Locally Determined Functions of Finite Simplicial Complexes that are Linear Combinations of the Numbers of Simplices in Each Dimension

TL;DR

This paper investigates which linear combinations of simplex counts in finite simplicial complexes are locally determined, showing that many important functions, including the Charney-Davis quantity, are not locally determined in certain cases.

Contribution

The paper demonstrates that not all functions defined as linear combinations of simplex counts are locally determined, highlighting limitations of local invariance in simplicial complex functions.

Findings

Euler characteristic is locally determined in both senses.

Charney-Davis quantity is not locally determined on certain complexes.

Many linear combinations of simplex counts lack local determination.

Abstract

The Euler characteristic, thought of as a function that assigns a numerical value to every finite simplicial complex, is locally determined in both a combinatorial sense and a geometric sense. In this note we show that not every function that assigns a numerical value to every finite simplicial complex via a linear combination of the numbers of simplices in each dimension is locally determined in either sense. In particular, the Charney-Davis quantity $\lambda (L)$ is not locally determined in either sense if it is defined on a set of simplicial complexes that includes all flag spheres of a given odd dimension.