We define a persistent cohomology invariant called persistent cup-length which is able to extract non trivial information about the evolution of the cohomology ring structure across a filtration. We also devise algorithms for the computation of this invariantand we furthermore show that the persistent cup-length is 2-Lipschitz continuous with respect to the homotopy interleaving and Gromov-Hausdorff distances .
Cohomological ideas have recently been injected into persistent homology and have been utilized for both enriching and accelerating the calculation of persistence diagrams. For instance, the software Ripser fundamentally exploits the computational advantages offered by cohomological ideas. The cup product operation which is available at cohomology level gives rise to a graded ring structure which extends the natural 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 efficient at discriminating spaces. In this paper, we lift the cup-length into the Persistent Cup-Length invariant for the purpose of extracting non-trivial information about the evolution of the cohomology ring structure across a filtration. We show that the Persistent Cup-Length can be computed from a family of representative cocycles and devise a polynomial time algorithm for the computation of the Persistent Cup-Length invariant. We furthermore show that this invariant is stable under suitable interleaving-type distances. Along the way, we identify an invariant which we call the Cup-Length Diagram, which is stronger than persistent cup-length but can still be computed efficiently .
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We define a persistent cohomology invariant called persistent cup-length which is
Cohomological ideas have recently been injected into persistent homology and have been utilized for both enriching and accelerating the calculation of persistence diagrams. For instance, the software Ripser fundamentally exploits the computational advantages offered by cohomological ideas. The cup product operation which is available at cohomology level gives rise to a graded ring structure which extends the natural vector space structure and is therefore
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non trivial information
and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the Cup-Length, which is efficient at discriminating spaces. In this paper, we lift the cup-length into the Persistent Cup-Length invariant for the purpose of extracting non-trivial information
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also devise algorithms
show that the Persistent Cup-Length can be computed from a family of representative cocycles and devise a polynomial time algorithm
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this invariantand we
the Persistent Cup-Length invariant. We
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the
this invariant is stable under suitable interleaving-type distances. Along the way, we identify an invariant which we call the Cup-Length Diagram, which is stronger than
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is 2-Lipschitz continuous with respect to the homotopy interleaving and Gromov-Hausdorff distances
but can still be computed efficiently