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

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.