A polyhedral transversality theorem for one-parameter fixed point theory

Abstract

The xed point set of a piecewise linear (PL )map h : PI! P is the set of points where h coincides with the projection : PI! P ; it is denoted by Fix(h) and is a subpolyhedron of PI. When P is a compact polyhedron, we show how to deform h (with appropriate control) to a new PL map h 0 so that Fix(h 0 ) is as nice as possible. Indeed it is not hard to arrange that Fix(h 0 ) have dimension 1 (Theorem A), but one would wish for a map h 0 such that Fix(h 0 ) is a manifold of dimension 1. This is achieved in Theorem B .I f Pis a PL manifold, Theorem B reduces to a standard PLtransversality theorem (Theorem C).