HOMOTOPY CONDITIONS WHICH DETECT SIMPLE HOMOTOPY EQUIVALENCES

Let X, Y, and K be compact polyhedra, let p: Y×K→Y be the projection map, and let f: X→Y×K be a homotopy equivalence which has a homotopy inverse g: Y×K→X along with homotopies fg≃id, gf≃id such that p(fg≃id) and pf(gf≃id) are small homotopies. In this paper we prove that if π1 of each component of K is free abelian, then f must be a simple homotopy equivalence. © 1979 by Pacific Journal of Mathematics.