The fiber of multiparameter persistent homology for simplicial complexes

Abstract

Vector-valued functions on finite simplicial complexes give rise to multiparameter sublevel set filtrations and corresponding persistent homology. We study the associated inverse problem for multiparameter persistent homology (MPH) of $n$-filters on a fixed finite simplicial complex. We endow both the space of filters and the moduli space of essentially finite persistence modules with stratifications, where strata are given by orbits of natural actions of order-isomorphisms of the unit $n$-cube. The MPH map is then shown to be equivariant and strongly stratified. Over each stratum in the image, the MPH map restricts to a trivial fiber bundle whose fiber is a polyhedral complex. We provide an upper bound on the dimension of the fibers in terms of multigraded Betti numbers, recovering as a special case the known one-parameter bound obtained by Leygonie and Tillmann (2022).