Homotopy conditions which detect simple homotopy equivalences

Abstract

Let X, Y, and K be compact polyhedra, let p: YxK-^Y be the projection map, and let /: X-^YxK be a homotopy equivalence which has a homotopy inverse g: YxK-^X along with homotopies //~id, gfc^i such that p(fg~ id) and pf(gf ~id) are small homotopies. In this paper we prove that if t of each component of K is free abelian, then / must be a simple homotopy equivalence.