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arXiv · 1408.2351

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

Ethan D. Bloch
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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 $λ(L)$ 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.

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Ethan D. Bloch. 2014-08-11. Locally Determined Functions of Finite Simplicial Complexes that are Linear Combinations of the Numbers of Simplices in Each Dimension. https://arxiv.org/abs/1408.2351

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