The Lefschetz Fixed Point Theorem for Involutions

Abstract

The purpose of this note is to show that the Lefschetz fixed point theorem holds for involutions on locally compact spaces. The Alexander-Spanier-Wallace cohomology with compact supports will be used. Let X be a locally compact space. The Lefschetz number Λ _f of a map f: X → X is defined by

$${A_{f}} = \sum\limits_{i} {{{\left( { - 1} \right)}^{i}}} Trace\left( {f*\left| {{H^{i}}\left( {X;R} \right)} \right.} \right)$$

where f*: H ⁱ(X; R) →H ⁱ(X; R) and R is the field of real numbers.

This work was supported by the National Science Foundation.