Euler calculus on spaces homeomorphic to definable sets and some applications

TL;DR

This paper extends Euler calculus to spaces homeomorphic to definable sets, generalizing sensor network results and providing new proofs for classical topological theorems involving Euler-Poincaré characteristic.

Contribution

It broadens the applicability of Euler calculus to more general topological spaces and offers new proofs for classical results, enhancing its utility in topology and applications.

Findings

Extended Euler calculus to spaces homeomorphic to definable sets

Generalized sensor network result for broader applications

Provided a new combinatorial proof for Euler-Poincaré characteristic of fiber bundles

Abstract

We show that integration with respect to the Euler-Poincar\'e characteristic can be extended from the setting of definable sets to the setting of topological spaces homeomorphic to definable sets. We use that extension to generalize a result regarding sensor networks due to Ghrist and Baryshnikov, in order to make it more flexible in applications. Finally, we obtain both an extension and a combinatorial proof of a classical result about the Euler-Poincar\'e characteristic of fiber bundles.