A polyhedral transversality theorem for one-parameter fixed point theory

Thomas Plavchak

Vol. 178 (1997), No. 1, 127–146

DOI: 10.2140/pjm.1997.178.127

Abstract

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

Milestones

Received: 20 November 1994

Revised: 19 February 1996

Published: 1 March 1997