A higher homotopic extension of persistent (co)homology
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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 a finite dimensional vector space 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_infinity-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_infinity-algebra structure stated before, refining the usual bottleneck metric, and which is also independent of the particular A_infinity-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 a recent article by V. de Silva, D. Morozov and M. Vejdemo-Johansson under some restricted assumptions, which we do not suppose.
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