Persistent Cup-Length

Ling Zhou, Facundo Memoli, Anastasios Stefanou, Marco Contessoto

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.