Lefschetz fixed point theorems for a new class of multi-valued maps

Michael J. Powers

Vol. 42 (1972), No. 1, 211–220

DOI: 10.2140/pjm.1972.42.211

Abstract

The Lefschetz fixed point theorem states that whenever the Lefschetz number Λ(f) of a map f : X → X is nonzero, then f must have a fixed point. The theorem is known to hold when X is an ANR and f is a compact continuous map. The theorem has been studied for compact, upper semi-continuous, acyclic multi-valued maps and is known to hold in this setting for topologically complete ANR’s.

A more general class of multi-valued maps is considered in this paper: the class of compact upper semi-continuous maps which can be written as a composition of acyclic maps. Using this class of maps, a theorem is proved which generates spaces for which the Lefschetz theorem holds. In particular, the Lefschetz theorem holds for all (metric) ANR’s.

Mathematical Subject Classification 2000

Primary: 55C20

Secondary: 54C60

Milestones

Received: 5 January 1971

Revised: 12 July 1971

Published: 1 July 1972