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

摘要

We study the stability of covers of simplicial complexes. Given a map f:Y?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 A3(????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.