The Nerve Theorem relates the topological type of a suitably nice space with the nerve of a good cover of that space. It has many variants, such as to consider acyclic covers and numerous applications in topology including applied and computational topology. The goal of this paper is to relax the notion of a good cover to an approximately good cover, or more precisely, we introduce the notion of an ε-acyclic cover. We use persistent homology to make this rigorous and prove tight bounds between the persistent homology of a space endowed with a function and the persistent homology of the nerve of an ε-acyclic cover of the space. Our approximations are stated in terms of interleaving distance between persistence modules. Using the Mayer-Vietoris spectral sequence, we prove upper bounds on the interleaving distance between the persistence module of the underlying space and the persistence module of the the nerve of the cover. To prove the best possible bound we must introduce special cases of interleavings between persistence modules called left and right interleavings. Finally, we provide examples which achieve the bound proving the lower bound and tightness of the result.
We present a new computing package Flagser, designed to construct the directed flag complex of a finite directed graph, and compute persistent homology for flexibly defined filtrations on the graph and the resulting complex. The persistent homology computation part of Flagser is based on the program Ripser [Bau18a], but is optimised specifically for large computations. The construction of the directed flag complex is done in a way that allows easy parallelisation by arbitrarily many cores. Flagser also has the option of working with undirected graphs. For homology computations Flagser has an Approximate option, which shortens compute time with remarkable accuracy. We demonstrate the power of Flagser by applying it to the construction of the directed flag complex of digital reconstructions of brain microcircuitry by the Blue Brain Project and several other examples. In some instances we perform computation of homology. For a more complete performance analysis, we also apply Flagser to some other data collections. In all cases the hardware used in the computation, the use of memory and the compute time are recorded.
The covering type of a space X is a numerical homotopy invariant that in some sense measures the homotopical size of X. It was first introduced by Karoubi and Weibel [9] as the minimal cardinality of a good cover of a space Y taken among all spaces Y that are homotopy equivalent to X. In this paper we give several estimates of the covering type in terms of other homotopy invariants of X, most notably the ranks of the homology groups of X, the multiplicative structure of the cohomology ring of X and the Lusternik-Schnirelmann category of X. In addition, we relate the covering type of a triangulable space to the number of vertices in its minimal triangulations. In this way we derive within a unified framework several estimates of vertex-minimal triangulations which are either new or extensions of results that have been previously obtained by ad hoc combinatorial arguments.2010 Mathematics Subject Classification. Primary 55M; Secondary 55M30, 57Q15, 57R05 .
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