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Topology"],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g.\u00a0the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell \\ge 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u2113<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and the other is the filtration parameter. This new persistence structure, called the <jats:italic>persistent cup module<\/jats:italic>, is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we obtain a 1-dimensional persistence module, called the persistent <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of a <jats:italic>persistent invariant<\/jats:italic>, which extends both the <jats:italic>rank invariant<\/jats:italic> (also referred to as <jats:italic>persistent Betti number<\/jats:italic>), Puuska\u2019s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-defined <jats:italic>persistent cup-length invariant<\/jats:italic>, and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called the <jats:italic>persistent LS-category invariant<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s41468-023-00138-5","type":"journal-article","created":{"date-parts":[[2023,10,7]],"date-time":"2023-10-07T20:33:45Z","timestamp":1696710825000},"page":"93-148","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Persistent cup product structures and related invariants"],"prefix":"10.1007","volume":"8","author":[{"given":"Facundo","family":"M\u00e9moli","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anastasios","family":"Stefanou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ling","family":"Zhou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,10,7]]},"reference":[{"issue":"1","key":"138_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2140\/pjm.2017.290.1","volume":"290","author":"M Adamaszek","year":"2017","unstructured":"Adamaszek, M., Adams, H.: The Vietoris-Rips complexes of a circle. 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