A higher homotopic extension of persistent (co)homology

Abstract

Our objective in this article is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of the (say) simplicial set embedded in R n induces a multiplicative filtration (which would not be a so harsh hypothesis in our setting) on the dg algebra given by the complex of simplicial cochains, we may use a result by T. Kadeishvili to get a unique (up to noncanonical equivalence) A∞-algebra structure on the complete persistent cohomology of the filtered simplicial (or topological) set. We then provide a construction of a (pseudo)metric on the set of all (generalized) barcodes (that is, of all cohomological degrees) enriched with the A∞-algebra structure stated before, refining the usual bottleneck metric, and which is also independent of the particular A∞-algebra structure chosen (among those equivalent to each other). We think that this distance might deserve some attention for topological data analysis, for it in particular can recognize different linking or foldings patterns, as in the Borromean rings. As an aside, we give a simple proof of a result relating the barcode structure between persistent homology and cohomology. This result was observed in [9] under some restricted assumptions, which we do not suppose.Mathematics subject classification 2010: 16E45, 16W70, 18G55, 55U10, 68U05.

Algorithms in A ∞-algebras

“…Herscovich [17] introduces a novel metric on persistent homology. Herscovich constructs a metric on locally finite Adams graded minimal A ∞ -algebras, and then quotients by quasi-isomorphism to establish a metric on persistent homology barcodes equipped with an A ∞ -algebra structure.…”

Spectral sequences associated to deformations

“…Moreover, another objective of the article is to explain the interesting constructions by Lapin in more structural terms. We would like to remark that we also became interested in the problem for its applications to persistent homology (see [5]).…”

Multiplicative persistent distances

“…Some approaches to use A ∞ -structures for persistence have already been considered in the literature, notably in the work of F. Belchí et al (see [GM14,BM15,Bel17,BS19]) and Herscovich (cf. [Her18]). In both cases, they consider transferred structures but do not consider the full either A ∞ or persistent structure and in particular do not define an associated interleaving distance.…”