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

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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)


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
  • cohomology
  • cup product
  • persistence
  • cup length
  • Gromov-Hausdorff distance


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  1. Henry Adams, Andrew Tausz, and Mikael Vejdemo-Johansson. javaplex: A research software package for persistent (co)homology. In Mathematical Software, ICMS 2014 - 4th International Congress, Proceedings, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), pages 129-136. Springer Verlag, 2014. 4th International Congress on Mathematical Software, ICMS 2014 ; Conference date: 05-08-2014 Through 09-08-2014. URL:
  2. Manuel Amann. Computational complexity of topological invariants. Proceedings of the Edinburgh Mathematical Society, 58(1):27-32, 2015. URL:
  3. HB Aubrey. Persistent cohomology operations. PhD thesis, Duke University, 2011. Google Scholar
  4. Ulrich Bauer. Ripser., 2016.
  5. Ulrich Bauer. Ripser: efficient computation of Vietoris-Rips persistence barcodes. Journal of Applied and Computational Topology, pages 1-33, 2021. URL:
  6. Francisco Belchí and Anastasios Stefanou. A-infinity persistent homology estimates detailed topology from point cloud datasets. Discrete & Computational Geometry, pages 1-24, 2021. URL:
  7. Andrew J Blumberg and Michael Lesnick. Universality of the homotopy interleaving distance. arXiv preprint, 2017. URL:
  8. Gunnar Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46(2):255-308, 2009. Google Scholar
  9. Gunnar Carlsson. Persistent homology and applied homotopy theory. In Handbook of Homotopy Theory, pages 297-329. Chapman and Hall/CRC, 2020. Google Scholar
  10. David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1):103-120, 2007. URL:
  11. Marco Contessoto, Facundo Mémoli, Anastasios Stefanou, and Ling Zhou. Persistent cup-length. arXiv preprint, 2021. URL:
  12. Luis Polanco Contreras and Jose Perea. Persistent cup product for quasi periodicity detection., 2021. Second Graduate Student Conference: Geometry and Topology meet Data Analysis and Machine Learning (GTDAML2021).
  13. Octavian Cornea, Gregory Lupton, John Oprea, Daniel Tanré, et al. Lusternik-Schnirelmann category. Number 103 in Mathematical Surveys and Monographs. American Mathematical Society, 2003. Google Scholar
  14. William Crawley-Boevey. Decomposition of pointwise finite-dimensional persistence modules. Journal of Algebra and its Applications, 14(05):1550066, 2015. URL:
  15. Vin de Silva, Dmitriy Morozov, and Mikael Vejdemo-Johansson. Dualities in persistent (co)homology. Inverse Problems, 27(12):124003, November 2011. URL:
  16. Vin De Silva, Dmitriy Morozov, and Mikael Vejdemo-Johansson. Persistent cohomology and circular coordinates. Discrete & Computational Geometry, 45(4):737-759, 2011. URL:
  17. Paweł Dłotko and Hubert Wagner. Simplification of complexes for persistent homology computations. Homology, Homotopy and Applications, 16(1):49-63, 2014. URL:
  18. Herbert Edelsbrunner and John Harer. Persistent homology-a survey. Contemporary mathematics, 453:257-282, 2008. Google Scholar
  19. Michael Farber. Topological complexity of motion planning. Discrete and Computational Geometry, 29(2):211-221, 2003. URL:
  20. Patrizio Frosini. A distance for similarity classes of submanifolds of a euclidean space. Bulletin of the Australian Mathematical Society, 42(3):407-415, 1990. URL:
  21. Patrizio Frosini. Measuring shapes by size functions. In Intelligent Robots and Computer Vision X: Algorithms and Techniques, volume 1607, pages 122-133. International Society for Optics and Photonics, 1992. URL:
  22. Rocío González Díaz and Pedro Real Jurado. Computation of cohomology operations of finite simplicial complexes. Homology, Homotopy and Applications (HHA), 5 (2), 83-93., 2003. Google Scholar
  23. Allen Hatcher. Algebraic topology. Cambridge Univ. Press, Cambridge, 2000. URL:
  24. Estanislao Herscovich. A higher homotopic extension of persistent (co)homology. Journal of Homotopy and Related Structures, 13(3):599-633, 2018. URL:
  25. Jonathan Huang. Cup products in computational topology, 2005. URL:
  26. Tomasz Kaczynski, Paweł Dłotko, and Marian Mrozek. Computing the cubical cohomology ring. Image-A: Applicable Mathematics in Image Engineering, 1 (3), 137-142, 2010. URL:
  27. Louis Kang, Boyan Xu, and Dmitriy Morozov. Evaluating state space discovery by persistent cohomology in the spatial representation system. Frontiers in Computational Neuroscience, 15, 2021. URL:
  28. Woojin Kim and Facundo Mémoli. Spatiotemporal persistent homology for dynamic metric spaces. Discrete & Computational Geometry, 66(3):831-875, 2021. URL:
  29. Umberto Lupo, Anibal M. Medina-Mardones, and Guillaume Tauzin. Persistence Steenrod modules. arXiv preprint, pages arXiv-1812, 2018. URL:
  30. Clément Maria, Jean-Daniel Boissonnat, Marc Glisse, and Mariette Yvinec. The Gudhi library: Simplicial complexes and persistent homology. In Hoon Hong and Chee Yap, editors, Mathematical Software - ICMS 2014, pages 167-174, Berlin, Heidelberg, 2014. Springer Berlin Heidelberg. URL:
  31. James R. Munkres. Elements of algebraic topology. Addison-Wesley, Menlo Park, CA, 1984. URL:
  32. Amit Patel. Generalized persistence diagrams. Journal of Applied and Computational Topology, 1(3):397-419, 2018. URL:
  33. Vanessa Robins. Towards computing homology from finite approximations. In Topology proceedings, volume 24, pages 503-532, 1999. Google Scholar
  34. Yuli B. Rudyak. On analytical applications of stable homotopy (the Arnold conjecture, critical points). Mathematische Zeitschrift, 230(4):659-672, 1999. URL:
  35. Yuli B Rudyak. On category weight and its applications. Topology, 38(1):37-55, 1999. URL:
  36. Parth Sarin. Cup length as a bound on topological complexity. arXiv preprint, 2017. URL:
  37. Felix Schmiedl. Computational aspects of the Gromov - Hausdorff distance and its application in non-rigid shape matching. Discrete Comput. Geom., 57(4):854-880, June 2017. URL:
  38. Steve Smale. On the topology of algorithms, I. Journal of Complexity, 3(2):81-89, 1987. URL:
  39. Andrew Yarmola. Persistence and computation of the cup product. Undergraduate honors thesis, Stanford University, 2010. Google Scholar
  40. Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. Discrete & Computational Geometry, 33(2):249-274, 2005. URL:
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