New Applications and Computations of the Lefschetz Number of Homeomorphisms and Open Maps

TL;DR

This paper proves the combinatorial Lefschetz number is a topological invariant, simplifying computations, extending fixed-point theorems, and generalizing to open maps, with significant implications for topological fixed-point theory.

Contribution

It establishes the topological invariance of the combinatorial Lefschetz number and extends fixed-point results to open maps, offering new computational and theoretical tools.

Findings

Combinatorial Lefschetz number is a topological invariant.

Simplified computation of Lefschetz numbers for certain maps.

New fixed-point theorems for unbounded spaces and open maps.

Abstract

We show that the combinatorial Lefschetz number is a topological invariant. This is an important result in itself; in order to point it out, we will also work here several relevant consequences in different directions. The first of them is a significant simplification of the computations involved in obtaining the Lefschetz number of certain maps, as well as some new Lefschetz fixed-point theorems for unbounded spaces. Indeed, these ideas allow us to obtain a clear lower bound for the Nielsen number of a triad in some spaces, such as, for example, the connected sum of two p-tori (p greater than 2). Another consequence, in the case of homeomorphisms, is that, in the classical axiomatic definition of the Lefschetz number, the wedge-of-circles axiom and the cofibration axiom can be replaced by the single axiom of topological invariance of the combinatorial Lefschetz number. Using the invariance of the combinatorial Lefschetz number we also generalize O'Neill's classical result about topological invariance of the fixed-point index and we prove a topological-invariance result for the relative Lefschetz number. We also generalize the combinatorial Lefschetz number from homeomorphisms to open maps and we obtain a new fixed-point theorem.