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

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.