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Journal of Homotopy and Related Structures

, Volume 13, Issue 3, pp 599–633 | Cite as

A higher homotopic extension of persistent (co)homology

  • Estanislao HerscovichEmail author
Article
  • 77 Downloads

Abstract

Our objective is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of a simplicial set embedded in \({\mathbb {R}}^{n}\) induces a multiplicative filtration on the dg algebra of simplicial cochains, we use a result by Kadeishvili to get a unique \(A_{\infty }\)-algebra structure on the complete persistent cohomology of the filtered simplicial set. We then construct of a (pseudo)metric on the set of all barcodes of all cohomological degrees enriched with the \(A_{\infty }\)-algebra structure stated before, refining the usual bottleneck metric, and which is also independent of the particular \(A_{\infty }\)-algebra structure chosen. We also compute this distance for some basic examples. As an aside, we give a simple proof of a result relating the barcode structure between persistent homology and cohomology, that was observed by de Silva, Morozov, and Vejdemo-Johansson under some restricted assumptions which we do not suppose.

Keywords

Persistent homology dg algebras \(A_{\infty }\)-algebras Bottleneck metric 

Mathematics Subject Classification

16E45 16W70 18G55 55U10 

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Copyright information

© Tbilisi Centre for Mathematical Sciences 2017

Authors and Affiliations

  1. 1.Institut FourierUniversité Grenoble AlpesGrenobleFrance
  2. 2.Departamento de Matemática, FCEyNUniversidad de Buenos AiresBuenos AiresArgentina

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