Homotopy conditions which detect simple homotopy equivalences

Thomas Ashland Chapman

Vol. 80 (1979), No. 1, 13–46

DOI: 10.2140/pjm.1979.80.13

Abstract

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.

Mathematical Subject Classification 2000

Primary: 57Q10

Milestones

Received: 7 March 1977

Revised: 13 June 1978

Published: 1 January 1979