Euler calculus on spaces homeomorphic to definable sets and some applications
E. Mac\'ias-Virg\'os, D. Mosquera-Lois

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.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Digital Image Processing Techniques · Medical Imaging Techniques and Applications
