Geometric groups and Whitehead torsion

Abstract

The purpose of this paper is to define "geometric groups" and to relate them to various problems in topology.This relation is exhibited through Conjectures I and II.It will be shown that Conjecture I implies Conjecture II and that Conjecture II implies the topological invariance of Whitehead torsion.Conjecture II is true for 2-complexes, and this implies that if K and N are finite connected complexes, Lc=-K is a subcomplex with dim F ^2, and /: K-> N is a homeomorphism with f\K-L p.w.l., then /is a simple homotopy equivalence.Another corollary is that if AT is a 2-complex contained in a p.w.l.manifold Mn, Un is a compact p.w.l.submanifold, 7C<= [/<= M, and U ^-deforms to K, then 7v<= U is a simple homotopy equivalence and, thus, if zz^6, F is a regular neighborhood of K. Finally, for any finitely presented group -n with Wh (tt)^O, 3 an /z-cobordism W with nx(W) = -n which is not topologically trivial.Geometric groups are related to other problems in topology and some of these are mentioned without proof in the appendix.For example, Conjecture I implies that compact ANRs of finite dimension have the homotopy type of finite complexes.Conjecture II has a noncompact analogue and since the difficulties are local, there is essentially nothing new here (Conjecture II is true for infinite 2-complexes).This noncompact form of Conjecture II implies the following: If/: Rn -> Rn is a homeomorphism (zz^5) such that fix Id : Rn x Rk -> Rn x Rk is stable, then / is stable.The final note of the appendix implies the following: Suppose (atJ) is an infinite matrix with integer entries, and that it and its inverse are band matrices, i.e., bounded about the diagonal.Then (au) can be diagonalized by row operations.This is a nongeometric analogue of Conjecture II for the infinite complex R1.Notation, definitions, and conjectures.Although the results of this paper hold for CW complexes, their use has been avoided whenever possible to simplify the proofs.Unless otherwise stated, the word "complex" means "simplicial complex".If K is a complex, K^Rn means that K is topologically embedded in Rn.This is important because in Lemma 5, K and N are homeomorphic complexes embedded in essentially the same manner.If K and N are combinatorially distinct, they cannot each be simplicially embedded.In general, K will be embedded in Rn simply because Rn is a pleasant metric space in which geometric construction may be performed.