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

TL;DR

The paper proves that the combinatorial Lefschetz number $\Lambda_{\mathrm{comb}}$ is a topological invariant and extends its definition to open maps, enabling a streamlined Lefschetz framework for both bounded and unbounded spaces. By establishing invariance, it derives new fixed-point results and Nielsen-number bounds, and connects these ideas to axiomatic formulations, including a replacement of certain Arkowitz axioms. The work provides practical computational tools through numerous examples and shows how invariance underpins generalizations of fixed-point index (O'Neill) and relative Lefschetz numbers. Overall, it advances fixed-point theory in definable and noncompact settings, offering both theoretical insight and actionable methods for computing Lefschetz numbers and related invariants.

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.

Figures (17)

• Figure 1: Complex $X$.
• Figure 2: $X_4$.
• Figure 3: Space $X$.
• Figure 4: Space $Y$.
• Figure 5: Space $X$.

Theorems & Definitions (67)

• Proposition 2.1: Inclusion-exclusion principle M-M1
• Remark 2.2
• Theorem 2.3: Definable-triangulation theorem Dries
• Remark 2.4
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• Corollary 3.4