Desuspending collapses

Abstract

Suppose X and Y are compact polyhedra and suppose the suspension of X collapses to the suspension of Y. This paper investigates the conditions under which it follows that X collapses to Y. If X is a piecewise linear manifold and the codimension of Y is greater than two, the only difficulty is with the fundamental group. The hypothesis in this case that the inclusion of Y into X induces an isomorphism of fundamental groups is sufficient to show that X collapses to Y. Codimension one and two require additional hypotheses. The major tool used is the piecewise linear s-cobordism theorem.