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An approximate nerve theorem. (English) Zbl 1400.55003
This paper describes a generalization of the classical theorem on the nerve of the cover of a topological space. This theorem says that under appropriate conditions the nerve of a cover where every non-empty intersection of finitely many sets in the cover is contractible, is homotopy equivalent to the space. The authors are interested in the case of the persistent homology of a space, motivated by the idea of using a set of sample points to approximate the space. Specifically the paper considers the persistent homology of a space as having the structure of a \(k[t]\)-module where \(k\) is a field. From this viewpoint \(t\) represents the time at which elements appear or disappear. The main theorem requires the notion of an \(\epsilon\)-acyclic cover, which is defined by the authors as an approximation to an acyclic cover. The main theorem relates the persistent homology of a space endowed with a function and the persistent homology of the nerve of an \(\epsilon\)-acyclic cover of the space. The notion of approximation in this context is based on the idea of interleaving distance between the persistence module of the underlying space and the persistence module of the nerve of the cover. The proof works by working through the Mayer-Vietoris spectral sequence to relate the persistent homology of the nerve to that of the space. The paper contains examples that show that the bounds on the interleaving distance in the main theorem are tight. There is also an application of the results to the case of a cover by small balls centered at sample points.

MSC:
55N99 Homology and cohomology theories in algebraic topology
55T99 Spectral sequences in algebraic topology
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