Geometric groups and Whitehead torsion

Abstract

The purpose of this paper is to define geometric 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, L' K is a subcomplex with dim L $2, and f: K -> N is a homeomorphism with f IK-L p.w.l., then f is a simple homotopy equivalence. Another corollary is that if K is a 2-complex contained in a p.w.l. manifold Mn, Un is a compact p.w.l. submanifold, K' U' M, and U e-deforms to K, then KG U is a simple homotopy equivalence and, thus, if n > 65 U is a regular neighborhood of K. Finally, for any finitely presented group with Wh (-a) #0, ] an h-cobordism W with -a (W) = 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 11 implies the following: If f: Rn -> Rn is a homeomorphism (n >5) such that fx Id: Rn x Rk Rn x Rk is stable, then f is stable. The final note of the appendix implies the following: Suppose (ai,j) is an infinite matrix with integer entries, and that it and its inverse are band matrices, i.e., bounded about the diagonal. Then (ai,j) can be diagonalized by row operations. This is a nongeometric analogue of Conjecture II for the infinite complex R1.