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Duality and the existence of differential cell filtrations for manifolds without boundary

Duality and the existence of differential cell filtrations for manifolds without boundary CHAPTER IX DUALITY AND THE EXISTENCE OF DIFFERENTIAL CELL FILTRATIONS FOR MANIFOLDS WITHOUT BOUNDARY The main theorem proved in this paper, which gives a strong sufficient condition for existence of a differenfiable cell filtration ~ of a differentiable manifold M such that.~' is neighborhood of a given cell filtration X, is the existence theorem of Chapter VII I. In general (for manifolds with boundary), it is easily seen that one cannot dispense with any the conditions of the theorem without the conclusion being false. If the manifold M possesses no boundary, there are ways of strengthening the existence theorem to apply more generally by making use of Poincar6 duality. This idea is originally due to Smale, and was used in his proof of the Poincar6 conjecture. Duality. -- The geometric manifestation of Poincar6 duality is that if you take a filtered manifold and turn it upside-down it still looks like a filtered manifold, with a rather different (dual) filtration. From the point of view of Morse theory, this fact can be expressed even more simply: Iff is a Morse function, so is (--f). In particular, iff had critical points Pa, ..., P, such that (i) f(p~) <f(P2) <-.. <f(P~) (ii) the index of p~ is nl, then the Morse function g(x) = --f(x) has critical points p,, p~_t, ..., Pl such that (i) g(p~)<g(p~_l)<... <g(p,) (ii) the index of Pi is n--n~ (where n = dim M). By virtue of the " equivalence " of Morse functions and differentiable cell filtrations, one may expect an analogous duality. Let M= (M0, ..., M~)" be a differentiable filtration, and let ~Pi : ~Dn~ � Dn-ni"+OMi-1 be the " attaching maps ", i= i,...,v. 56 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 57 Let M~ ----1VI~--M~_~ be the closures of the complements of the M,,_~, i= o, ..., ,~. Then 9 9 u~p~ Dn(.,_~) M,: == M~ _ 1 X D n n(,_i) - where q~ : Dn(~-i) � oDn-n(v-i) -~ OM~_ 1 is the inclusion map in M i. Therefore M*---- (g0, ..., M~*)" is a filtration called the dual filtration to M, where n i ~--- n-- n(~_i) , % : ODn~ � Dn-m _~ 0Mi_ 1. Of course, if the Morse function f corresponds to the differentiable cell filtration M, the Morse function (--f) will correspond to M*. Geometric k-skeletons. -- Let M be a differentiable manifold without boundary. By a geometric k-skeleton Of M, I will mean a differentiable submanifold U cM such that U is a compact manifold obtainable as the closure of an open set in M, and if V_cM is the closed complement of U in M, then geom dim U<k geom dim V< n--k-- I where n = dim M. It is easily seen that geometric k-skeletons always exist; they are not unique. If X is a properly ordered cell filtration, then X (k), the k-skeleton of X, will mean the subfiltration of X consisting of all cells of X of dimension less than or equal to k. If the differentiable cell filtration-s is a neighborhood of X, then ,/t '(k), the k-skeleton 0f~', will denote the part of ~ lying above X (~). If Jr' is a properly ordered differentiable filtration of M, then the submanifold U c M cut out by the k-skeleton ~,(k) of~' is a geometric k-skeleton of M. To see this, first notice that geom dim U<k, by definition. Further, if V is the closed complement of U in M, then V is easily seen to be cut out by (~(~-k-t), the (n-- k-- t)-skeleton of de', the dual filtration of dr'. Thus geom dim V<n-- k-- i, and U is therefore a geometric k-skeleton of M. Construction of differentiable cell filtrations for manifolds without boundary. -- Any geometric k-skeleton U c M decomposes M as the union of two submanifolds M=UuV. If 2<k<n--2, then geomdimU<n 3 geom dim V<n 3, 8 5 8 BARRY MAZUR which means that both U and V satisfy the requirements of existence theorem, and one may construct differentiable cell filtrations of ~, r by " simple-homotopy-theoretic means ". Therefore, to obtain differentiable cell filtrations of M one must have a method for compounding differentiable filtrations ~, ~ of U, V into a differentiable filtration ~' of M. This is an easy matter: Proposition (9. x ). -- Let M be a differentiable manifold without boundary, and M=UuV Un V = 0U = 0V where U, V are submanifolds of M. If ~ is a differentiable cell filtration of U, and ~r is a differentiable cell filtration of V, then there is a differentiable cell filtration ~' of M such that ~/ is a subfiltration of dr', and ~ is a subfiltrafion of ~. The proof of (9- i ) may be given most succinctly in the language of Morse functions. The differentiable filtrations ~, ~ give rise to Morse functionsfu,f v on U, V respectively and if we take I � (UnV)cM as a tubular neighborhood of UnV=0U=0V in M (with {-- I} � (UnV) cU, {--~ I} X (U~V) cV, {o}x (Uf'IV) =UrlV) we may assume the normalization: if t<o, xeUnV fu(t, x)=t if t>o, xeUnV. fv(t, x)=--t Let f: M-+R denote the following differentiable function: a) f[ U = f, b) f[V=--fv. By virtue of the normalizations, f is again a Morse function. It gives rise to a diffe- rentiable cell filtration ~ of M, which has the properties desired (i.e. it is easily verified that ~' satisfies a), b), c), d)). Proposition (9. i) succeeds in constructing differentiable cell filtrations of a manifold without boundary, solely on the basis of simple homotopy theoretic information, once one is provided with some geometric k-skeleton (2<k<n--2). It is much weaker than the most general existence theorem that one might conjecture: Let f : M~X be a simple homotopy equivalence between the differentiable manifold M (without boundary) and the cell filtration X. Then there is a differentiable cell filtration ~ of M which is a neighborhood of X such that f : M~X is a projection map. A weakness of (9- i) is that so far, no homotopy-theoretic criterion has been given for the existence of geometric k-skeletons. (Of course, once a geometric k-skeleton has been given, all other information necessary is simple-homotopy-theoretic.) The next paragraph will provide certain homotopy criteria for the construction of nice geometric k-skeletons. 58 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 59 Construction of geometric k-skeletons of k-connected manifolds. Lemma (9.2). -- Let X be a cell filtration which is k-connected and (k + i)-dimen- sional. Then X is simple-homotopy equivalent with S~+IV ..... ,,/.~k+ q 1 where q is the rank of Hk+I(X ). Proof. -- a) If k = o, then X is a I-dimensional connected complex which is clearly of the simple homotopy type of S~V...VS~. b) If k> I, then X is simply connected and of the homotopy type of S~ + 1V... VS~ + ~, and one may apply (5.4). Proposition (9.3)- -- Let M ~ be a k-connected manifold without boundary, such that k<n--4. Then a differentiably imbedded n-cell, D~cM " is a geometric k-skeleton of M ~. Proof. -- Let ~ be a properly ordered differentiable cell filtration of M, and let dt '(k+l) be its (k ~-I )-skeleton. Let ~,(k+l) be a filtration of the submanifold U_cM. Let dg (k+l) be a neighborhood of X (J'+l), the (k+ 1)-skeleton of X, the cell filtration of which ~ is a neighborhood. Then ~q(X (~+1)) ~ ~q(X)~ ~q(M)= {o} if q~k. The first isomorphism can be seen from the homotopy sequence for the pair (X, X (k+l)) and the fact that X--X (k+l) is made up only of cells of dimension greater than k~ I. Therefore X (k+ 1) satisfies the requirements of Lemma 9. ~ and is of the simple homotopy type of S~ + IV ...VSq k+ 1 . Thus U, the submanifold of M cut out by ..s is a geometric (k-+- i)-skeleton of M, k 4- i <n--2, and there is a simple homotopy equi- valence ~ : U~S~+IV. ,,/.~k+t The existence theorem applies, yielding a differen- .... q * tiable cell filtration #/of U such that ~e~(St~+*V...VSqk+l). (By S~+~V. . .VSqk+t I refer to the cell filtration: DO,, nk+~,, nk+t,, k+t.) "o-"-'J_ ~o~2 ~o- - - uoDq Let ~y-=.~(~-~-2) be the (n--k--2)-skeleton of.~"~ the dual filtration of~'. Then r is a filtration of V, the closure of the complement of U. Proposition 9- i applies and we may conglomerate the filtrations #/on U, $/" on V to obtain a differentiable filtration #" on M. It follows that S#-(k) = ag(k) is a neighborhood of D O . That is, ~(k)= D O x D". Therefore the submanifold Dn== D O � D'C M ~ is a geometric k-skeleton of M ~, proving Proposition 9.2. 59 6o BARRY MAZUR Another way of stating (9.2) is: Corollary (9.3). -- Let M n be a k-connected differentiable manifold without boundary (k<n--4). Let M, be the bounded differentiable manifold obtained by removing the interior of an n-cell from M n. Then geom dim M~,~n--k - i. Corollary (9.4). -- Let M" be a 2-connected manifold without boundary, n>6. Let M, be as in (9.3). Let f : M,~X be a (simple) homotopy equivalence between M, and a properly ordered q-dimensional cell filtration X (q<n--3). Then there is a diffe- rentiable filtration MI of M, which is a neighborhood of X such that f: M,~X is a projection. Proof. -- Since geom dim M~,~n--k - I, the existence theorem applies. From these considerations one may prove a theorem of Smale's: Corollary (9.5)- -- Let M" be a k-connected manifold without boundary, k< 4. Then there exists a Morse function on M" with a unique maximum, a unique minimum, and all of "whose critical points pi have indices Ji such that k<j~<n--k. Or, equivalently, if M. is M" with the interior of an n-cell removed, then M. admits a differentiable cell filtration ~r which is a neighborhood of a cell filtration X such that (i) dim X<n--k. (ii) X has a unique o-cell, D ~ (iii) X possesses no cells of dimension less than or equal to k. Proof. -- We shall prove the second version of the above proposition. By Proposition 9.3, geom dim M < n-- k-- I. Therefore there is a differentiable filtration of M. such that zcf. is a neighborhood of a properly ordered cell filtration Y of dimension less than n-- k. Giving D" the trivial differentiable filtration, and expressing M = M. uD n, (9.I) applies to give a differentiable filtration ~ of M. Then ~r cuts out a submanifold U of M, and by the same reasoning as in (9-3), y~(k+l) is a neighborhood of X (k+l), a (k+ i)-dimensional k-connected cell filtration. By (9.2) X (k+l) is of the simple homotopy type of S~+aV...VS~ +l for some q. Again, applying the existence theorem, U admits a differenfiable cell filtration d// which is a neighborhood of S~+IV V S k +l If V is the closed complement of U in M, then .... q " ,~ =~(n--k--2) is a differentiable cell filtration of V. Combining the differentiable filtrations q/of U, of V to obtain a filtration ~ of M (by means of (9. i)), it is immediately seen that~ possesses the required properties for 9.5. More precisely, dr' possesses a unique " thickened " n-celh x D O ./r =..d/,uD" and ~', is a differentiable cell filtration of M, obeying (i), (ii), (iii). 60 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 6I Manifolds of the same homotopy type as SL -- The general Poincar4 conjecture for differentiable manifolds of the same homotopy type as the n-sphere was originally proved by Smale [ii] for n_>5: Theorem of Smale. -- Let M ~ be a differentiable manifold such that n_> 5 ; M'*"~ S~- Then M ~ D 1 uvD 2 where q~ :~D~0D~ is a difl~rentiable isomorphism. Other equivalent statements of Smale's theorem are: A) There is a function f on M n possessing precisely two non-degenerate critical points (a maximum and a minimum). B) M'eP ~ which is the group of differentiable n-spheres defined by Milnor [7]. C) There is a piece-wise differentiable homeomorphism g:M'~S n. D) M" possesses a differentiable cell filtrationdt' which is a neighborhood of D~ n. Corollary 9.5 implies Smale's theorem for n_> 6. This may be seen since M n, by hypothesis, isk-connected for k = n-- 4. According to 9.5, M, admits a differentiable filtration dr'. which is a neighborhood of a cell filtration X containing a single o-cell, no cells of dimension less than k + i, and dim X< n--k-- I. It follows that X = D ~ and consequently ,//. = D O � D" if n--k--i<k+I. This happens if n>6. If n=6, X=D~ uD~. Since H3(X ) ~ H3(M.) ~ {o} it follows that q-- o, X = D ~ and M. ---- D O � DL The last isomorphims may be seen, for instance, using the Mayer-Vietoris sequence for the decomposition Mn=M, uD ~, making use of the fact: H.(M ~) = H.(sn). Smale's theorem for n = 5 cannot be directly obtained from Corollary 9.5. The best that one may easily obtain from (9.5) is that there is a differentiable filtration of M ~ x I, Jr', which is a neighborhood of X = D o u D 5. From this fact, and the differenfiable Schoenflies theorem (in dimension 5) [4], [9], one may obtain that M 5 is homeomorphic with S 5. Homotopy Skeletons. -- The problem of the existence of geometric k-skeletons will be reduced to a homotopy question (see Proposition 9.8). For this, we will need a lemma of simple homotopy theory (Lemma 15, page 4 6 of [14] ). Lemma (9.6). -- Let K be a subfiltration of the cell filtration X. Let r:~(X, K) =% n= I, ..., r. Then there is a cell filtration Y satisfying these properties: a) There is an inclusion ~ : KeY. b) All cells of Y--K are of dimension greater than r. 61 62 BARRY MAZUR c) There is a simple homotopy equivalence f: X-*Y which is the identity on K. d) dim Y< max {r q- 2, dim X}. Definition (9.7). --- By a homotopy k-skeleton of the differentiable manifold M, I shall mean a continuous map f: X-*M of the cell filtration X into M such that a) dim X<k b) z~q(M, X)= o q = I, ..., k. If f : X-*M ~ is an arbitrary continuous map of the cell filtration X into the differentiable manifold M, I will say that the compact submanifold U"_cM" is a neighborhood of the mapping f if a) f(X) cU. b) There is a differentiable cell filtration ~ of U which is a neighborhood of X, for which f: X-*U is a cross-section. Proposition (9.8). -- Let M" be a compact differentiable manifold without boundary. Let k~n-- 5 and f: X-*M" a homotopy k-skeleton for M"; assume X properly ordered. Then there is a neighborhood U"_cM n of the mapping f, which is a geometric k-skeleton of ML Conversely, every geometric k-skeleton of M" comes from a homotopy k-skeleton, f: X-*M", in this manner. After Proposition 9.8, the construction of geometric k-skeletons in manifolds of high enough dimension is a homotopy theoretic matter. Let M/"-~l be some geometric (n--5)-skeleton of M; M=MI"-5/oM f5). Thus there is a differentiable cell filtration of M I"-~), Jt '/"-5), which is a neighborhood of Y, a cell filtration, dim Y<_n-- 5. Since dim X=k<_n--5, we may deform f: X-*M up to homotopy, and assume that f(X)cM("-5)) cM. Let g : X-*Y be the continuous map X t M(._5) where r~ is a projection map for the differentiable cell filtration ~,(n-sl. Let W be the cell filtration consisting in the mapping cylinder of g: w = x � 62 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 6 3 (the right-hand side may naturally be given the structure of a cell filtration). There is a " projection map " p W-~Y : which is a simple homotopy equivalence, XcW as a subfiltration, and the diagram: XcW is commutative. (This is standard.) Since =q(M, X)=o, q= I, ..., k we have: ~q(W, X) =o q= I, ..., k. dim W< max{k + I, n-- 5}~ n--4- Consequently, according to Lemma 9.6, there is a cell filtration Y', such that XcY', dim Y'< n-- 3 and all cells of Y'--X are of dimensions greater than k, and there is a simple homotopy equivalence f: W-+Y' which is identity on X. The cell filtration Y' may be taken to be properly ordered. Let q0 : Y'-+Y be a simple homotopy equivalence making w~Y y, homotopy-commutative. Since ~(n-5)eJU"(Y) and max{dimY, dimY'}<n--2 the nonstable neigh- borhood theorem applies, yielding a differentiable cell filtration Jt"esU '~ (Y') corres- ponding to dg (~-5) under 6- Hence .A" is also X filtration of the differentiable manifold M ('-5). By means of the decomposition, M = M(~-5) u M (~), one may compound a differentiable cell filtration ~ on M using dr" on M (~-5) and dt '(~) on M (5). Then ~t//" is a differentiable cell filtration of M such that Jr" c~g/'. If ~(k) is the k-skeleton of ~, it is immediate that ~(k) is a neighborhood of X and U, the submanifold cut out by ~(k), is a neighborhood of the mapping f : X-~ M. Clearly U is a geometric k-skeleton of M, proving 9-8. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Duality and the existence of differential cell filtrations for manifolds without boundary

Publications mathématiques de l'IHÉS , Volume 15 (1) – Aug 4, 2007
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Springer Journals
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Copyright © 1963 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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0073-8301
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10.1007/BF02684285
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CHAPTER IX DUALITY AND THE EXISTENCE OF DIFFERENTIAL CELL FILTRATIONS FOR MANIFOLDS WITHOUT BOUNDARY The main theorem proved in this paper, which gives a strong sufficient condition for existence of a differenfiable cell filtration ~ of a differentiable manifold M such that.~' is neighborhood of a given cell filtration X, is the existence theorem of Chapter VII I. In general (for manifolds with boundary), it is easily seen that one cannot dispense with any the conditions of the theorem without the conclusion being false. If the manifold M possesses no boundary, there are ways of strengthening the existence theorem to apply more generally by making use of Poincar6 duality. This idea is originally due to Smale, and was used in his proof of the Poincar6 conjecture. Duality. -- The geometric manifestation of Poincar6 duality is that if you take a filtered manifold and turn it upside-down it still looks like a filtered manifold, with a rather different (dual) filtration. From the point of view of Morse theory, this fact can be expressed even more simply: Iff is a Morse function, so is (--f). In particular, iff had critical points Pa, ..., P, such that (i) f(p~) <f(P2) <-.. <f(P~) (ii) the index of p~ is nl, then the Morse function g(x) = --f(x) has critical points p,, p~_t, ..., Pl such that (i) g(p~)<g(p~_l)<... <g(p,) (ii) the index of Pi is n--n~ (where n = dim M). By virtue of the " equivalence " of Morse functions and differentiable cell filtrations, one may expect an analogous duality. Let M= (M0, ..., M~)" be a differentiable filtration, and let ~Pi : ~Dn~ � Dn-ni"+OMi-1 be the " attaching maps ", i= i,...,v. 56 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 57 Let M~ ----1VI~--M~_~ be the closures of the complements of the M,,_~, i= o, ..., ,~. Then 9 9 u~p~ Dn(.,_~) M,: == M~ _ 1 X D n n(,_i) - where q~ : Dn(~-i) � oDn-n(v-i) -~ OM~_ 1 is the inclusion map in M i. Therefore M*---- (g0, ..., M~*)" is a filtration called the dual filtration to M, where n i ~--- n-- n(~_i) , % : ODn~ � Dn-m _~ 0Mi_ 1. Of course, if the Morse function f corresponds to the differentiable cell filtration M, the Morse function (--f) will correspond to M*. Geometric k-skeletons. -- Let M be a differentiable manifold without boundary. By a geometric k-skeleton Of M, I will mean a differentiable submanifold U cM such that U is a compact manifold obtainable as the closure of an open set in M, and if V_cM is the closed complement of U in M, then geom dim U<k geom dim V< n--k-- I where n = dim M. It is easily seen that geometric k-skeletons always exist; they are not unique. If X is a properly ordered cell filtration, then X (k), the k-skeleton of X, will mean the subfiltration of X consisting of all cells of X of dimension less than or equal to k. If the differentiable cell filtration-s is a neighborhood of X, then ,/t '(k), the k-skeleton 0f~', will denote the part of ~ lying above X (~). If Jr' is a properly ordered differentiable filtration of M, then the submanifold U c M cut out by the k-skeleton ~,(k) of~' is a geometric k-skeleton of M. To see this, first notice that geom dim U<k, by definition. Further, if V is the closed complement of U in M, then V is easily seen to be cut out by (~(~-k-t), the (n-- k-- t)-skeleton of de', the dual filtration of dr'. Thus geom dim V<n-- k-- i, and U is therefore a geometric k-skeleton of M. Construction of differentiable cell filtrations for manifolds without boundary. -- Any geometric k-skeleton U c M decomposes M as the union of two submanifolds M=UuV. If 2<k<n--2, then geomdimU<n 3 geom dim V<n 3, 8 5 8 BARRY MAZUR which means that both U and V satisfy the requirements of existence theorem, and one may construct differentiable cell filtrations of ~, r by " simple-homotopy-theoretic means ". Therefore, to obtain differentiable cell filtrations of M one must have a method for compounding differentiable filtrations ~, ~ of U, V into a differentiable filtration ~' of M. This is an easy matter: Proposition (9. x ). -- Let M be a differentiable manifold without boundary, and M=UuV Un V = 0U = 0V where U, V are submanifolds of M. If ~ is a differentiable cell filtration of U, and ~r is a differentiable cell filtration of V, then there is a differentiable cell filtration ~' of M such that ~/ is a subfiltration of dr', and ~ is a subfiltrafion of ~. The proof of (9- i ) may be given most succinctly in the language of Morse functions. The differentiable filtrations ~, ~ give rise to Morse functionsfu,f v on U, V respectively and if we take I � (UnV)cM as a tubular neighborhood of UnV=0U=0V in M (with {-- I} � (UnV) cU, {--~ I} X (U~V) cV, {o}x (Uf'IV) =UrlV) we may assume the normalization: if t<o, xeUnV fu(t, x)=t if t>o, xeUnV. fv(t, x)=--t Let f: M-+R denote the following differentiable function: a) f[ U = f, b) f[V=--fv. By virtue of the normalizations, f is again a Morse function. It gives rise to a diffe- rentiable cell filtration ~ of M, which has the properties desired (i.e. it is easily verified that ~' satisfies a), b), c), d)). Proposition (9. i) succeeds in constructing differentiable cell filtrations of a manifold without boundary, solely on the basis of simple homotopy theoretic information, once one is provided with some geometric k-skeleton (2<k<n--2). It is much weaker than the most general existence theorem that one might conjecture: Let f : M~X be a simple homotopy equivalence between the differentiable manifold M (without boundary) and the cell filtration X. Then there is a differentiable cell filtration ~ of M which is a neighborhood of X such that f : M~X is a projection map. A weakness of (9- i) is that so far, no homotopy-theoretic criterion has been given for the existence of geometric k-skeletons. (Of course, once a geometric k-skeleton has been given, all other information necessary is simple-homotopy-theoretic.) The next paragraph will provide certain homotopy criteria for the construction of nice geometric k-skeletons. 58 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 59 Construction of geometric k-skeletons of k-connected manifolds. Lemma (9.2). -- Let X be a cell filtration which is k-connected and (k + i)-dimen- sional. Then X is simple-homotopy equivalent with S~+IV ..... ,,/.~k+ q 1 where q is the rank of Hk+I(X ). Proof. -- a) If k = o, then X is a I-dimensional connected complex which is clearly of the simple homotopy type of S~V...VS~. b) If k> I, then X is simply connected and of the homotopy type of S~ + 1V... VS~ + ~, and one may apply (5.4). Proposition (9.3)- -- Let M ~ be a k-connected manifold without boundary, such that k<n--4. Then a differentiably imbedded n-cell, D~cM " is a geometric k-skeleton of M ~. Proof. -- Let ~ be a properly ordered differentiable cell filtration of M, and let dt '(k+l) be its (k ~-I )-skeleton. Let ~,(k+l) be a filtration of the submanifold U_cM. Let dg (k+l) be a neighborhood of X (J'+l), the (k+ 1)-skeleton of X, the cell filtration of which ~ is a neighborhood. Then ~q(X (~+1)) ~ ~q(X)~ ~q(M)= {o} if q~k. The first isomorphism can be seen from the homotopy sequence for the pair (X, X (k+l)) and the fact that X--X (k+l) is made up only of cells of dimension greater than k~ I. Therefore X (k+ 1) satisfies the requirements of Lemma 9. ~ and is of the simple homotopy type of S~ + IV ...VSq k+ 1 . Thus U, the submanifold of M cut out by ..s is a geometric (k-+- i)-skeleton of M, k 4- i <n--2, and there is a simple homotopy equi- valence ~ : U~S~+IV. ,,/.~k+t The existence theorem applies, yielding a differen- .... q * tiable cell filtration #/of U such that ~e~(St~+*V...VSqk+l). (By S~+~V. . .VSqk+t I refer to the cell filtration: DO,, nk+~,, nk+t,, k+t.) "o-"-'J_ ~o~2 ~o- - - uoDq Let ~y-=.~(~-~-2) be the (n--k--2)-skeleton of.~"~ the dual filtration of~'. Then r is a filtration of V, the closure of the complement of U. Proposition 9- i applies and we may conglomerate the filtrations #/on U, $/" on V to obtain a differentiable filtration #" on M. It follows that S#-(k) = ag(k) is a neighborhood of D O . That is, ~(k)= D O x D". Therefore the submanifold Dn== D O � D'C M ~ is a geometric k-skeleton of M ~, proving Proposition 9.2. 59 6o BARRY MAZUR Another way of stating (9.2) is: Corollary (9.3). -- Let M n be a k-connected differentiable manifold without boundary (k<n--4). Let M, be the bounded differentiable manifold obtained by removing the interior of an n-cell from M n. Then geom dim M~,~n--k - i. Corollary (9.4). -- Let M" be a 2-connected manifold without boundary, n>6. Let M, be as in (9.3). Let f : M,~X be a (simple) homotopy equivalence between M, and a properly ordered q-dimensional cell filtration X (q<n--3). Then there is a diffe- rentiable filtration MI of M, which is a neighborhood of X such that f: M,~X is a projection. Proof. -- Since geom dim M~,~n--k - I, the existence theorem applies. From these considerations one may prove a theorem of Smale's: Corollary (9.5)- -- Let M" be a k-connected manifold without boundary, k< 4. Then there exists a Morse function on M" with a unique maximum, a unique minimum, and all of "whose critical points pi have indices Ji such that k<j~<n--k. Or, equivalently, if M. is M" with the interior of an n-cell removed, then M. admits a differentiable cell filtration ~r which is a neighborhood of a cell filtration X such that (i) dim X<n--k. (ii) X has a unique o-cell, D ~ (iii) X possesses no cells of dimension less than or equal to k. Proof. -- We shall prove the second version of the above proposition. By Proposition 9.3, geom dim M < n-- k-- I. Therefore there is a differentiable filtration of M. such that zcf. is a neighborhood of a properly ordered cell filtration Y of dimension less than n-- k. Giving D" the trivial differentiable filtration, and expressing M = M. uD n, (9.I) applies to give a differentiable filtration ~ of M. Then ~r cuts out a submanifold U of M, and by the same reasoning as in (9-3), y~(k+l) is a neighborhood of X (k+l), a (k+ i)-dimensional k-connected cell filtration. By (9.2) X (k+l) is of the simple homotopy type of S~+aV...VS~ +l for some q. Again, applying the existence theorem, U admits a differenfiable cell filtration d// which is a neighborhood of S~+IV V S k +l If V is the closed complement of U in M, then .... q " ,~ =~(n--k--2) is a differentiable cell filtration of V. Combining the differentiable filtrations q/of U, of V to obtain a filtration ~ of M (by means of (9. i)), it is immediately seen that~ possesses the required properties for 9.5. More precisely, dr' possesses a unique " thickened " n-celh x D O ./r =..d/,uD" and ~', is a differentiable cell filtration of M, obeying (i), (ii), (iii). 60 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 6I Manifolds of the same homotopy type as SL -- The general Poincar4 conjecture for differentiable manifolds of the same homotopy type as the n-sphere was originally proved by Smale [ii] for n_>5: Theorem of Smale. -- Let M ~ be a differentiable manifold such that n_> 5 ; M'*"~ S~- Then M ~ D 1 uvD 2 where q~ :~D~0D~ is a difl~rentiable isomorphism. Other equivalent statements of Smale's theorem are: A) There is a function f on M n possessing precisely two non-degenerate critical points (a maximum and a minimum). B) M'eP ~ which is the group of differentiable n-spheres defined by Milnor [7]. C) There is a piece-wise differentiable homeomorphism g:M'~S n. D) M" possesses a differentiable cell filtrationdt' which is a neighborhood of D~ n. Corollary 9.5 implies Smale's theorem for n_> 6. This may be seen since M n, by hypothesis, isk-connected for k = n-- 4. According to 9.5, M, admits a differentiable filtration dr'. which is a neighborhood of a cell filtration X containing a single o-cell, no cells of dimension less than k + i, and dim X< n--k-- I. It follows that X = D ~ and consequently ,//. = D O � D" if n--k--i<k+I. This happens if n>6. If n=6, X=D~ uD~. Since H3(X ) ~ H3(M.) ~ {o} it follows that q-- o, X = D ~ and M. ---- D O � DL The last isomorphims may be seen, for instance, using the Mayer-Vietoris sequence for the decomposition Mn=M, uD ~, making use of the fact: H.(M ~) = H.(sn). Smale's theorem for n = 5 cannot be directly obtained from Corollary 9.5. The best that one may easily obtain from (9.5) is that there is a differentiable filtration of M ~ x I, Jr', which is a neighborhood of X = D o u D 5. From this fact, and the differenfiable Schoenflies theorem (in dimension 5) [4], [9], one may obtain that M 5 is homeomorphic with S 5. Homotopy Skeletons. -- The problem of the existence of geometric k-skeletons will be reduced to a homotopy question (see Proposition 9.8). For this, we will need a lemma of simple homotopy theory (Lemma 15, page 4 6 of [14] ). Lemma (9.6). -- Let K be a subfiltration of the cell filtration X. Let r:~(X, K) =% n= I, ..., r. Then there is a cell filtration Y satisfying these properties: a) There is an inclusion ~ : KeY. b) All cells of Y--K are of dimension greater than r. 61 62 BARRY MAZUR c) There is a simple homotopy equivalence f: X-*Y which is the identity on K. d) dim Y< max {r q- 2, dim X}. Definition (9.7). --- By a homotopy k-skeleton of the differentiable manifold M, I shall mean a continuous map f: X-*M of the cell filtration X into M such that a) dim X<k b) z~q(M, X)= o q = I, ..., k. If f : X-*M ~ is an arbitrary continuous map of the cell filtration X into the differentiable manifold M, I will say that the compact submanifold U"_cM" is a neighborhood of the mapping f if a) f(X) cU. b) There is a differentiable cell filtration ~ of U which is a neighborhood of X, for which f: X-*U is a cross-section. Proposition (9.8). -- Let M" be a compact differentiable manifold without boundary. Let k~n-- 5 and f: X-*M" a homotopy k-skeleton for M"; assume X properly ordered. Then there is a neighborhood U"_cM n of the mapping f, which is a geometric k-skeleton of ML Conversely, every geometric k-skeleton of M" comes from a homotopy k-skeleton, f: X-*M", in this manner. After Proposition 9.8, the construction of geometric k-skeletons in manifolds of high enough dimension is a homotopy theoretic matter. Let M/"-~l be some geometric (n--5)-skeleton of M; M=MI"-5/oM f5). Thus there is a differentiable cell filtration of M I"-~), Jt '/"-5), which is a neighborhood of Y, a cell filtration, dim Y<_n-- 5. Since dim X=k<_n--5, we may deform f: X-*M up to homotopy, and assume that f(X)cM("-5)) cM. Let g : X-*Y be the continuous map X t M(._5) where r~ is a projection map for the differentiable cell filtration ~,(n-sl. Let W be the cell filtration consisting in the mapping cylinder of g: w = x � 62 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 6 3 (the right-hand side may naturally be given the structure of a cell filtration). There is a " projection map " p W-~Y : which is a simple homotopy equivalence, XcW as a subfiltration, and the diagram: XcW is commutative. (This is standard.) Since =q(M, X)=o, q= I, ..., k we have: ~q(W, X) =o q= I, ..., k. dim W< max{k + I, n-- 5}~ n--4- Consequently, according to Lemma 9.6, there is a cell filtration Y', such that XcY', dim Y'< n-- 3 and all cells of Y'--X are of dimensions greater than k, and there is a simple homotopy equivalence f: W-+Y' which is identity on X. The cell filtration Y' may be taken to be properly ordered. Let q0 : Y'-+Y be a simple homotopy equivalence making w~Y y, homotopy-commutative. Since ~(n-5)eJU"(Y) and max{dimY, dimY'}<n--2 the nonstable neigh- borhood theorem applies, yielding a differentiable cell filtration Jt"esU '~ (Y') corres- ponding to dg (~-5) under 6- Hence .A" is also X filtration of the differentiable manifold M ('-5). By means of the decomposition, M = M(~-5) u M (~), one may compound a differentiable cell filtration ~ on M using dr" on M (~-5) and dt '(~) on M (5). Then ~t//" is a differentiable cell filtration of M such that Jr" c~g/'. If ~(k) is the k-skeleton of ~, it is immediate that ~(k) is a neighborhood of X and U, the submanifold cut out by ~(k), is a neighborhood of the mapping f : X-~ M. Clearly U is a geometric k-skeleton of M, proving 9-8.

Journal

Publications mathématiques de l'IHÉS – Springer Journals

Published: Aug 4, 2007

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