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J. H. C. Whitehead’s concept: Simple homotopy type

J. H. C. Whitehead’s concept: Simple homotopy type CHAPTER V J. H. C. WHITEHEAD'S CONCEPT : SIMPLE HOMOTOPY TYPE In studying invariants finer than homotopy type, Whitehead [I4] introduced the notion of simple homotopy type. The conclusion of his theory is that although the notion of simple homotopy equivalence is quite " geometrically " defined, there is a purely " algebraic " criterion to determine whether any homotopy equivalence f : X-+Y is a simple homotopy equivalence. More precisely, Whitehead defines a " torsion group " W(G) which is an abelian group assignable to any group G. If X is a topological space he defines W(X) =W(~z l(x)). Whitehead then constructs an " obstruction " v(f)~W for any homotopy equivalence f : X-+Y such that f is a simple homotopy equivalence if and only if -:(f)=o. This clearly establishes the notion of simple homotopy type as an " algebraic " concept. In many cases, W(G) may be computed to be trivial. (For example: G={o}, Z2, Za, Z4, Z). Thus, for spaces X having as fundamental group such a G, the concepts of simple homotopy equivalence and homotopy equivalence coincide. If M is a differentiable manifold, it is classical that there is a unique underlying combinatorial structure of M; that is, there exists a smooth triangulation of M which is unique as a simplicial complex up to rectilinear subdivision. Since Whitehead showed that the simple homotopy type of a finite complex is independent of subdivision, it follows that the simple homotopy type of a differentiable manifold is a well-defined notion. " Unthickening " a differentiable cell filtration ..dr" loses much of the differential- topological structure of Mr', however the simple homotopy type of Jr' may still be recovered from the resulting cell filtration p~gt'. It turns out to be quite natural to define the simple homotopy type of a cell filtration. In terms of this concept we may provide a strong necessary condition to the problem posed in Chapter IV; if f: M-+X is a homotopy class of continuous maps of a differentiable manifold M to a cell filtration X: When does there exist a differentiable 28 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 29 cell filtration dr' of M which is a neighborhood of X such that f : M--->X is a projection map ? A necessary condition on f for the existence of such an ~ is that f be a simple homotopy equivalence. It is therefore to be expected that extensions of theorems of Smale to non simply connected situations will involve this notion. This chapter is devoted to defining simple homotopy equivalence for cell filtrations and quoting results of Whitehead's theory to be used in the sequel. Let K be a finite simplicial complex, A an n-simplex. Let J0 be the union of all faces but one in the boundary of A. Let n : A--->J0 be a projection of A onto J0 such that ,z]J 0 is the identity map. Let K* be the simplicial complex K*:K%~ Then, in this circumstance, i : K-+K*, the inclusion map is called an elementary expansion, and r~ : K*~ K, the map defined by b) ~ I K : I K is called an elementary contraction. We shall recall Whitehead's well-known definition (Formal deformation, w I3, [I4]): Definition (5. I). -- Let K, L be finite simplicial complexes. Then a simplicial map f: K-+L is called a simple homotopy equivalence if there is a sequence s : K=K0~ K1 -+t, K2 -+"t. "Kz:L ofsimplicial complexes and maps fl, .. -,f such thatf is either an elementary inclusion or contraction for i = I, ..., l, and the map f is the composite, f :fof_,o. . . of~. Two simplicial complexes related in this way will be said to be of the same simple homotopy type. The dimension of a simple homotopy equivalence, f, denoted dim f, is defined to be: dim f< max dim K i = dim o9' i = o, ..., I. (It is taken to be the minimum dimension of all sequences ~ exhibiting f as a simple homotopy equivalence.) As it stands, the simple homotopy type of a simplicial complex K is dependent upon the particular simplicial structure of[K [, the underlying topological space. It is unknown whether simple homotopy type is a topological invariant. It is proved in [I3] (Corollary to theorem 7) that the inclusion map i : K'-->K 29 3 ~ BARRY MAZUR of a subdivision K' of K is a simple homotopy equivalence. (This proof apparently contains a flaw; el. C. Zeeman, Unknotting combinatorial balls, but the result is nevertheless true. The reader is referred to Zeeman's paper (yet unpublished) for a variant to the Whitehead definition of simple homotopy equivalence (formal deformations) more suitable to general polyhedral spaces (i.e. not tied down to particular triangulations)). I shall define an analogous notion for the category ~" of cell filtrations. Let X~ ~-. Let X*~ be any element of ~- which is reordering-equivalent to Xu~D"-Iu+D ", where q~ : ~D"-I-+ x0eX is theconstantmap, and + : 0Dn--~XuvD n-1 is the " identification" + : OD n = S n-~ -+ Dn-1/D"-lc__Xu~Dn-1. Let /:X-+X* be the inclusion map. Then i is called an elementary expansion. Consider the filtration-non-preserving map 7: :X*-+X such that r~oi=i, and 7:(D" u D "-1) = x0eX. rc is called an elementary contraction. If i : X-+X* is an elementary expansion, then D "-I is called the free face of the expansion i, and D" is called the cell of the expansion i. Again, if X o -+ X 1 -+ X~ -+... -+ X, l, t, t, ft is a sequence of elementary expansions and contractions in if, then the composite f = f of_~ o... of~ : X 0-+ Xz is called a simple homotopy equivalence in the category ~'. The definition of dim f is a duplication of the definition of the analogous concept in the category of simplicial complexes. To relate the notion of simple homotopy type in the category of simplicial complexes to the notion of simple homotopy type in o~-, one needs the following proposition (to be proved in Chapter XI): (5.2) (Proposition r I .8) The simplicial complexes K, L are of the same simple homotopy type if and only if their induced cell filtrations K, L are of the same simple homotopy type. More exactly, as a consequence of Chapter XI, the underlying combinatorial manifold of a neighborhood of X~ is some regular neighborhood of some simplicial complex K which gives rise to X. Actually the nature of the proofs occurring in Whitehead's paper are such that they quite easily carry over to cell filtrations. Proposition (5.3). -- If K, L are simplicial complexes which are simply connected, and f : K-+L is a homotopy equivalence, then it is a simple homotopy equivalence. Proof. -- See [I4] (As mentioned in the beginning of this chapter, this result is also valid if 7:1 (K)= Z2, Z3, Z4, Z.) Proposition (5.4). -- Let f: K-+L be a simple homotopy equivalence of two simplicial complexes. Thenfis homotopic to g : K-+L such that: dim g< Max{dim K + I, dim L, 3} + I. 30 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 31 Proof. -- See the " addendum " on Page 4 8 of [I4]. Proposition (5.5). -- Let f: X~Y be a simple homotopy equivalence between two cell filtrations X, Y. Then there is a sequence h f~ X=X0-~ Xl->...-~ X=Y such that a) f~ is an elementary expansion for o<i<~ b) f~ is an elementary contraction for ~x<i<~ c) dim f0< dimf~ <... < dimf~_l d) dimf~> dimf~+l> ... > dimf~ e) dim f< Max {dim K-l- I, dim L, 3}-t- i The above proposition provides a convenient normal form for simple homotopy equivalences. Proposition (5.5) insures that one can find a sequence {f} such that e) is satisfied. Conditions a), b), c) and d) are brought by rearranging the order of occurrence of elementary expansions and contractions. Clearly one may arrange it so that all elementary expansions come first in order of increasing dimension. This achieves a), b) and c). The last condition may be achieved by finding a " regular " cell decomposition X k which represents Xk, up to rearrangement. (A regular cell decomposition is a properly ordered cell decomposition Y such that Y----- (Y0, .-., Y~) ; Yi =Yi-~u~iDn~ where q~i(0Dni) is contained in the (n i- x)-skeleton of Y, for i= I, ..., v. It is quite easy to see that any properly ordered cell filtration is represented by a regular cell decomposition.) Clearly in the model X k one may rearrange the sequence of elementary contractions to achieve d). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

J. H. C. Whitehead’s concept: Simple homotopy type

Publications mathématiques de l'IHÉS , Volume 15 (1) – Aug 4, 2007

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Publisher
Springer Journals
Copyright
Copyright © 1963 by Publications mathématiques de l’I.H.É.S
Subject
Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN
0073-8301
eISSN
1618-1913
DOI
10.1007/BF02684281
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Abstract

CHAPTER V J. H. C. WHITEHEAD'S CONCEPT : SIMPLE HOMOTOPY TYPE In studying invariants finer than homotopy type, Whitehead [I4] introduced the notion of simple homotopy type. The conclusion of his theory is that although the notion of simple homotopy equivalence is quite " geometrically " defined, there is a purely " algebraic " criterion to determine whether any homotopy equivalence f : X-+Y is a simple homotopy equivalence. More precisely, Whitehead defines a " torsion group " W(G) which is an abelian group assignable to any group G. If X is a topological space he defines W(X) =W(~z l(x)). Whitehead then constructs an " obstruction " v(f)~W for any homotopy equivalence f : X-+Y such that f is a simple homotopy equivalence if and only if -:(f)=o. This clearly establishes the notion of simple homotopy type as an " algebraic " concept. In many cases, W(G) may be computed to be trivial. (For example: G={o}, Z2, Za, Z4, Z). Thus, for spaces X having as fundamental group such a G, the concepts of simple homotopy equivalence and homotopy equivalence coincide. If M is a differentiable manifold, it is classical that there is a unique underlying combinatorial structure of M; that is, there exists a smooth triangulation of M which is unique as a simplicial complex up to rectilinear subdivision. Since Whitehead showed that the simple homotopy type of a finite complex is independent of subdivision, it follows that the simple homotopy type of a differentiable manifold is a well-defined notion. " Unthickening " a differentiable cell filtration ..dr" loses much of the differential- topological structure of Mr', however the simple homotopy type of Jr' may still be recovered from the resulting cell filtration p~gt'. It turns out to be quite natural to define the simple homotopy type of a cell filtration. In terms of this concept we may provide a strong necessary condition to the problem posed in Chapter IV; if f: M-+X is a homotopy class of continuous maps of a differentiable manifold M to a cell filtration X: When does there exist a differentiable 28 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 29 cell filtration dr' of M which is a neighborhood of X such that f : M--->X is a projection map ? A necessary condition on f for the existence of such an ~ is that f be a simple homotopy equivalence. It is therefore to be expected that extensions of theorems of Smale to non simply connected situations will involve this notion. This chapter is devoted to defining simple homotopy equivalence for cell filtrations and quoting results of Whitehead's theory to be used in the sequel. Let K be a finite simplicial complex, A an n-simplex. Let J0 be the union of all faces but one in the boundary of A. Let n : A--->J0 be a projection of A onto J0 such that ,z]J 0 is the identity map. Let K* be the simplicial complex K*:K%~ Then, in this circumstance, i : K-+K*, the inclusion map is called an elementary expansion, and r~ : K*~ K, the map defined by b) ~ I K : I K is called an elementary contraction. We shall recall Whitehead's well-known definition (Formal deformation, w I3, [I4]): Definition (5. I). -- Let K, L be finite simplicial complexes. Then a simplicial map f: K-+L is called a simple homotopy equivalence if there is a sequence s : K=K0~ K1 -+t, K2 -+"t. "Kz:L ofsimplicial complexes and maps fl, .. -,f such thatf is either an elementary inclusion or contraction for i = I, ..., l, and the map f is the composite, f :fof_,o. . . of~. Two simplicial complexes related in this way will be said to be of the same simple homotopy type. The dimension of a simple homotopy equivalence, f, denoted dim f, is defined to be: dim f< max dim K i = dim o9' i = o, ..., I. (It is taken to be the minimum dimension of all sequences ~ exhibiting f as a simple homotopy equivalence.) As it stands, the simple homotopy type of a simplicial complex K is dependent upon the particular simplicial structure of[K [, the underlying topological space. It is unknown whether simple homotopy type is a topological invariant. It is proved in [I3] (Corollary to theorem 7) that the inclusion map i : K'-->K 29 3 ~ BARRY MAZUR of a subdivision K' of K is a simple homotopy equivalence. (This proof apparently contains a flaw; el. C. Zeeman, Unknotting combinatorial balls, but the result is nevertheless true. The reader is referred to Zeeman's paper (yet unpublished) for a variant to the Whitehead definition of simple homotopy equivalence (formal deformations) more suitable to general polyhedral spaces (i.e. not tied down to particular triangulations)). I shall define an analogous notion for the category ~" of cell filtrations. Let X~ ~-. Let X*~ be any element of ~- which is reordering-equivalent to Xu~D"-Iu+D ", where q~ : ~D"-I-+ x0eX is theconstantmap, and + : 0Dn--~XuvD n-1 is the " identification" + : OD n = S n-~ -+ Dn-1/D"-lc__Xu~Dn-1. Let /:X-+X* be the inclusion map. Then i is called an elementary expansion. Consider the filtration-non-preserving map 7: :X*-+X such that r~oi=i, and 7:(D" u D "-1) = x0eX. rc is called an elementary contraction. If i : X-+X* is an elementary expansion, then D "-I is called the free face of the expansion i, and D" is called the cell of the expansion i. Again, if X o -+ X 1 -+ X~ -+... -+ X, l, t, t, ft is a sequence of elementary expansions and contractions in if, then the composite f = f of_~ o... of~ : X 0-+ Xz is called a simple homotopy equivalence in the category ~'. The definition of dim f is a duplication of the definition of the analogous concept in the category of simplicial complexes. To relate the notion of simple homotopy type in the category of simplicial complexes to the notion of simple homotopy type in o~-, one needs the following proposition (to be proved in Chapter XI): (5.2) (Proposition r I .8) The simplicial complexes K, L are of the same simple homotopy type if and only if their induced cell filtrations K, L are of the same simple homotopy type. More exactly, as a consequence of Chapter XI, the underlying combinatorial manifold of a neighborhood of X~ is some regular neighborhood of some simplicial complex K which gives rise to X. Actually the nature of the proofs occurring in Whitehead's paper are such that they quite easily carry over to cell filtrations. Proposition (5.3). -- If K, L are simplicial complexes which are simply connected, and f : K-+L is a homotopy equivalence, then it is a simple homotopy equivalence. Proof. -- See [I4] (As mentioned in the beginning of this chapter, this result is also valid if 7:1 (K)= Z2, Z3, Z4, Z.) Proposition (5.4). -- Let f: K-+L be a simple homotopy equivalence of two simplicial complexes. Thenfis homotopic to g : K-+L such that: dim g< Max{dim K + I, dim L, 3} + I. 30 DIFFERENTIAL TOPOLOGY FROM THE POINT OF VIEW OF SIMPLE HOMOTOPY THEORY 31 Proof. -- See the " addendum " on Page 4 8 of [I4]. Proposition (5.5). -- Let f: X~Y be a simple homotopy equivalence between two cell filtrations X, Y. Then there is a sequence h f~ X=X0-~ Xl->...-~ X=Y such that a) f~ is an elementary expansion for o<i<~ b) f~ is an elementary contraction for ~x<i<~ c) dim f0< dimf~ <... < dimf~_l d) dimf~> dimf~+l> ... > dimf~ e) dim f< Max {dim K-l- I, dim L, 3}-t- i The above proposition provides a convenient normal form for simple homotopy equivalences. Proposition (5.5) insures that one can find a sequence {f} such that e) is satisfied. Conditions a), b), c) and d) are brought by rearranging the order of occurrence of elementary expansions and contractions. Clearly one may arrange it so that all elementary expansions come first in order of increasing dimension. This achieves a), b) and c). The last condition may be achieved by finding a " regular " cell decomposition X k which represents Xk, up to rearrangement. (A regular cell decomposition is a properly ordered cell decomposition Y such that Y----- (Y0, .-., Y~) ; Yi =Yi-~u~iDn~ where q~i(0Dni) is contained in the (n i- x)-skeleton of Y, for i= I, ..., v. It is quite easy to see that any properly ordered cell filtration is represented by a regular cell decomposition.) Clearly in the model X k one may rearrange the sequence of elementary contractions to achieve d).

Journal

Publications mathématiques de l'IHÉSSpringer Journals

Published: Aug 4, 2007

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