The Lefschetz-Hopf theorem and axioms for the Lefschetz number

Martin Arkowitz,Robert F Brown

doi:10.1155/s1687182004308120

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 .

Highlights

Let X be a finite polyhedron and denote by H∗(X) its reduced homology with rational coefficients
Let Ꮿ be the collection of spaces X of the homotopy type of a finite, connected CWcomplex
We look at the left end of diagram (2.3) and get 0 = Tr hN+1 = Tr fN − Tr gN + Tr hN Im βN, (2.10)

Summary

Introduction

Let X be a finite polyhedron and denote by H∗(X) its reduced homology with rational coefficients. In an unpublished dissertation [10], Hoang extended Watts’s axioms to characterize the reduced Lefschetz number for basepoint-preserving self-maps of finite polyhedra. His list of axioms is different from, but similar to, those in Theorem 1.1. The Lefschetz-Hopf theorem follows from the normalization property by the additivity property of the fixed-point index. The homotopy and additivity properties of the fixed-point index imply that the normalization property follows from the Lefschetz-Hopf theorem

Lefschetz numbers and exact sequences

Lefschetz number

Characterization of the Lefschetz number

The normalization property