Axioms for the Lefschetz number as a lattice valuation

TL;DR

This paper introduces new axioms for the Lefschetz number based on lattice valuation principles, providing a unique characterization that extends from simplicial complexes to continuous maps with homotopy invariance.

Contribution

It establishes a novel axiomatic framework for the Lefschetz number, connecting it to valuation theory and extending classical results to continuous maps.

Findings

Unique axiomatic characterization of the Lefschetz number.

Extension of axioms from simplicial complexes to continuous maps.

Homotopy invariance can be weakened while preserving the axiomatic framework.

Abstract

We give new axioms for the Lefschetz number based on Hadwiger's characterization of the Euler characteristic as the unique lattice valuation on polyhedra which takes value 1 on simplices. In the setting of maps on abstract simplicial complexes, we show that the Lefschetz number is unique with respect to a valuation axiom and an axiom specifying the value on a simplex. These axioms lead naturally to the classical computation of the Lefschetz number as a trace in homology. We then extend this approach to continuous maps of polyhedra, assuming an extra homotopy invariance axiom. We also show that this homotopy axiom can be weakened.