Functions of finite simplicial complexes that are not locally determined

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 λ(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.