A Hadwiger Theorem for Simplicial Maps

TL;DR

This paper introduces valuations on simplicial maps, generalizing the Lefschetz number, and proves a Hadwiger-style classification theorem for these valuations in geometric realizations.

Contribution

It defines valuations on simplicial maps and establishes a classification theorem analogous to Hadwiger's theorem for these valuations.

Findings

Defined Lefschetz volumes as valuations on simplicial maps

Generalized the Lefschetz number to a broader valuation framework

Proved a Hadwiger-style classification theorem for these valuations

Abstract

We define the notion of valuation on simplicial maps between geometric realizations of simplicial complexes in $\mathbb{R}^n$. Valuations on simplicial maps are analogous to valuations on sets. In particular, we define the Lefschetz volumes, which are analogous to the intrinsic volumes of subsets of $\mathbb{R}^n$. Our definition not only provides a generalization of the Lefschetz number, but also yields a Hadwiger-style classification theorem for all such valuations.