Open Access

The Lefschetz-Hopf theorem and axioms for the Lefschetz number

Fixed Point Theory and Applications20042004:465090

DOI: 10.1155/S1687182004308120

Received: 28 August 2003

Published: 3 March 2004

Abstract

The reduced Lefschetz number, that is, where denotes the Lefschetz number, is proved to be the unique integer-valued function on self-maps of compact polyhedra which is constant on homotopy classes such that (1) for and ; (2) if is a map of a cofiber sequence into itself, then ; (3) , where is a self-map of a wedge of circles, is the inclusion of a circle into the th summand, and is the projection onto the th summand. If is a self-map of a polyhedron and is the fixed point index of on all of , then we show that satisfies the above axioms. This gives a new proof of the normalization theorem: if is a self-map of a polyhedron, then equals the Lefschetz number of . This result is equivalent to the Lefschetz-Hopf theorem: if is a self-map of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of is the sum of the indices of all the fixed points of .

Authors’ Affiliations

(1)
Department of Mathematics, Dartmouth College
(2)
Department of Mathematics, University of California

Copyright

© Arkowitz and Brown 2004