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Dualities in Multiparameter Persistence

MCML Authors

Ulrich Bauer

Prof. Dr.

Principal Investigator

Applied Topology and Geometry

Abstract

In the theory of persistent homology, a well known duality relates the barcodes of the absolute homology and relative cohomology of a one-parameter simplicial filtration. Motivated by the problem of computing free presentations of the (co)homology of multiparameter Rips filtrations, we give a multiparameter generalization of this duality. Considering two duality functors on multiparameter persistence modules, the pointwise dual (−)∗ and the global dual (−)†, we show that Hq(C)∗≅HN+q(C†) for chain complexes C of free N-parameter persistence modules with acyclic colimit. We give an elementary and accessible proof based on a long exact sequence argument, and also give an alternate proof that casts the result as a special case of multigraded Grothendieck local duality. As a corollary, we recover a simple correspondence between minimal free resolutions of a persistence module M and those of its pointwise dual M∗, a result previously obtained by Miller, 2000. These results form the foundation of a state-of-the-art algorithm for computing free resolutions of the homology of Vietoris--Rips bifiltrations, described in a forthcoming paper.

misc BLL26