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Persistent Cup-Length

Authors Marco Contessoto, Facundo Mémoli, Anastasios Stefanou, Ling Zhou

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Author Details

Marco Contessoto
  • Department of Mathematics, São Paulo State University - UNESP, Brazil
Facundo Mémoli
  • Department of Mathematics and Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, US
Anastasios Stefanou
  • Department of Mathematics and Computer Science, University of Bremen, Germany
Ling Zhou
  • Department of Mathematics, The Ohio State University, Columbus, OH, US

Funding

  • Contessoto, Marco: MC was supported by FAPESP through grants 2016/24707-4, 2017/ 25675-1 and 2019/22023-9.
  • Mémoli, Facundo: FM was partially supported by the NSF through grants RI-1901360, CCF-1740761, and CCF-1526513, and DMS-1723003.
  • Stefanou, Anastasios: AS was supported by NSF through grants CCF-1740761, DMS-1440386, RI-1901360, and the Dioscuri program initiated by the Max Planck Society, jointly managed with the National Science Centre (Poland), and mutually funded by the Polish Ministry of Science and Higher Education and the German Federal Ministry of Education and Research.
  • Zhou, Ling: LZ was partially supported by the NSF through grants RI-1901360, CCF-1740761, and CCF-1526513, and DMS-1723003.

Cite As Get BibTex

Marco Contessoto, Facundo Mémoli, Anastasios Stefanou, and Ling Zhou. Persistent Cup-Length. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.31

Abstract

Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser.
The cup product operation which is available at cohomology level gives rise to a graded ring structure that extends the usual vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the cup-length, which is useful for discriminating spaces.
In this paper, we lift the cup-length into the persistent cup-length function for the purpose of capturing ring-theoretic information about the evolution of the cohomology (ring) structure across a filtration. We show that the persistent cup-length function can be computed from a family of representative cocycles and devise a polynomial time algorithm for its computation. We furthermore show that this invariant is stable under suitable interleaving-type distances.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Computational geometry
  • Mathematics of computing → Topology
Keywords
  • cohomology
  • cup product
  • persistence
  • cup length
  • Gromov-Hausdorff distance

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