A_∞ persistent homology estimates the topology from pointcloud datasets

2019/02/25 by Francisco Belchí, Belchí, Francisco, Anastasios Stefanou +1
Mathematics · #55N #55P #55U #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT #msc:55N #msc:55P #msc:55U

paper · pdf · doi:10.48550/arxiv.1902.09138

26 pages

arxiv created 2019/02/25 · arxiv updated 2019/02/26

Abstract

Let X be a closed subspace of a metric space M. Under mild hypotheses, one can estimate the Betti numbers of X from a finite set P ⊂ 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. These properties are computed via A_∞-structures, and are therefore related to the cup and Massey products of X, its loop space ΩX, its formality, linking numbers, etc. Additionally, we study the following setting: given a continuous function f \colon Y \longrightarrow \mathbb R on a topological space Y, A_∞ persistent homology builds a family of barcodes presenting a highly detailed description of some geometric and topological properties of Y. We prove here that under mild assumptions, these barcodes are stable: small perturbations in the function f imply at most small perturbations in the barcodes.

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