Fixed Point Theory and Applications
Volume 2004 (2004), Issue 1, Pages 1-11
doi:10.1155/S1687182004308120

The Lefschetz-Hopf theorem and axioms for the Lefschetz number

Martin Arkowitz1 and Robert F. Brown2

1Department of Mathematics, Dartmouth College, Hanover 03755-1890, NH, USA
2Department of Mathematics, University of California, Los Angeles 90095-1555, CA, USA

Received 28 August 2003

Copyright © 2004 Martin Arkowitz and Robert F. Brown. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The reduced Lefschetz number, that is, L(⋅)−1 where L(⋅) denotes the Lefschetz number, is proved to be the unique integer-valued function λ on self-maps of compact polyhedra which is constant on homotopy classes such that (1) λ(fg)=λ(gf) for f:X→Y and g:Y→X; (2) if (f1,f2,f3) is a map of a cofiber sequence into itself, then λ(f1)=λ(f1)+λ(f3); (3) λ(f)=−(deg(p1fe1)+⋯+deg(pkfek)), where f is a self-map of a wedge of k circles, er is the inclusion of a circle into the rth summand, and pr is the projection onto the rth summand. If f:X→X is a self-map of a polyhedron and I(f) is the fixed point index of f on all of X, then we show that I(⋅)−1 satisfies the above axioms. This gives a new proof of the normalization theorem: if f:X→X is a self-map of a polyhedron, then I(f) equals the Lefschetz number L(f) of f. This result is equivalent to the Lefschetz-Hopf theorem: if f:X→X 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 f is the sum of the indices of all the fixed points of f.