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 Rn induces a multiplicative filtration on the dg algebra of simplicial cochains, we use a result by Kadeishvili to get a unique A∞-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∞-algebra structure stated before, refining the usual bottleneck metric, and which is also independent of the particular A∞-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. © 2017, Tbilisi Centre for Mathematical Sciences.
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Citas:
---------- APA ----------
(2018)
. A higher homotopic extension of persistent (co)homology. Journal of Homotopy and Related Structures, 13(3), 599-633.
http://dx.doi.org/10.1007/s40062-017-0195-x---------- CHICAGO ----------
Herscovich, E.
"A higher homotopic extension of persistent (co)homology"
. Journal of Homotopy and Related Structures 13, no. 3
(2018) : 599-633.
http://dx.doi.org/10.1007/s40062-017-0195-x---------- MLA ----------
Herscovich, E.
"A higher homotopic extension of persistent (co)homology"
. Journal of Homotopy and Related Structures, vol. 13, no. 3, 2018, pp. 599-633.
http://dx.doi.org/10.1007/s40062-017-0195-x---------- VANCOUVER ----------
Herscovich, E. A higher homotopic extension of persistent (co)homology. J. Homotopy Related Struct. 2018;13(3):599-633.
http://dx.doi.org/10.1007/s40062-017-0195-x