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

Irit Dinur; Roy Meshulam

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arXiv · 1909.08507

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

Irit Dinur·Roy Meshulam

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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.

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Irit Dinur, Roy Meshulam. 2019-09-18. Near Coverings and Cosystolic Expansion -- an example of topological property testing. https://arxiv.org/abs/1909.08507

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