A Combinatorial Study of the Fixed Point Index

TL;DR

This paper develops a new combinatorial framework for the fixed point index that broadens its applicability and introduces an integration theory, improving upon previous Lefschetz-based methods.

Contribution

It introduces a combinatorial adaptation of the fixed point index that extends the Lefschetz number and enhances invariance and integration capabilities.

Findings

Broader applicability of fixed point index without restrictive assumptions

Extension of the Lefschetz number to a combinatorial setting

New invariance results and integration methods for fixed points

Abstract

We introduce a theory of integration with respect to the fixed point index, offering a substantial improvement over previous approaches based on the Lefschetz number. This framework eliminates several restrictive assumptions -- such as the need for definability, openness, or f-invariance of subspaces -- thereby allowing broader applicability. We also present a natural combinatorial adaptation of the fixed point index that extends the combinatorial Lefschetz number. This extension yields new topological and homotopical invariance results and facilitates the integration of real-valued functions with respect to fixed points.