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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. Introduction. In this paper we use infinite-dimensional topology to prove that Whitehead torsion is a topological invariant for compact connected CW-complexes. This answers affirmatively a question raised by Whitehead in [12]. Our techniques are motivated by the work of Kirby-Siebenmann [7], where handle straightening was used to prove the invariance of torsion for compact connected PL manifolds. This type of approach very strongly uses the fact that PL manifolds have nice neighborhoods of each point, a property which is not generally satisfied for CW-complexes. Our proof of torsion invariance uses some recent results concerning Hilbert cube manifolds (or Q-manifolds), where a Q-manifold is a separable metric manifold modeled on the Hilbert cube Q. The first key idea involved in the proof is the following result of West [10]: If X is any compact CW-complex, then X x Q is a Q-manifold. This has the effect of converting a CW-complex into a space which has nice neighborhoods of each point. Indeed any point in a Q-manifold lies in an open set which is homeomorphic to 9 x [0,1) [2]. The second key idea in the proof is an infinite-dimensional version of the finite-dimensional handle straightening idea which was used in [7]. This infinitedimensional handle straightening technique is the main result of [4] and issummarized in Section 2 of this paper. Combining these ideas we obtain the following characterization of simple homotopy equivalences in terms of homeomorphisms of Q-manifolds. We remark that the "only if" part of this characterization is essentially Corollary 3 of [10] (see our Lemma 2.2). MAIN THEOREM. Let X and Y be compact connected CW-complexes and let f: X--Y be a map (i.e. a continuous function). Then f is a simple homotopyReceived July 21, 1972.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. Let M', N' be topological n-manifolds, n > 5, and let f: M->N be a proper nap (i.e., a map such that inverse images of compacta are compact). The purpose of this paper is to answer the following question: When is f close to a homeomorphism? Our answer is phrased in terms of local homotopy restrictions on f which give us necessary and sufficient conditions for f to be close to a homeomorphism.Here is the basic definition. If a is an open cover of N, then the proper map f:M-*N is said to be an a-equivalence provided that for some map g: N->M there are homotopies (t from fg to the identity on N, and Pt from gf to the identity on M, such that (1) for each mEM, there is a U E-containing {fqt(m)IO6t'6 1}, (2) for each nEN, there is a UEa containing {(t(n)10 5. For every open cover a of N there is an open cover ,B of N such that any /3-equivalence f: Mn--Nn which is already a homeomorphism from aM to aN is a-close to a homeomorphism h: M->N (i.e., for each mE M, there is a U Ea containing f(m) and h(m)). An a-approximation theorem was first proved for Q-manifolds (Q = Hilbert cube). In [5] the second author proved that if NQ is a Q-manifold, then there is an open cover a of NQ such that any a-equivalence f: M9Q-NQ is close to a homeomorphism h: M--N. The proof given there involved global mapping cylinder constiructions which do not seem to have analogs in finite-dimensional topology. By means of these mapping cylinder constructions it was shown in [5]that the homeomorphism h: M->N can be chosen to depend continuously on thus leading to a proof that the homeomorphism group of a compact Q-manifold is an ANR. The proof we give here is a handle-by-handle approach in which we lose the continuous dependence of h on f. In [13]Siebenmann proved that if f:Mn-->Nn is a proper surjection of n-manifolds, n > 5, which is already a homeomorphism from aM to aN, and for which each point inverse fl(n) has the Borsuk shape of a point, then f can be approximated arbitrarily closely by homeomorphisms. (Such maps f are called CE.) It follows immediately from Lacher [10] that any CE map f: M->N is an a-equivalence, for any a. Thus our a-Approximation Theorem generalizes the CE approximation theorem of [13].Our proof of the a-Approximat...

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