvature, if the universal covering of M is spin, and if n = ^{M), then for all a e H^BTT, %), <A(M) uf\a), [M]> = o. COROLLARY (TO THE CONJECTURE) 2.3. — No closed K(TT, i) -manifold can admit a metric of positive scalar curvature. 412 C*-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 201 PROOF OF 2.3 FROM 2.2. — Let M be a closed K(7r, i)-manifold of positive scalar curvature. If M is not orientable, pass to a double covering and replace TC by a sub- group of index 2. We can take M == ETC and f == id, and of course the universal covering of M is spin since it is contractible. Taking a + o in the top-degree coho- mology of M, we get <fl, [M]> = o, a contradiction. It seems that Conjecture 2. i (for a fixed finitely presented group 71) is closely related to the (generalized) Novikov Conjecture for the same group, which we state here for reference: NOVIKOV CONJECTURE. — Among the class of closed oriented manifolds M with funda- mental group 7Ti(M) ^ TT, if f: M -^BTC denotes the classifying map, then the " higher signa- tures" <L(M) uf*{a), [M]>, a eH*(B7r, QJ, are oriented homotopy invariants. Despite the obvious similarity between higher signatures and higher A-genera, we know of no direct way of deducing Conjecture 2. i from the Novikov Conjecture. However, it seems that for most groups for which one can prove one of these, one can prove the other as well. It is therefore worth briefly summarizing some of the ideas of [17]. § 9- If TT is any countable group, one can choose the space BTT to be a countable GW-complex (but usually not to be a finite complex). Following Kasparov, we denote by RK* representable topological K-theory, the representable cohomology theory coinciding with K* on compact spaces. For a countable CW-complex X with finite skeletons X^, one can also define LK*(X) = HmK^XJ, and there is a surjection RK(X) -> LK^X) (with kernel given by lim^ K^-^XJ [20]). We denote by K, the homology theory on compact metrizable spaces dual to K*, and let RK^(X) = lim K^(X^) for X as above. For a separable G*-algebra A, we denote by K*(A) the group ELK* (A, C) < ( dual9 5 to K,(A) == KK*(C, A). (Caution: our convention about raised and lowered indices is reversed from that of [16] and [17], in order to agree with the usual convention that K* should denote a contravariant functor. Thus K^^X))^ K^(X).) As in [i6], ®A denotes the "intersection product59 K^(A) ®K*(A) ->K^(C) == Z, gene- ralizing the Kronecker pairing between K-homology and K-cohomology. If C^TT) denotes the group C*-algebra of n (the completion of L^TC) in the greatest G*-norm), Kasparov defines maps a : K^G^TT)) ->RK*(BTT) and (3 : RK,(BTT) -> K,(G*(7c)) which are dual to one another in the sense that if S' denotes the projection of a into LK^BTC), then a(;c)®B^ == ^®c^)P(jO {^ x e K^G^)), ^(=RK,(BTI). Consider now the following possible properties of TT: PROPERTY SNG i. — The image of 0' (after tensoring with QJ is dense in the projective limit topology on LK*(BTI) ®Q,. 26 209 JONATHAN ROSENBER G PROPERTY SNG 2. — The map (B is injective after tensoring with Q^. PROPERTY SNC 3. — The map (B is injective (even without tensoring with QJ. PROPERTY SNG 4. — The map (3 is an isomorphism from RK,(BTT) -> K^C^TC)). Here the abbreviation " SNG " stands for " Strong Novikov Conjecture ", terminology which is defensible because of the string of implications (SNG 4) => (SNG 3) => (SNG 2) <= (SNC i), and the theorem ofKasparov ([17], § 9) that ifrc is finitely presented (so that the Novikov Conjecture for TT makes sense), then SNG i or SNG 2 implies the Novikov Conjecture for TT (in the form stated above). This can be explained as follows. Suppose M is a closed oriented manifold with fundamental group TT. Then we can form the <c universal flat G* (71;)-vector bundle " V == C*(7r) x^ M over M, where M is the universal covering of M and TT acts on C*(7c) by right translation and on M by covering transfor- mations. The generalized signature of M is defined by <L(M) u ch[y^], [M]>; this may be thought of as the G*(TI:)-index of the signature operator with coefficients in i^ (in the case where dimM =. o (mod 4)) (1). By [22] or by [17], § 9, Theorem 2, this generalized signature is an oriented homotopy invariant. Applying ®c*(7r)^ ^or classes y eK*(C*(7r)), it follows that <L(M) uf*(a), [M]> is an oriented homotopy invariant if a e H^BTC, QJ lies in the image (under the Chern character) of the image of a, and hence SNG implies the Novikov Conjecture. In the remainder of this section we list a number of situations where various forms of SNG are known, and then broaden the list a bit by discussing various stability pro- perties of the class of groups with SNC. (One may view this as a C*-algebraic counter- part to [6].) Then we describe consequences for the positive-scalar-curvature problem. The fundamental (and deep) results are the following: THEOREM (Migcenko, see [12]). — Suppose there exists a closed orientable K(TT, i)- manifold admitting a Riemannian metric with non-positive sectional curvatures. Then SNC i holds for TV. THEOREM (Kasparov, see [17], § 9, Theorem i). — Suppose n can be embedded as a discrete subgroup in a connected Lie group. Then SNG i holds for n (even without tensoring with QJ. One should note that MiScenko's theorem is stated in terms of existence of certain Fredholm representations, but the above formulation is equivalent. Also, these results appear to lend support to Conjecture 2.3, since there do not seem to be any published examples of closed oriented K(7r, i)-manifolds for which the hypotheses of both theorems (l) We are cheating slightly. One should really replace the Hirzebruch class L(M) here by the Atiyah- Singer modification ([4], § 6) of it, -§?(M), and then multiply in front by a suitable power of 2. However, for purposes of formulating the Novikov Conjecture, it does not matter whether one uses L or -S?. 4U C*-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 203 fail. However, M. Gromov has suggested that there are probably many K(TT, i)- manifolds not satisfying these conditions, so that we are still far from a proof of 2.3. For purposes of proving stability properties of SNG, there is a considerable technical advantage to working with SNC2-SNG4 rather than with SNG i. The reason is that the functors RK, and (for G*-algebras) K^ commute with direct limits, whereas RK* and K* do not. This makes it possible to avoid dealing with the Milnor lim^ sequence and with problems arising from I-adic completion. PROPOSITION 2.4. — Suppose n is a countable direct limit n = lim^ and the groups n^ all satisfy SNG 2, SNC 3, or SNG 4. Then the same holds for TV. PROOF. — We have BTT == lim BTT^ and C*(7c) = limG*(7rJ (G*-algebra inductive limit). So use naturality of (B and the fact that RK, and K, commute with direct limits. PROPOSITION 2.5. — Suppose n is an extension i ->7ri—>-7c->Z—>- i and SNG 4 holds/or the group TC^. Then SNG 4 holds for n. PROOF (cf. [26], Theorem 3.3). — Since Z is free, the group extension must split, and we can write n == TT^ X[ Z, where the semidirect product structure is determined by an automorphism 6 of TTr This gives a corresponding decomposition of C*(TC) as a crossed product C^TT^) XIQ Z and of BTT as a fibre bundle over S1 = BZ with fibres ^ BTT^. Then we may compute K,(G*(7r)) from K,(G'''(7ri}) via the Pimsner- Voiculescu exact sequence [25] and RK,(B7r) as the homology of the mapping torus of the self-map 6, ofBTTi. As in the proof of [26], Theorem 3.3, we obtain a commu- tative diagram with exact rows K,(G*(7ri)) -^ K,(G*(7r,)) K,(G-(7r)) K.-i(G*(^)) K,-i(C*(^)) e. Pi 01 Pi RK<(B7ti) -^ RK,(B7ti) ——^ RK,(B7t) ——> RK,_i(B7ti) RK,_i(B^) Thus if the maps (Bi are isomorphisms, so is p, by the 5-Lemma, REMARK. — Because of the need to use the 5-Lemma, it is clear that the proof of 2.5 breaks down if we replace SNG 4 by SNG 3 or SNG 2. THEOREM a. 6. — If n is a countable solvable group having a composition series in which the composition factors are torsion-free abelian, then SNG 4 holds for TC. PROOF. — We argue by induction on the length of such a composition series. Suppose TCI < TC and TC/TCI is torsion-free abelian. We may assume that the theorem 204 JONATHANR.OSENBER G already holds for ^. (To start the induction, take ^ ={1 } if TT is abelian) Now T^ is the direct limit of its finitely generated subgroups, hence is a countable direct limit of free abelian groups. Thus n = Hm^, where ^ < ^ for each n and where ^ is free abelian. By 2.4, it is enough to prove the theorem for TC . For this it is enough to use the inductive hypothesis and 2.5 (repeated a finite number of times). REMARK. — Theorem 2.6 applies to some cases not covered by Kasparov's theorem since there are many groups (even finitely presented) satisfying the hypothesis of 2 6 but not polycyclic. PROPOSITION 2.7. — Suppose TT contains a subgroup ^ of finite index for which SNC 2 holds. Then SNG 2 holds for -K. PROOF. — The group ^ acts freely on ETC so we obtain a finite covering E7t/^~B7^E7t/7t=B7t. (If ^ is normal then the covering is Galois, but this is irrelevant for our purposes.) We also have an inclusion i: C*(^) ^ c^) making C^n) into a finitely generated free C^) -module. Thus we have maps t,:K.(G*(7Ci))-^K.(G*(TC)) as well as a transfer map t: K (G*(7c)) -»K (C*(TC )) (coming from the forgetful functor from finitely generated projective G*(7t)-modules to finitely generated projective G-(^)-modules). Note that tai, is multiplication by [7r:TCj, hence i, is split mono after tensoring with Q (to invert [w • TT ]) Arguing similarly with the transfer in K-homology (see, e.g., [i], Ch. 4), we 'see A : RK.(B^) -^. RK.(BTC) is split epi after tensoring with %. The conclusion follows from consideration of the commutative diagram RK.(B7Ti) ® Q, -^ K.(C*(^)) ® % P« RK.(B7t) ® % -^ K.(C*(7t)) ® Q,. PROPOSITION a. 8. — Suppose TC contains a finite normal subgroup ^ and SNC 2 holds for TT/TCI. Then SNC 2 holds for TC. PROOF. — Consider the commutative diagram RK.(B7T)®Q," K,(G*(7r))®% RK.(B(7r/7ti)) ® % -^ K.(G*(7r/^)) ® Q, 416 G*-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 205 If P' is mono, then to show (3 is mono, it is enough to show that RK,(BTC) ® % -> RK,(B(7c/7i:i)) ® % is mono, or that H,(BTT, %) ->• H,(B(7T/7Ti), %) is mono. But this follows from collapsing of the Lyndon-Hochschild-Serre spectral sequence H^TT/TTI, H^i, QJ) => H^TT, QJ. PROPOSITION a. 9. — Suppose 7^ onrf 7^ ar^ groups for which SNG 2 (^^., SNG 4^ Ao/A, aW fl&o suppose C^i) z'j m ^ category of [27] for which the Kunneth theorem holds (this is the case, for instance, ifn^ satisfies the hypothesis of 2.6, but is never the case ifn-^ is not amenable). Then SNG 2 (resp., SNG4J holds for the direct product ^ x T^. PROOF. — Using the Kunneth theorems for K, of C^-algebras [27] and for RK, of spaces (which follows by Spanier-Whitehead duality and passage to limits from the Kunneth theorem [2] for K* of finite complexes), we obtain a commutative diagram with exact rows o —> RK,(B7Ti) ® RK^BTtg) ——> RK,(B(7Ti x rrj) -> Tor(RK,(B^), RK^B^)) ——> o Pi®3« 3 Pi* 3. o -> K^G^)) ® K,(e(^)) -> K^G^^ x T^)) —> Tor(K,(G*(7Ti)), K^G^^))) -> o The result for SNG 4 follows by diagram chasing. The case of SNG 2 is even easier since we can disregard the Tor terms. Note, however, that the argument breaks down for SNG 3, since ^ and ^ mono does not imply [B^ ® pg is mono. PROPOSITION a. 10. — Suppose TT^ and TC^ are groups for which SNG 2 (resp., SNG 3, SNG 4; holds. Then SNC 2 (res?., SNG 3, SNC 4; holds for the free product T^*^. PROOF. — We have, of course, B(T^ * ^) == B^ v B^ and C*(^ * 71:3) = G^) * C^) (the free product of the G*-algebras amalgamated over the common identity element). The result now follows trivially from [8]. REMARK. — Undoubtedly there are results like 2.9 and 2.10 for direct products of more general groups and for certain amalgamated free products. However, one encounters serious technical difficulties (such as non-uniqueness of C*-cross-norms on tensor products of non-nuclear algebras); and in the case of amalgamated products, [6] suggests the problems may be more than technical. It should be evident from the Mi§cenko and Kasparov theorems, as well as from 2.4-2.10, that the Strong Novikov Conjecture holds for quite a large class of groups. 417 206 JONATHAN ROSENBER G We proceed to the applications to manifolds of positive scalar curvature. With appro- priate conditions on TT, we obtain slightly more than Conjecture 2.1. THEOREM a. 11 (cf. [10], Theorem 13.8). — Assume the Strong Novikov Conjecture (SNG 2) holds for TT, and suppose M is a closed spin manifold and g : M -> ETC a continuous map. Then if M admits a metric of positive scalar curvature, ^(A(M) n [M]) == o in H^(BTC, Q,), or equivalent for all a GH^BTC, QJ, <A(M)u^),[M]>=o . In particular, if &,([M]) + o in H^BTT, QJ, then M (or even any closed manifold homotopy equivalent to M.} does not admit a metric of positive scalar curvature. NOTE. — In Theorem 3.5 below, we will show that the condition that w^(M) = o can be weakened to the assumption that M has a spin covering. PROOF. — Suppose M admits a metric of positive scalar curvature and g is as indicated. Then g is determined up to homotopy by a homomorphism Tr^(M) -> TT, and hence the pull-back under g of the universal G^TT)-bundle i^ = C*(n) X ETC may 7t be viewed as a flat bundle over M, which can be taken to be smooth. First assume M is even-dimensional. Applying Theorem i. i to g*{^) with A == C*(TT), we get <A(M) u^(ch[^]),[M]>=o, or o = <^(ch|yi), A(M) n [M] > == <ch[y], ^(A(M) n [M]) >, i.e. pQjr,(ch~l(A(M) n [M]))). But ch is an isomorphism modulo torsion so this together with SNG 2 says ^(A(M) n [M]) = o in H,(BTC, %). If ^([M]) + o, this is impossible, since A(M) == i + (terms of degree^ 4), and in fact only the homotopy type of M is relevant in this case (since we've used the orientation of M but not the Pontrjagin classes). If M is odd-dimensional, consider instead N = M x S1 with the product metric obtained from a metric of positive scalar curvature on M and the flat metric on S1. We replace g by g X id: M x S1 -^ BTT x S1 = B(TC x Z). By 2.9, SNG 2 holds for ye x Z, so by the case already considered, (g x irf),(A(N) n [N]) = o. However, ^ • •. . ' • << ' _ A(N) is the external product ofA(M) and of i e H°(S1), and [N] is the external product of [M] and [S1]. Since id^: H»(S1) -> H,(S1) is an isomorphism, we obtain ^(A(M)n[M])=o . 418 C*-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 207 3. Refinements A) Taking the torsion into account As we mentioned previously, there is good evidence in [10] that if M is a closed spin manifold with fundamental group n and classifying map (for the universal covering) /: M ->BTT;, then the condition for existence of a metric of positive scalar curvature on M should be the vanishing of the image of the class of (M,/) under the j^-homo- morphism Q^BTr) -^ RKO^BTT:). Our methods so far have verified one direction of this conjecture (positive scalar curvature => vanishing of the " j^-genus "), modulo torsion, in the case where the Strong Novikov Conjecture holds for TT. Our objective now is to improve this result by taking the torsion in RKO,(BTC) into account. It is clear (from [n], §§ 4.1-4.4) that to obtain the best possible results about 2-primary torsion, it will be necessary to deal with the KR- or KO-index of real Dirac operators. This already causes serious difficulties, since to carry out the same program we have discussed above, it would be necessary to formulate an analogue of the MiSfenko- Fomenko index theorem using real or "Real" G*-algebras. However, [16] suggests that this is feasible. A more serious obstacle is that it appears that the obvious analogues of SNC 3 and SNC 4 in real K-theory are not likely to be true in very many cases, so that some essentially new idea will be required. We therefore leave this as a difficult but promising area for future work. On the other hand, since KO®Z[i/2] is a direct summand in KU®Z[i/2], it is natural to try to deal with ^-primary torsion (p =(= 2) by using complex K-theory. For this, the methods we have used so far <c almost work "—the only difficulty is that the MiSfienko-Fomenko theorem is not quite optimal; it only computes the index of an A-elliptic operator in Ko(A) ® %, instead of computing " on the nose " in Ko(A). For purposes of analogy, one might compare the case of the index theorem for families—the K-theoretic version in [5] takes torsion into account, whereas the homological formu- lation in [28] does not. We begin, therefore, with a K-theoretic version of the Mi§cenko-Fomenko index theorem. Since this is probably well-known to Kasparov and Gonnes, who may publish more complete accounts elsewhere, we shall omit some details. It is necessary first to recall the version of the index theorem formulated in [7] (first four pages). If X is a closed manifold, not necessarily orientable, then the cotangent bundle T*X always carries a canonical Spin^structure. This defines a class, which we shall call ^ for " index ", in Ko(T*X) = KK(Co(T*X), C). This is essentially the class of the Dirac operator defined by the Spin^structure, or in the language of [7], /!, where /: T*X -> pt. Now if D is an elliptic operator over X (more accurately, taking sections of one vector bundle over X to sections of another), then D defines a class [D] e Ko(X) = KK(G(X), C). On the other hand, the symbol <rof D defines a class [a] e K°(T*X). The index theorem of [7] (or of [14], § 7) says that ind D ===^([D]) eKo(pt) = Z (where ^:X-^pt ) 419 2o8 JONATHAN ROSEN B ER G can also be computed as [a] ®Co(T*x) ^* ^ ^ is itself an even-dimensional Spin0- manifold and D == DE" is the Dirac operator with coefficients in a vector bundle E over X, then ^!eKo(X) is also defined, and the index formula simplifies to ind DE- = [E] ®c(x)^!. We are ready for THEOREM 3.1 (K-theoretic MiScenko-Fomenko theorem). — Let X be a closed manifold^ let A. be a C*-algebra with unit, and let D be a pseudodijferential elliptic ^.-operator over X with symbol class [a] e K°(T*X, A). Then D defines a class [D] in KK(G(X), A) and ind^ D = A([D]) == M ®co(T*x) ^ ^ KK(C, A) = Ko(A). Here p : X -> pt and ^ is the index class in K.o(T*X) as above. If X is an even-dimensional Spin°-manifold and D == D^ is the Dirac operator with coefficients in an A-vector bundle E, then also indAD==[E]OOc(x)^! (where pi is defined as in [7] using the SpW-structure on X). PROOF (Sketch). — Suppose D is an n-th order operator from r°°(E) to r°°(F), where E and F are A-vector bundles over X. Then ^ = H^X, E) ©HP-^X, F) is a graded Hilbert A-module. Without loss of generality, we may assume n == o and s == o, in which case the Sobolev spaces become just L^X, E) and L^X, F) (inter- preted in terms of the A-valued inner product). Then we get a representation /o m . 9 : C(X) -> oS^(J^) by pointwise multiplication, and T = | ) is a self-adjoint operator in -^(^) (of degree i with respect to the grading), which because of the ( < A-Fredholmness theorem " of Miscenko-Fomenko ([23], Theorem 3.4) is invertible modulo Jf(J^). The commutator of D with multiplication by any ^eC^X) is pseudo-differential of order — i; thus [T, y(C(X))] <= ^(^) ^ 1>3L Lemma 3.3. Thus according to the prescription of [i6], § 4, the triple (e^, 9, T) defines an element of KK(C(X), A). (To put it in standard form, we need T2 — i ejT(^), but by [21], Gh. i, this can be arranged by replacing D by a partial isometry with the same A-index.) From the definition ofp^ in [i6], § 4, one can easily check that ^*([D]) coincides with the A-index of D as defined in [23]. As in [7], one can now check that [D] == [o] ®C,(T*X)^X with ^x eKoCPX X X) as defined by Connes-Skandalis. Then the index formula follows from [7], Gorollaire 4, since ind^D ==A([<y]®c^x)^x) == M ®c,(T*x)A(^x) = M®C.(T*X)^- In the case where X is an even-dimensional Spin^manifold, the same argument of [7], p. 872, that works when E is an ordinary vector bundle basically works here. THEOREM 3.2 (revised version of Theorem i. i). — Let M be a closed even-dimensional spin manifold, and suppose M admits a Riemannian metric whose scalar curvature is everywhere 420 G^-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 209 non-negative and strictly positive at at least one point. Let A be any C*-algebra with unit and 1^ aflat K-vector bundle over M. Then \i^} ®c(x)^ 1 = ° m ^(A), where p : M ->pt and p\ is defined using the spin structure on M. PROOF. — The proof of Theorem i. i applies word-for-word, except that one must replace the Miscenko-Fomenko index calculation by the second part of Theorem 3.1. THEOREM 3.3 (revised version of Theorem 2.11) . — Let M. be a closed spin manifold admitting a metric of positive scalar curvature. Let n == •^(M) and let f: M ->BTC be the classifying map for the universal covering of M. If the Strong Novikov Conjecture (SNC 3) holds for n, then f^{p\) = o in RK^BTT). (Here p\ is the class of the Dirac operator in Ko(M) if M is even-dimensional or in Ki(M) if M is odd-dimensional.) PROOF. — First suppose M is even-dimensional, and apply Theorem 3.2 with V == G*(7c) Xn M the universal C^TC)-bundle over M. We conclude that DO ®C(X}P^ = °- However, V is the pull-back under / of the universal G*(7r)- bundle ^^ over BTT used to define the Kasparov map p. So by the naturality property of the intersection product, /'([^J) ®G(X)^! = D^] ®B./^'). ^. P(/^')) = o ([i6], § 4, Theorem 4). But p is injective by hypothesis. If M is odd-dimensional, repeat the argument with M X S1 as in the proof of 2. n , then " de-suspend ". Combining 3.3 with Theorem 2.6, we get one direction of the Gromov-Lawson conjecture modulo 2-primary torsion for a large class of solvable fundamental groups. (The map [(M,/)] \->f^p^^) gives the analogue of the Milnor j^-homomorphism in complex K-theory: O^B^) -^ RK,(BTT).) B) Weakening the spin condition The notion of enlargeability used in [10] (as opposed to that of [9]) suggests that we ought to be able to weaken the spin condition on M in Theorem 2.11, and assume only that the universal covering M of M should carry a spin structure. (For instance, there might conceivably be non-spin K(TT, i)-manifolds with TT not having any finite quotients, in which case this improvement might be needed in order to prove non- existence of positive-scalar-curvature metrics on such manifolds.) It turns out that techniques using operator algebras are very convenient for handling this extra generality. (It is possible one could even take torsion into account at the same time, as in § 3A above, but we are not sure how to do this at the moment.) The essential idea is the following. Suppose M is a closed oriented even-dimensional manifold with fundamental group re, whose universal covering M (usually non-compact, of course) carries a spin structure. Also let E be some A-vector bundle over M. We would like to compute the <c index of the Dirac operator D^ with coefficients in E>9 ? over M, but of course this does not make any sense. However, we have half-spinor /^/ f^/ bundles y± and Dirac operators D^ defined over M, and the action ofjron M by covering 27 210 JONATHAN ROSENBER G transformations <( almost " commutes with D^, in the sense that we have a " projective action " of n on spinors and the Dirac operator, or a true action of some double cove- ring ft of TT on ^7±, commuting with D±. If Pgo denotes the principal SO-bundle of orthonormal oriented frames on M, and Pgp^ denotes the principal spin bundle, then TT acts on Pgo (since M is oriented—of course we choose the orientation of £l compatible with that on M), and the group ft is constructed by " pull-back ": f, -^ Aut(Pg^) 7T ——> Aut(Pgo). Then ft acts on the spinor bundle «^, and preserves the splitting into half-spinor bundles since this is defined by the orientation. To see ft commutes with D4', just compute in local coordinates. Now suppose M carries a metric of positive scalar curvature, and for the moment take E to be trivial. If the Dirac operator D4' existed on M, we would know by [19] that A(M) = <A(M), [M]> == o. By the L^index theorem ofAtiyah [3], we also know that indj^ D4' ===0 o L2 — indg D4" = o. However, even if D4' does not make sense on M, one can still show that there are no L^harmonic spinors on M (because the scalar curvature is strictly bounded below). Mikhael Gromov and Blaine Lawson were kind enough to inform me that they had used this fact together with a slight modification of Atiyah's proof to conclude that A(M) = o even when M does not have a spin structure. (Here the L^-index theorem is being used the reverse of the way it is usually applied—data about the non-compact manifold M yields information about M!) However, this argument does not generalize to the case where E is a flat G*-vector bundle, so we prefer to construct a genuine elliptic operator on M and apply the MiS^enko-Fomenko theorem. For this, note that we have a split exact sequence of G*-algebras o^CT(ft)^^ 0^)^0(7^0, where G^ft)^ is universal for unitary representations of ft that are non-trivial on the kernel of the covering map ft ->• TC. The trace coming from the regular representation of ft defines a non-zero finite trace a on C^ft)^? hence a non-zero map Ko(C*(ft)^) -> R. Observe now that since the cocycle of the group extension {l}-^{ ± l}-^ft-^7T-^{l } exactly cancels that for the (c action " of TC on ^±, the G*(ft)^ ® A-vector bundles (G*(ft)odd ^ ^±) ® E actually make sense on M, and we obtain a genuine operator D4'. Now we obtain 422 C--ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 211 THEOREM 3.4 (revised version of Theorem i. i). — Let M be a closed oriented even- dimensional manifold whose universal covering is spin, and suppose M admits a metric of positive scalar curvature. Let A be any C*-algebra with unit and E aflat A-vector bundle over M. Then <A(M) u ch[E], [M]> = o in Ko(A) ® %. PROOF. — Since we are still dealing with flat bundles, the index of the Dirac operator D^ constructed above is zero in K.o(Gllt(7^)^d0A). But from the trace a on C'^TT)^ we obtain a " partial trace " a ® id : G*(TT)^d®A -^A and thus a map K^G^fc)^ ® A) ->-Ko(A)®R (l). Applying this to the index formula now gives ^<A(M ) u ch[E], [M]> = o in Ko(A) ®R, where c > o is some number which we do not need to compute, but which comes out to i if we normalize a correctly. The same arguments as in § 2 now give a revised version of Theorem 2.11, and proofs of Conjectures 2.2 and 2.3, assuming n satisfies SNG 2. THEOREM 3.5. — Assume the Strong Novikov Conjecture (SNG 2) holds for TT, and suppose M is a closed oriented manifold whose universal covering has a spin structure. Let g: M -> BTT be any continuous map. Then ij"M admits a metric of'positive scalarcurvature, ^,(A(M) n [M]) == o in H^(BTT, QJ, or equivalently, for all a e H^BTC, QJ, <A(M) u^),[M]>=o. In particular, if ^»([M]) + o in H^(BTT, Q^), then no manifold homotopy-equivalent to M admits a metric of positive scalar curvature. REFERENCES [i] J. F. ADAMS, Infinite Loop Spaces, Annals of Math. Studies, no. 90, Princeton, Princeton Univ. Press, 1978. [a] M. F. ATIYAH, Vector bundles and the Klinneth formula, Topology 1 (1962), 245-248. [3] M. F. ATIYAH, Elliptic operators, discrete groups and von Neumann algebras, Asterisque, no. 32-33 (1976), 43-72. [4] M. F. ATIYAH and I. M. SINGER, The index of elliptic operators: III, Arm. of Math. (2) 87 (1968), 546-604. [5] M. F. ATIYAH and I. M. SINGER, The index of elliptic operators: IV, Ann. of Math. (2) 93 (1971), 119-138. [6] S. E. CAPPELL, On homotopy invariance of higher signatures. Invent. Math. 33 (1976), 171-179. [7] A. CONNES and G. SKANDALIS, Theoreme de Findice pour les feuilletages, C. R. Acad. Sci. Paris, Ser. I, 292 (1981), 871-876. [8] J. CUNTZ, The K-groups for free products ofC'-algebras, in Operator Algebras and Applications, R. V. KADISON, ed., Proc. Symp. Pure Math., vol. 38, Amer. Math. Soc., Providence, R. I., 1982, Part I, 81-84. (1) At least when the Klinneth theorem of [27] holds. However, we do not need to worry about this here, since the Chem character of the symbol of D+ clearly lies in H;(T*M, Q) ®K,(C*(7r)odd) ®K,(A). 423 ai2 JONATHAN ROSENBER G [9] M. GROMOV and H. B. LAWSON, Jr., Spin and scalar curvature in the presence of a fundamental group, I, Ann. of Math. (2), 111 (1980), 209-230. [10] M. GROMOV and H. B. LAWSON, Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, this volume, 83-196. [ii] N. HITCHIN, Harmonic spinors. Adv. in Math, 14 (1974), I-55. [12] W. C. HSIANG and H. D. REES, Miscenko^s work on Novikov's conjecture, in Operator Algebras and K-Theory, R. G. DOUGLAS and C. SCHOCHET, eds.. Contemporary Math., vol. 10, Amer. Math. Soc., Providence, R. I., 1982, 77-98. [13] M. KAROUBI, K-Theory: An Introduction, Grundlehren der math. Wissenschaften, no. 226, Heidelberg and New York, Springer-Verlag, 1978. [14] G. G. KASPAROV, Topological invariants of elliptic operators, I: K-homology, Izv. Akad. Nauk SSSR, Ser. Mat. 39 (i975)> 796-838 == Math. USSR-Izv. 9 (1975), 75I-792. [15] G. G. KASPAROV, Hilbert C'-modules: theorems of Stinespring and Voiculescu, /. Operator Theory 4 (1980), I33-I50. [16] G. G. KASPAROV, The operator K-functor and extensions of C*-algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980), 571-636 == Math. USSR-Izv. 16 (1981), 513-572. [17] G. G. KASPAROV, K-theory. group C*-algebras, and higher signatures (conspectus), preprint, Chernogolovka, 1981. 118] J. L. KAZDAN and F. W. WARNER, Scalar curvature and conformal deformation of Riemannian structure, J . Differential Geometry 10 (1975), 113-134. [19] A. LICHNEROWICZ, Spineurs harmoniques, C. R. Acad. Sci. Paris, Ser. A-B 257 (1963), 7-9. [20] J. MILNOR, On axiomatic homology theory. Pacific J . Math. 12 (1962), 337-34I. [21] J. A. MINGO, K-theory and multipliers of stable C*-algebras, Ph. D. Thesis, Halifax, N.S., Dalhousie Univ., 1982. [22] A. S. MiScENKO, C*-algebras and K-theory, in Algebraic Topology, Aarhus 1978, J. L. DupONrandH. MADSEN, eds., Lecture Notes in Math., no. 763, Heidelberg and New York, Springer-Verlag, 1979, 262-274. [23] A. S. MI§CENKO and A. T. FOMENKO, The index of elliptic operators over G*-algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 43 (1979), 831-859 == Math. USSR-Izv. 15 (1980), 87-112. [24] A. S. MI§CENKQ and J. P. SOLOV'EV, On infinite-dimensional representations of fundamental groups and formulas of Hirzebruch type, Dokl. Akad. Nauk SSSR 234 (1977), 761-764 = Soviet Math. Dokl. 18 (1977), 767-771. [25] M. PIMSNER and D. VOIGULESCU, Exact sequences for K-groups and Ext-groups of certain cross-product C*-algebras, /. Operator Theory 4 (1980), 93-118. [26] J. ROSENBERG, Group C*-algebras and topological invariants, in Operator Algebras and Group Representations, Proc. Conf. in Neptun, Romania, 1980, London, Pitman, 1983, to appear. [27] C. SCHOCHET, Topological methods for C*-algebras, II: geometric resolutions and the Kunneth formula, Pacific J . Math. 98 (1982), 443-458. [28] W. SHIH, Fiber cobordism and the index of a family of elliptic differential operators. Bull. Amer. Math. Soc. 72 (1966), 984-991- Department of Mathematics, University of Maryland, College Park, Maryland 20742 Manuscrit refu Ie 28 mars 1983. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

C*-algebras, positive scalar curvature, and the Novikov conjecture

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Springer Journals
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Copyright © 1983 by Publications Mathématiques de l’I.H.É.S.
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Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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C-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE by JONATHAN ROSENBERG (1) This note, an appendix to the foregoing paper [10], is based on the philosophy of [9] that topological obstructions to existence of metrics of positive scalar curvature on non-simply connected manifolds should be closely related to the Novikov conjecture on " higher signatures ". I am very grateful to Mikhael Gromov and to Blaine Lawson for patiently explaining their work to me, for suggesting that I examine its relation to the work of Kasparov and of MiScenko et a/., and for helping with many details of this project. I would also like to thank Alain Connes, Paul Baum, Ron Douglas, Jerry Kaminker, John Miller and Larry Taylor for helpful discussions concerning index theorems and/or the Novikov conjecture, and Alain Connes and Mikhael Gromov for useful suggestions about improvements in the exposition of this paper. We shall show below that certain manifolds do not admit Riemannian metrics of positive scalar curvature. In most situations involving closed manifolds of practical geometric interest, our results of this type will be the same as those of [9] and of Corol- lary A and Theorem J of [10]. Nevertheless, it is not clear that their results always imply ours or vice versa. More significantly, we hope to illustrate another way in which operator algebras can be applied to problems of geometry and topology, and to demons- trate a closer relationship between the positive scalar curvature problem and the Novikov conjecture than may be apparent from the Gromov-Lawson method of attack. Our main results are Theorems 3.3 and 3.5 below. However, their proofs involve technical complications that tend to obscure the main idea, so we have first given the proof of an easier special case, which we have formulated as Theorem 2.11. The auxiliary results 2.4-2.10 on the Novikov conjecture (especially Theorem 2.6) may be of independent interest, i. The main construction As in [9] and [10], our tool will be an analysis of the index of the Dirac operator with coefficients in a bundle, together with the idea of the vanishing theorems of [19] (1) Sloan Foundation Fellow. Partially supported by NSF Grant no. MCS-Sa-ooyoG. 409 198 JONATHAN ROSENBER G and [i i]. But whereas Gromov and Lawson use for coefficients ordinary vector bundles of small curvature, we shall use flat coefficient bundles {i.e. bundles of zero curvature), but only at the expense of making the fibres of the bundles infinite-dimensional. This can be explained by the fact that (by Ghem-Weil theory) ordinary flat bundles have trivial rational characteristic classes, whereas infinite-dimensional flat bundles sometimes do not. Our main technical tool will therefore be not the Atiyah-Singer index theorem but rather the index theorem of MiScenko and Fomenko [23]. Thus we begin with a review of the key definitions and concepts in the Mi§cenko-Fomenko theory. Let A be a (complex) G^-algebra with unit. If X is a compact space, by an A-vector bundle over X we shall mean a locally trivial (Banach) vector bundle E over X, in which the fibres have the structure of finitely generated projective (left) A-modules. Morphisms of such bundles will be required to preserve the A-module structure on the fibres. The Grothendieck group of formal differences of equivalence classes of such bundles (with addition coming from the Whitney sum operation) will be denoted K°(X,A); it coincides with the algebraic K-group Ko(C(X)®A), where ® means the (spatial) CT-tensor product. (Of course, C(X)®A^ C(X,A). For all this see not only [23] but also [13], Exercise 11.6.14, and [21], Ch. i.) By the Kunneth theorem of [27], or rather by a relatively easy special case thereof, for compact metrizable X there is a natural isomorphism K°(X, A) ® Q^ (K°(X) ® Ko(A) 0 QJ C (Ri(X) ® K^A) ® QJ (where the tensor products are taken over Z). Thus using the ordinary Ghern character K^X) -^H*(X, %), we obtain the Chern character of Miscenko-Solov'ev [24]: ch : K°(X, A) -> H^X, %) ® Ko(A) C H^X, %) ® Ki(A), which is an isomorphism modulo torsion. Now suppose X is a closed manifold (for us this will always mean a compact connected G00 manifold without boundary), E and F are smooth A-vector bundles over X, and D is an elliptic pseudodifferential A-operator of order n in the sense of [23], § 3, taking smooth sections of E to smooth sections of F. One can define Sobolev spaces IP(X,E) and H^X, F), which are Hilbert G*-modules (see [23], § i, or [15]) over A. The fundamental theorem of [23] asserts that D defines a bounded operator IP(X, E) -^IP'^X, F), that this operator is an A-module map, with adjoint, which is A-Fredholm {i.e. invertible modulo A-compact operators), and that the " A-index " of D is independent of the choice ofj. One must be careful with the definition of index here; the kernel and cokernel of D are not necessarily finitely generated and A-projective, but this is true of D + K for some A-compact operator K. The A-index of D is then the formal difference (in Ko(A)): [ker(D + K)] - [coker(D + K)], which turns out to be independent of the choice of K ([23], § i, and [21], Gh. i). Finally, the A-index of D depends only on the class of the principal symbol of D in K°(T*X, A) 410 C*-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 199 (where T*X is the cotangent bundle of X, and where for locally compact spaces, K* denotes K-theory with compact supports). The A-index is, modulo torsion, given by a formula formally identical to that of Atiyah-Singer [4], the only difference being that the Ghern character must be interpreted as just explained above. We now apply this machinery to the case of spin manifolds of positive scalar curvature. If n is a discrete group, we denote by ETC a classifying space for TT, i.e. a K(TC, i)-space. For any (connected) manifold M with fundamental group TT, the universal covering of M is associated to a classifying map /: M ->BTT, well-defined up to homotopy and inducing the identity on TT^. THEOREM 1.1 . — Let M be a compact smooth even-dimensional spin manifold without boundary, and suppose M admits a Riemannian metric whose scalar curvature is everywhere non- negative and strictly positive at at least one point. Let A be any G-algebra with unit and i^ a flat (i.e. pulled back from ETC, where TC = n^{M)} A-vector bundle over M. Then <A(M) u chl/H, [M]> = o in Ko(A) ®%. PROOF. — By Proposition 3.8 of [i8], we may assume that M has been given a Riemannian metric whose scalar curvature K is everywhere strictly positive, hence bounded below by a constant KQ > o. Let <^4" and y~ be the two half-spinor bundles of M. We may give V a flat connection and thus define Dirac operators (y : r^^®^) -^r00^-®^) ID- : r°°(^- ® v) -> r^.^ ® -T) which are formal adjoints of one another with respect to the A-valued inner products. These are elliptic A-operators, and the usual calculation gives Th^^M) u chlXD4-)]) == A(M) u ch[^], A. ^ where A(M) is the total A-class of M, a certain polynomial in the rational Pontrjagin classes, and where Th : H*(M, %) ->H^(T*M, QJ is the Thorn isomorphism. Thus by the MiScenko-Fomenko Theorem, we get ind^ D4- = <A(M) u ch^], [M]> e Ko(A) ® %. We claim, however, that D4' has A-index zero. Indeed, with respect to the A-valued L^inner products on I^^®^), we have DTD± = V*V + K, by exactly the same calculation as in [9], § i, and in [10], § i. (There is no ^ term since ^ is flat.) We can conclude from this (together with the fact that K >_ KQ > o) that D'^'D- and D~D'1" are one-to-one with dense ranges and bounded inverses. Here 411 200 JONATHANROSENBERG we must use spectral theory and the fact that if a and b are positive self-adjoint elements of a G*-algebra, then a + b >, b and hence \\a + b\\ ^ \\b\\. To justify this step precisely, one can for instance first embed i^ as a direct summand in a trivial bundle of the form M X A" (using [21], Proposition 4.3). The operator 1C v<tv + - extends in an obvious way to F^^ ^A^, and since the coefficient bundle is trivial, we can use ordinary spectral theory on the first factor. Since VV+^^ i Ko/4, we conclude that D^D' and D~D'1' have bounded inverses with respect to the L^norms. A compact perturbation of D'1" will then be invertible as an operator from H84"1^4") to IP(^-), and tensoring with the identity operator on A", we see that D-^- has A-index zero. Hence <A(M) u ch[V], [M]> = o. a. C'-algebras and the Novikov Conjecture The usefulness of Theorem i. i depends on the possibilities for ch[^], i.e. for the characteristic classes of flat A-vector bundles over M. The best one could hope for is that these might generate all the <c characteristic cohomology 39 of M, i.e. that if /: M ->BTT; is the classifying map for the universal covering of M, one has CONJECTURE a. i (Vanishing of higher A-genera). — If M is a closed spin mamfold admitting a metric of positive scalar curvature, and if TT ^ TCi(M), then for all a e H*(B7r, %), <A(M) u/*(a), [M]> = o. Note that Conjecture 2.1 can be rephrased in the form /,(A(M) n [M]) == o. Modulo torsion, this is the same as the conjecture in [10] that the image of the class of (M,/) must vanish under the " J^-homomorphism " Q^BTT) ->KO,(B7r). We shall see now that Conjecture 2. i follows from the same intermediate step needed by Kasparov and by Migcenko in their proofs of the Novikov Conjecture for certain fundamental groups TC. Thus Conjecture 2. i is valid for many (and conceivably all) finitely presented groups TC. One can deduce from this that spin manifolds of certain homotopy types do not admit metrics of positive scalar curvature, and that within other homotopy types of manifolds, such metrics may exist but only subject to certain cons- traints on the Pontrjagin classes. We shall also see in § gB below that the somewhat annoying spin condition in 2. i can be relaxed to give CONJECTURE 2.2. — If M. is a closed oriented manifold admitting a metric of positive scalar curvature, if the universal covering of M is spin, and if n = ^{M), then for all a e H^BTT, %), <A(M) uf\a), [M]> = o. COROLLARY (TO THE CONJECTURE) 2.3. — No closed K(TT, i) -manifold can admit a metric of positive scalar curvature. 412 C*-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 201 PROOF OF 2.3 FROM 2.2. — Let M be a closed K(7r, i)-manifold of positive scalar curvature. If M is not orientable, pass to a double covering and replace TC by a sub- group of index 2. We can take M == ETC and f == id, and of course the universal covering of M is spin since it is contractible. Taking a + o in the top-degree coho- mology of M, we get <fl, [M]> = o, a contradiction. It seems that Conjecture 2. i (for a fixed finitely presented group 71) is closely related to the (generalized) Novikov Conjecture for the same group, which we state here for reference: NOVIKOV CONJECTURE. — Among the class of closed oriented manifolds M with funda- mental group 7Ti(M) ^ TT, if f: M -^BTC denotes the classifying map, then the " higher signa- tures" <L(M) uf*{a), [M]>, a eH*(B7r, QJ, are oriented homotopy invariants. Despite the obvious similarity between higher signatures and higher A-genera, we know of no direct way of deducing Conjecture 2. i from the Novikov Conjecture. However, it seems that for most groups for which one can prove one of these, one can prove the other as well. It is therefore worth briefly summarizing some of the ideas of [17]. § 9- If TT is any countable group, one can choose the space BTT to be a countable GW-complex (but usually not to be a finite complex). Following Kasparov, we denote by RK* representable topological K-theory, the representable cohomology theory coinciding with K* on compact spaces. For a countable CW-complex X with finite skeletons X^, one can also define LK*(X) = HmK^XJ, and there is a surjection RK(X) -> LK^X) (with kernel given by lim^ K^-^XJ [20]). We denote by K, the homology theory on compact metrizable spaces dual to K*, and let RK^(X) = lim K^(X^) for X as above. For a separable G*-algebra A, we denote by K*(A) the group ELK* (A, C) < ( dual9 5 to K,(A) == KK*(C, A). (Caution: our convention about raised and lowered indices is reversed from that of [16] and [17], in order to agree with the usual convention that K* should denote a contravariant functor. Thus K^^X))^ K^(X).) As in [i6], ®A denotes the "intersection product59 K^(A) ®K*(A) ->K^(C) == Z, gene- ralizing the Kronecker pairing between K-homology and K-cohomology. If C^TT) denotes the group C*-algebra of n (the completion of L^TC) in the greatest G*-norm), Kasparov defines maps a : K^G^TT)) ->RK*(BTT) and (3 : RK,(BTT) -> K,(G*(7c)) which are dual to one another in the sense that if S' denotes the projection of a into LK^BTC), then a(;c)®B^ == ^®c^)P(jO {^ x e K^G^)), ^(=RK,(BTI). Consider now the following possible properties of TT: PROPERTY SNG i. — The image of 0' (after tensoring with QJ is dense in the projective limit topology on LK*(BTI) ®Q,. 26 209 JONATHAN ROSENBER G PROPERTY SNG 2. — The map (B is injective after tensoring with Q^. PROPERTY SNC 3. — The map (B is injective (even without tensoring with QJ. PROPERTY SNG 4. — The map (3 is an isomorphism from RK,(BTT) -> K^C^TC)). Here the abbreviation " SNG " stands for " Strong Novikov Conjecture ", terminology which is defensible because of the string of implications (SNG 4) => (SNG 3) => (SNG 2) <= (SNC i), and the theorem ofKasparov ([17], § 9) that ifrc is finitely presented (so that the Novikov Conjecture for TT makes sense), then SNG i or SNG 2 implies the Novikov Conjecture for TT (in the form stated above). This can be explained as follows. Suppose M is a closed oriented manifold with fundamental group TT. Then we can form the <c universal flat G* (71;)-vector bundle " V == C*(7r) x^ M over M, where M is the universal covering of M and TT acts on C*(7c) by right translation and on M by covering transfor- mations. The generalized signature of M is defined by <L(M) u ch[y^], [M]>; this may be thought of as the G*(TI:)-index of the signature operator with coefficients in i^ (in the case where dimM =. o (mod 4)) (1). By [22] or by [17], § 9, Theorem 2, this generalized signature is an oriented homotopy invariant. Applying ®c*(7r)^ ^or classes y eK*(C*(7r)), it follows that <L(M) uf*(a), [M]> is an oriented homotopy invariant if a e H^BTC, QJ lies in the image (under the Chern character) of the image of a, and hence SNG implies the Novikov Conjecture. In the remainder of this section we list a number of situations where various forms of SNG are known, and then broaden the list a bit by discussing various stability pro- perties of the class of groups with SNC. (One may view this as a C*-algebraic counter- part to [6].) Then we describe consequences for the positive-scalar-curvature problem. The fundamental (and deep) results are the following: THEOREM (Migcenko, see [12]). — Suppose there exists a closed orientable K(TT, i)- manifold admitting a Riemannian metric with non-positive sectional curvatures. Then SNC i holds for TV. THEOREM (Kasparov, see [17], § 9, Theorem i). — Suppose n can be embedded as a discrete subgroup in a connected Lie group. Then SNG i holds for n (even without tensoring with QJ. One should note that MiScenko's theorem is stated in terms of existence of certain Fredholm representations, but the above formulation is equivalent. Also, these results appear to lend support to Conjecture 2.3, since there do not seem to be any published examples of closed oriented K(7r, i)-manifolds for which the hypotheses of both theorems (l) We are cheating slightly. One should really replace the Hirzebruch class L(M) here by the Atiyah- Singer modification ([4], § 6) of it, -§?(M), and then multiply in front by a suitable power of 2. However, for purposes of formulating the Novikov Conjecture, it does not matter whether one uses L or -S?. 4U C*-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 203 fail. However, M. Gromov has suggested that there are probably many K(TT, i)- manifolds not satisfying these conditions, so that we are still far from a proof of 2.3. For purposes of proving stability properties of SNG, there is a considerable technical advantage to working with SNC2-SNG4 rather than with SNG i. The reason is that the functors RK, and (for G*-algebras) K^ commute with direct limits, whereas RK* and K* do not. This makes it possible to avoid dealing with the Milnor lim^ sequence and with problems arising from I-adic completion. PROPOSITION 2.4. — Suppose n is a countable direct limit n = lim^ and the groups n^ all satisfy SNG 2, SNC 3, or SNG 4. Then the same holds for TV. PROOF. — We have BTT == lim BTT^ and C*(7c) = limG*(7rJ (G*-algebra inductive limit). So use naturality of (B and the fact that RK, and K, commute with direct limits. PROPOSITION 2.5. — Suppose n is an extension i ->7ri—>-7c->Z—>- i and SNG 4 holds/or the group TC^. Then SNG 4 holds for n. PROOF (cf. [26], Theorem 3.3). — Since Z is free, the group extension must split, and we can write n == TT^ X[ Z, where the semidirect product structure is determined by an automorphism 6 of TTr This gives a corresponding decomposition of C*(TC) as a crossed product C^TT^) XIQ Z and of BTT as a fibre bundle over S1 = BZ with fibres ^ BTT^. Then we may compute K,(G*(7r)) from K,(G'''(7ri}) via the Pimsner- Voiculescu exact sequence [25] and RK,(B7r) as the homology of the mapping torus of the self-map 6, ofBTTi. As in the proof of [26], Theorem 3.3, we obtain a commu- tative diagram with exact rows K,(G*(7ri)) -^ K,(G*(7r,)) K,(G-(7r)) K.-i(G*(^)) K,-i(C*(^)) e. Pi 01 Pi RK<(B7ti) -^ RK,(B7ti) ——^ RK,(B7t) ——> RK,_i(B7ti) RK,_i(B^) Thus if the maps (Bi are isomorphisms, so is p, by the 5-Lemma, REMARK. — Because of the need to use the 5-Lemma, it is clear that the proof of 2.5 breaks down if we replace SNG 4 by SNG 3 or SNG 2. THEOREM a. 6. — If n is a countable solvable group having a composition series in which the composition factors are torsion-free abelian, then SNG 4 holds for TC. PROOF. — We argue by induction on the length of such a composition series. Suppose TCI < TC and TC/TCI is torsion-free abelian. We may assume that the theorem 204 JONATHANR.OSENBER G already holds for ^. (To start the induction, take ^ ={1 } if TT is abelian) Now T^ is the direct limit of its finitely generated subgroups, hence is a countable direct limit of free abelian groups. Thus n = Hm^, where ^ < ^ for each n and where ^ is free abelian. By 2.4, it is enough to prove the theorem for TC . For this it is enough to use the inductive hypothesis and 2.5 (repeated a finite number of times). REMARK. — Theorem 2.6 applies to some cases not covered by Kasparov's theorem since there are many groups (even finitely presented) satisfying the hypothesis of 2 6 but not polycyclic. PROPOSITION 2.7. — Suppose TT contains a subgroup ^ of finite index for which SNC 2 holds. Then SNG 2 holds for -K. PROOF. — The group ^ acts freely on ETC so we obtain a finite covering E7t/^~B7^E7t/7t=B7t. (If ^ is normal then the covering is Galois, but this is irrelevant for our purposes.) We also have an inclusion i: C*(^) ^ c^) making C^n) into a finitely generated free C^) -module. Thus we have maps t,:K.(G*(7Ci))-^K.(G*(TC)) as well as a transfer map t: K (G*(7c)) -»K (C*(TC )) (coming from the forgetful functor from finitely generated projective G*(7t)-modules to finitely generated projective G-(^)-modules). Note that tai, is multiplication by [7r:TCj, hence i, is split mono after tensoring with Q (to invert [w • TT ]) Arguing similarly with the transfer in K-homology (see, e.g., [i], Ch. 4), we 'see A : RK.(B^) -^. RK.(BTC) is split epi after tensoring with %. The conclusion follows from consideration of the commutative diagram RK.(B7Ti) ® Q, -^ K.(C*(^)) ® % P« RK.(B7t) ® % -^ K.(C*(7t)) ® Q,. PROPOSITION a. 8. — Suppose TC contains a finite normal subgroup ^ and SNC 2 holds for TT/TCI. Then SNC 2 holds for TC. PROOF. — Consider the commutative diagram RK.(B7T)®Q," K,(G*(7r))®% RK.(B(7r/7ti)) ® % -^ K.(G*(7r/^)) ® Q, 416 G*-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 205 If P' is mono, then to show (3 is mono, it is enough to show that RK,(BTC) ® % -> RK,(B(7c/7i:i)) ® % is mono, or that H,(BTT, %) ->• H,(B(7T/7Ti), %) is mono. But this follows from collapsing of the Lyndon-Hochschild-Serre spectral sequence H^TT/TTI, H^i, QJ) => H^TT, QJ. PROPOSITION a. 9. — Suppose 7^ onrf 7^ ar^ groups for which SNG 2 (^^., SNG 4^ Ao/A, aW fl&o suppose C^i) z'j m ^ category of [27] for which the Kunneth theorem holds (this is the case, for instance, ifn^ satisfies the hypothesis of 2.6, but is never the case ifn-^ is not amenable). Then SNG 2 (resp., SNG4J holds for the direct product ^ x T^. PROOF. — Using the Kunneth theorems for K, of C^-algebras [27] and for RK, of spaces (which follows by Spanier-Whitehead duality and passage to limits from the Kunneth theorem [2] for K* of finite complexes), we obtain a commutative diagram with exact rows o —> RK,(B7Ti) ® RK^BTtg) ——> RK,(B(7Ti x rrj) -> Tor(RK,(B^), RK^B^)) ——> o Pi®3« 3 Pi* 3. o -> K^G^)) ® K,(e(^)) -> K^G^^ x T^)) —> Tor(K,(G*(7Ti)), K^G^^))) -> o The result for SNG 4 follows by diagram chasing. The case of SNG 2 is even easier since we can disregard the Tor terms. Note, however, that the argument breaks down for SNG 3, since ^ and ^ mono does not imply [B^ ® pg is mono. PROPOSITION a. 10. — Suppose TT^ and TC^ are groups for which SNG 2 (resp., SNG 3, SNG 4; holds. Then SNC 2 (res?., SNG 3, SNC 4; holds for the free product T^*^. PROOF. — We have, of course, B(T^ * ^) == B^ v B^ and C*(^ * 71:3) = G^) * C^) (the free product of the G*-algebras amalgamated over the common identity element). The result now follows trivially from [8]. REMARK. — Undoubtedly there are results like 2.9 and 2.10 for direct products of more general groups and for certain amalgamated free products. However, one encounters serious technical difficulties (such as non-uniqueness of C*-cross-norms on tensor products of non-nuclear algebras); and in the case of amalgamated products, [6] suggests the problems may be more than technical. It should be evident from the Mi§cenko and Kasparov theorems, as well as from 2.4-2.10, that the Strong Novikov Conjecture holds for quite a large class of groups. 417 206 JONATHAN ROSENBER G We proceed to the applications to manifolds of positive scalar curvature. With appro- priate conditions on TT, we obtain slightly more than Conjecture 2.1. THEOREM a. 11 (cf. [10], Theorem 13.8). — Assume the Strong Novikov Conjecture (SNG 2) holds for TT, and suppose M is a closed spin manifold and g : M -> ETC a continuous map. Then if M admits a metric of positive scalar curvature, ^(A(M) n [M]) == o in H^(BTC, Q,), or equivalent for all a GH^BTC, QJ, <A(M)u^),[M]>=o . In particular, if &,([M]) + o in H^BTT, QJ, then M (or even any closed manifold homotopy equivalent to M.} does not admit a metric of positive scalar curvature. NOTE. — In Theorem 3.5 below, we will show that the condition that w^(M) = o can be weakened to the assumption that M has a spin covering. PROOF. — Suppose M admits a metric of positive scalar curvature and g is as indicated. Then g is determined up to homotopy by a homomorphism Tr^(M) -> TT, and hence the pull-back under g of the universal G^TT)-bundle i^ = C*(n) X ETC may 7t be viewed as a flat bundle over M, which can be taken to be smooth. First assume M is even-dimensional. Applying Theorem i. i to g*{^) with A == C*(TT), we get <A(M) u^(ch[^]),[M]>=o, or o = <^(ch|yi), A(M) n [M] > == <ch[y], ^(A(M) n [M]) >, i.e. pQjr,(ch~l(A(M) n [M]))). But ch is an isomorphism modulo torsion so this together with SNG 2 says ^(A(M) n [M]) = o in H,(BTC, %). If ^([M]) + o, this is impossible, since A(M) == i + (terms of degree^ 4), and in fact only the homotopy type of M is relevant in this case (since we've used the orientation of M but not the Pontrjagin classes). If M is odd-dimensional, consider instead N = M x S1 with the product metric obtained from a metric of positive scalar curvature on M and the flat metric on S1. We replace g by g X id: M x S1 -^ BTT x S1 = B(TC x Z). By 2.9, SNG 2 holds for ye x Z, so by the case already considered, (g x irf),(A(N) n [N]) = o. However, ^ • •. . ' • << ' _ A(N) is the external product ofA(M) and of i e H°(S1), and [N] is the external product of [M] and [S1]. Since id^: H»(S1) -> H,(S1) is an isomorphism, we obtain ^(A(M)n[M])=o . 418 C*-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 207 3. Refinements A) Taking the torsion into account As we mentioned previously, there is good evidence in [10] that if M is a closed spin manifold with fundamental group n and classifying map (for the universal covering) /: M ->BTT;, then the condition for existence of a metric of positive scalar curvature on M should be the vanishing of the image of the class of (M,/) under the j^-homo- morphism Q^BTr) -^ RKO^BTT:). Our methods so far have verified one direction of this conjecture (positive scalar curvature => vanishing of the " j^-genus "), modulo torsion, in the case where the Strong Novikov Conjecture holds for TT. Our objective now is to improve this result by taking the torsion in RKO,(BTC) into account. It is clear (from [n], §§ 4.1-4.4) that to obtain the best possible results about 2-primary torsion, it will be necessary to deal with the KR- or KO-index of real Dirac operators. This already causes serious difficulties, since to carry out the same program we have discussed above, it would be necessary to formulate an analogue of the MiSfenko- Fomenko index theorem using real or "Real" G*-algebras. However, [16] suggests that this is feasible. A more serious obstacle is that it appears that the obvious analogues of SNC 3 and SNC 4 in real K-theory are not likely to be true in very many cases, so that some essentially new idea will be required. We therefore leave this as a difficult but promising area for future work. On the other hand, since KO®Z[i/2] is a direct summand in KU®Z[i/2], it is natural to try to deal with ^-primary torsion (p =(= 2) by using complex K-theory. For this, the methods we have used so far <c almost work "—the only difficulty is that the MiSfienko-Fomenko theorem is not quite optimal; it only computes the index of an A-elliptic operator in Ko(A) ® %, instead of computing " on the nose " in Ko(A). For purposes of analogy, one might compare the case of the index theorem for families—the K-theoretic version in [5] takes torsion into account, whereas the homological formu- lation in [28] does not. We begin, therefore, with a K-theoretic version of the Mi§cenko-Fomenko index theorem. Since this is probably well-known to Kasparov and Gonnes, who may publish more complete accounts elsewhere, we shall omit some details. It is necessary first to recall the version of the index theorem formulated in [7] (first four pages). If X is a closed manifold, not necessarily orientable, then the cotangent bundle T*X always carries a canonical Spin^structure. This defines a class, which we shall call ^ for " index ", in Ko(T*X) = KK(Co(T*X), C). This is essentially the class of the Dirac operator defined by the Spin^structure, or in the language of [7], /!, where /: T*X -> pt. Now if D is an elliptic operator over X (more accurately, taking sections of one vector bundle over X to sections of another), then D defines a class [D] e Ko(X) = KK(G(X), C). On the other hand, the symbol <rof D defines a class [a] e K°(T*X). The index theorem of [7] (or of [14], § 7) says that ind D ===^([D]) eKo(pt) = Z (where ^:X-^pt ) 419 2o8 JONATHAN ROSEN B ER G can also be computed as [a] ®Co(T*x) ^* ^ ^ is itself an even-dimensional Spin0- manifold and D == DE" is the Dirac operator with coefficients in a vector bundle E over X, then ^!eKo(X) is also defined, and the index formula simplifies to ind DE- = [E] ®c(x)^!. We are ready for THEOREM 3.1 (K-theoretic MiScenko-Fomenko theorem). — Let X be a closed manifold^ let A. be a C*-algebra with unit, and let D be a pseudodijferential elliptic ^.-operator over X with symbol class [a] e K°(T*X, A). Then D defines a class [D] in KK(G(X), A) and ind^ D = A([D]) == M ®co(T*x) ^ ^ KK(C, A) = Ko(A). Here p : X -> pt and ^ is the index class in K.o(T*X) as above. If X is an even-dimensional Spin°-manifold and D == D^ is the Dirac operator with coefficients in an A-vector bundle E, then also indAD==[E]OOc(x)^! (where pi is defined as in [7] using the SpW-structure on X). PROOF (Sketch). — Suppose D is an n-th order operator from r°°(E) to r°°(F), where E and F are A-vector bundles over X. Then ^ = H^X, E) ©HP-^X, F) is a graded Hilbert A-module. Without loss of generality, we may assume n == o and s == o, in which case the Sobolev spaces become just L^X, E) and L^X, F) (inter- preted in terms of the A-valued inner product). Then we get a representation /o m . 9 : C(X) -> oS^(J^) by pointwise multiplication, and T = | ) is a self-adjoint operator in -^(^) (of degree i with respect to the grading), which because of the ( < A-Fredholmness theorem " of Miscenko-Fomenko ([23], Theorem 3.4) is invertible modulo Jf(J^). The commutator of D with multiplication by any ^eC^X) is pseudo-differential of order — i; thus [T, y(C(X))] <= ^(^) ^ 1>3L Lemma 3.3. Thus according to the prescription of [i6], § 4, the triple (e^, 9, T) defines an element of KK(C(X), A). (To put it in standard form, we need T2 — i ejT(^), but by [21], Gh. i, this can be arranged by replacing D by a partial isometry with the same A-index.) From the definition ofp^ in [i6], § 4, one can easily check that ^*([D]) coincides with the A-index of D as defined in [23]. As in [7], one can now check that [D] == [o] ®C,(T*X)^X with ^x eKoCPX X X) as defined by Connes-Skandalis. Then the index formula follows from [7], Gorollaire 4, since ind^D ==A([<y]®c^x)^x) == M ®c,(T*x)A(^x) = M®C.(T*X)^- In the case where X is an even-dimensional Spin^manifold, the same argument of [7], p. 872, that works when E is an ordinary vector bundle basically works here. THEOREM 3.2 (revised version of Theorem i. i). — Let M be a closed even-dimensional spin manifold, and suppose M admits a Riemannian metric whose scalar curvature is everywhere 420 G^-ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 209 non-negative and strictly positive at at least one point. Let A be any C*-algebra with unit and 1^ aflat K-vector bundle over M. Then \i^} ®c(x)^ 1 = ° m ^(A), where p : M ->pt and p\ is defined using the spin structure on M. PROOF. — The proof of Theorem i. i applies word-for-word, except that one must replace the Miscenko-Fomenko index calculation by the second part of Theorem 3.1. THEOREM 3.3 (revised version of Theorem 2.11) . — Let M. be a closed spin manifold admitting a metric of positive scalar curvature. Let n == •^(M) and let f: M ->BTC be the classifying map for the universal covering of M. If the Strong Novikov Conjecture (SNC 3) holds for n, then f^{p\) = o in RK^BTT). (Here p\ is the class of the Dirac operator in Ko(M) if M is even-dimensional or in Ki(M) if M is odd-dimensional.) PROOF. — First suppose M is even-dimensional, and apply Theorem 3.2 with V == G*(7c) Xn M the universal C^TC)-bundle over M. We conclude that DO ®C(X}P^ = °- However, V is the pull-back under / of the universal G*(7r)- bundle ^^ over BTT used to define the Kasparov map p. So by the naturality property of the intersection product, /'([^J) ®G(X)^! = D^] ®B./^'). ^. P(/^')) = o ([i6], § 4, Theorem 4). But p is injective by hypothesis. If M is odd-dimensional, repeat the argument with M X S1 as in the proof of 2. n , then " de-suspend ". Combining 3.3 with Theorem 2.6, we get one direction of the Gromov-Lawson conjecture modulo 2-primary torsion for a large class of solvable fundamental groups. (The map [(M,/)] \->f^p^^) gives the analogue of the Milnor j^-homomorphism in complex K-theory: O^B^) -^ RK,(BTT).) B) Weakening the spin condition The notion of enlargeability used in [10] (as opposed to that of [9]) suggests that we ought to be able to weaken the spin condition on M in Theorem 2.11, and assume only that the universal covering M of M should carry a spin structure. (For instance, there might conceivably be non-spin K(TT, i)-manifolds with TT not having any finite quotients, in which case this improvement might be needed in order to prove non- existence of positive-scalar-curvature metrics on such manifolds.) It turns out that techniques using operator algebras are very convenient for handling this extra generality. (It is possible one could even take torsion into account at the same time, as in § 3A above, but we are not sure how to do this at the moment.) The essential idea is the following. Suppose M is a closed oriented even-dimensional manifold with fundamental group re, whose universal covering M (usually non-compact, of course) carries a spin structure. Also let E be some A-vector bundle over M. We would like to compute the <c index of the Dirac operator D^ with coefficients in E>9 ? over M, but of course this does not make any sense. However, we have half-spinor /^/ f^/ bundles y± and Dirac operators D^ defined over M, and the action ofjron M by covering 27 210 JONATHAN ROSENBER G transformations <( almost " commutes with D^, in the sense that we have a " projective action " of n on spinors and the Dirac operator, or a true action of some double cove- ring ft of TT on ^7±, commuting with D±. If Pgo denotes the principal SO-bundle of orthonormal oriented frames on M, and Pgp^ denotes the principal spin bundle, then TT acts on Pgo (since M is oriented—of course we choose the orientation of £l compatible with that on M), and the group ft is constructed by " pull-back ": f, -^ Aut(Pg^) 7T ——> Aut(Pgo). Then ft acts on the spinor bundle «^, and preserves the splitting into half-spinor bundles since this is defined by the orientation. To see ft commutes with D4', just compute in local coordinates. Now suppose M carries a metric of positive scalar curvature, and for the moment take E to be trivial. If the Dirac operator D4' existed on M, we would know by [19] that A(M) = <A(M), [M]> == o. By the L^index theorem ofAtiyah [3], we also know that indj^ D4' ===0 o L2 — indg D4" = o. However, even if D4' does not make sense on M, one can still show that there are no L^harmonic spinors on M (because the scalar curvature is strictly bounded below). Mikhael Gromov and Blaine Lawson were kind enough to inform me that they had used this fact together with a slight modification of Atiyah's proof to conclude that A(M) = o even when M does not have a spin structure. (Here the L^-index theorem is being used the reverse of the way it is usually applied—data about the non-compact manifold M yields information about M!) However, this argument does not generalize to the case where E is a flat G*-vector bundle, so we prefer to construct a genuine elliptic operator on M and apply the MiS^enko-Fomenko theorem. For this, note that we have a split exact sequence of G*-algebras o^CT(ft)^^ 0^)^0(7^0, where G^ft)^ is universal for unitary representations of ft that are non-trivial on the kernel of the covering map ft ->• TC. The trace coming from the regular representation of ft defines a non-zero finite trace a on C^ft)^? hence a non-zero map Ko(C*(ft)^) -> R. Observe now that since the cocycle of the group extension {l}-^{ ± l}-^ft-^7T-^{l } exactly cancels that for the (c action " of TC on ^±, the G*(ft)^ ® A-vector bundles (G*(ft)odd ^ ^±) ® E actually make sense on M, and we obtain a genuine operator D4'. Now we obtain 422 C--ALGEBRAS, POSITIVE SCALAR CURVATURE, AND THE NOVIKOV CONJECTURE 211 THEOREM 3.4 (revised version of Theorem i. i). — Let M be a closed oriented even- dimensional manifold whose universal covering is spin, and suppose M admits a metric of positive scalar curvature. Let A be any C*-algebra with unit and E aflat A-vector bundle over M. Then <A(M) u ch[E], [M]> = o in Ko(A) ® %. PROOF. — Since we are still dealing with flat bundles, the index of the Dirac operator D^ constructed above is zero in K.o(Gllt(7^)^d0A). But from the trace a on C'^TT)^ we obtain a " partial trace " a ® id : G*(TT)^d®A -^A and thus a map K^G^fc)^ ® A) ->-Ko(A)®R (l). Applying this to the index formula now gives ^<A(M ) u ch[E], [M]> = o in Ko(A) ®R, where c > o is some number which we do not need to compute, but which comes out to i if we normalize a correctly. The same arguments as in § 2 now give a revised version of Theorem 2.11, and proofs of Conjectures 2.2 and 2.3, assuming n satisfies SNG 2. THEOREM 3.5. — Assume the Strong Novikov Conjecture (SNG 2) holds for TT, and suppose M is a closed oriented manifold whose universal covering has a spin structure. Let g: M -> BTT be any continuous map. Then ij"M admits a metric of'positive scalarcurvature, ^,(A(M) n [M]) == o in H^(BTT, QJ, or equivalently, for all a e H^BTC, QJ, <A(M) u^),[M]>=o. In particular, if ^»([M]) + o in H^(BTT, Q^), then no manifold homotopy-equivalent to M admits a metric of positive scalar curvature. REFERENCES [i] J. F. ADAMS, Infinite Loop Spaces, Annals of Math. Studies, no. 90, Princeton, Princeton Univ. Press, 1978. [a] M. F. ATIYAH, Vector bundles and the Klinneth formula, Topology 1 (1962), 245-248. [3] M. F. ATIYAH, Elliptic operators, discrete groups and von Neumann algebras, Asterisque, no. 32-33 (1976), 43-72. [4] M. F. ATIYAH and I. M. SINGER, The index of elliptic operators: III, Arm. of Math. (2) 87 (1968), 546-604. [5] M. F. ATIYAH and I. M. SINGER, The index of elliptic operators: IV, Ann. of Math. (2) 93 (1971), 119-138. [6] S. E. CAPPELL, On homotopy invariance of higher signatures. Invent. Math. 33 (1976), 171-179. [7] A. CONNES and G. SKANDALIS, Theoreme de Findice pour les feuilletages, C. R. Acad. Sci. Paris, Ser. I, 292 (1981), 871-876. [8] J. CUNTZ, The K-groups for free products ofC'-algebras, in Operator Algebras and Applications, R. V. KADISON, ed., Proc. Symp. Pure Math., vol. 38, Amer. Math. Soc., Providence, R. I., 1982, Part I, 81-84. (1) At least when the Klinneth theorem of [27] holds. However, we do not need to worry about this here, since the Chem character of the symbol of D+ clearly lies in H;(T*M, Q) ®K,(C*(7r)odd) ®K,(A). 423 ai2 JONATHAN ROSENBER G [9] M. GROMOV and H. B. LAWSON, Jr., Spin and scalar curvature in the presence of a fundamental group, I, Ann. of Math. (2), 111 (1980), 209-230. [10] M. GROMOV and H. B. LAWSON, Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, this volume, 83-196. [ii] N. HITCHIN, Harmonic spinors. Adv. in Math, 14 (1974), I-55. [12] W. C. HSIANG and H. D. REES, Miscenko^s work on Novikov's conjecture, in Operator Algebras and K-Theory, R. G. DOUGLAS and C. SCHOCHET, eds.. Contemporary Math., vol. 10, Amer. Math. Soc., Providence, R. I., 1982, 77-98. [13] M. KAROUBI, K-Theory: An Introduction, Grundlehren der math. Wissenschaften, no. 226, Heidelberg and New York, Springer-Verlag, 1978. [14] G. G. KASPAROV, Topological invariants of elliptic operators, I: K-homology, Izv. Akad. Nauk SSSR, Ser. Mat. 39 (i975)> 796-838 == Math. USSR-Izv. 9 (1975), 75I-792. [15] G. G. KASPAROV, Hilbert C'-modules: theorems of Stinespring and Voiculescu, /. Operator Theory 4 (1980), I33-I50. [16] G. G. KASPAROV, The operator K-functor and extensions of C*-algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980), 571-636 == Math. USSR-Izv. 16 (1981), 513-572. [17] G. G. KASPAROV, K-theory. group C*-algebras, and higher signatures (conspectus), preprint, Chernogolovka, 1981. 118] J. L. KAZDAN and F. W. WARNER, Scalar curvature and conformal deformation of Riemannian structure, J . Differential Geometry 10 (1975), 113-134. [19] A. LICHNEROWICZ, Spineurs harmoniques, C. R. Acad. Sci. Paris, Ser. A-B 257 (1963), 7-9. [20] J. MILNOR, On axiomatic homology theory. Pacific J . Math. 12 (1962), 337-34I. [21] J. A. MINGO, K-theory and multipliers of stable C*-algebras, Ph. D. Thesis, Halifax, N.S., Dalhousie Univ., 1982. [22] A. S. MiScENKO, C*-algebras and K-theory, in Algebraic Topology, Aarhus 1978, J. L. DupONrandH. MADSEN, eds., Lecture Notes in Math., no. 763, Heidelberg and New York, Springer-Verlag, 1979, 262-274. [23] A. S. MI§CENKO and A. T. FOMENKO, The index of elliptic operators over G*-algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 43 (1979), 831-859 == Math. USSR-Izv. 15 (1980), 87-112. [24] A. S. MI§CENKQ and J. P. SOLOV'EV, On infinite-dimensional representations of fundamental groups and formulas of Hirzebruch type, Dokl. Akad. Nauk SSSR 234 (1977), 761-764 = Soviet Math. Dokl. 18 (1977), 767-771. [25] M. PIMSNER and D. VOIGULESCU, Exact sequences for K-groups and Ext-groups of certain cross-product C*-algebras, /. Operator Theory 4 (1980), 93-118. [26] J. ROSENBERG, Group C*-algebras and topological invariants, in Operator Algebras and Group Representations, Proc. Conf. in Neptun, Romania, 1980, London, Pitman, 1983, to appear. [27] C. SCHOCHET, Topological methods for C*-algebras, II: geometric resolutions and the Kunneth formula, Pacific J . Math. 98 (1982), 443-458. [28] W. SHIH, Fiber cobordism and the index of a family of elliptic differential operators. Bull. Amer. Math. Soc. 72 (1966), 984-991- Department of Mathematics, University of Maryland, College Park, Maryland 20742 Manuscrit refu Ie 28 mars 1983.

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