Near Coverings and Cosystolic Expansion -- an example of topological property testing

TL;DR

This paper links the stability of simplicial complex covers to cosystolic expansion, providing a new topological property testing perspective and demonstrating cover-stability in specific high-dimensional structures.

Contribution

It establishes the equivalence between cover-stability and cosystolic expansion, introducing a novel topological property testing framework for high-dimensional complexes.

Findings

Cover-stability is equivalent to cosystolic expansion.

The 2-dimensional spherical building $A_{3}( ext{F}_q)$ is cover-stable.

Introduces the concept of topological property testing for complexes.

Abstract

We study the stability of covers of simplicial complexes. Given a map $f:Y\to X$ that satisfies almost all of the local conditions of being a cover, is it close to being a genuine cover of $X$? Complexes $X$ for which this holds are called cover-stable. We show that this is equivalent to $X$ being a cosystolic expander with respect to non-abelian coefficients. This gives a new combinatorial-topological interpretation to cosystolic expansion which is a well studied notion of high dimensional expansion. As an example, we show that the $2$-dimensional spherical building $A_{3}(\mathbb{F}_q)$ is cover-stable. We view this work as a possibly first example of "topological property testing", where one is interested in studying stability of a topological notion that is naturally defined by local conditions.