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\(A_\infty \) Persistent Homology Estimates Detailed Topology from Pointcloud Datasets

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Abstract

Let X be a closed subspace of a metric space M. It is well known that, under mild hypotheses, one can estimate the Betti numbers of X from a finite set \(P\subset M\) of points approximating X. In this paper, we show that one can also use P to estimate much more detailed topological properties of X. We achieve this by proving the stability of \(A_\infty \)-persistent homology. In its most general case, this stability means that given a continuous function \(f:Y\rightarrow {\mathbb {R}}\) on a topological space Y, small perturbations in the function f imply at most small perturbations in the family of \(A_\infty \)-barcodes. This work can be viewed as a proof of the stability of cup-product and generalized-Massey-products persistence. The technical key of this paper consists of figuring out a setting which makes \(A_\infty \)-persistence functorial.

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Notes

  1. Note that \(\check{{\mathcal {C}}}_\epsilon (X)\) is sometimes defined using the radius \(\epsilon /2\).

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Acknowledgements

We would like to thank Justin Curry for valuable feedback on previous versions of this paper. F. Belchí was partially supported by the EPSRC grant EPSRC EP/N014189/1 (Joining the dots) to the University of Southampton and by the Spanish State Research Agency through the María de Maeztu Seal of Excellence to IRI (MDM-2016-0656). A. Stefanou was partially supported by the National Science Foundation through grants CCF-1740761 (TRIPODS TGDA@OSU) and DMS-1440386 (Mathematical Biosciences Institute at the Ohio State University).

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  1. Institut de Robòtica i Informàtica Industrial, CSIC-UPC, Llorens i Artigas 4-6, 08028, Barcelona, Spain

    Francisco Belchí

  2. Department of Mathematics, Mathematical Biosciences Institute, Ohio State University, Columbus, USA

    Anastasios Stefanou

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  1. Francisco Belchí
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  2. Anastasios Stefanou
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Correspondence to Francisco Belchí.

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Belchí, F., Stefanou, A. \(A_\infty \) Persistent Homology Estimates Detailed Topology from Pointcloud Datasets. Discrete Comput Geom 68, 274–297 (2022). https://doi.org/10.1007/s00454-021-00319-y

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