Approximating Homotopy Equivalences by Homeomorphisms

Abstract

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<tl}. Thus an a-equivalence is a special type of homotopy equivalence in which we place restrictions on the size of the homotopies fg?id and gfz-id. It is easy to show that if f is close to a homeomorphism, then f must be an a-equivalence (for a a fine open cover). Here is our main result. a-APPROXIMATION THEOREM. Let Nn be an n-manifold n > 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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