A property that characterizes Euler characteristic among invariants of combinatorial manifolds

TL;DR

This paper proves that among invariants of compact combinatorial manifolds depending solely on the counts of simplices, the Euler characteristic is uniquely determined and characterizes such invariants.

Contribution

The paper establishes that the Euler characteristic is the only invariant depending only on the number of simplices in each dimension for compact combinatorial manifolds.

Findings

Euler characteristic uniquely determined by simplex counts

Invariant depends only on Euler characteristic and boundary

Characterization of invariants based on simplex counts

Abstract

If a real value invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension on the manifold, then the invariant is completely determined by Euler characteristics of the manifold and its boundary. So essentially, Euler characteristic is the unique invariant of this type.