Finiteness of rank invariants of multidimensional persistent homology groups

TL;DR

This paper establishes a precise condition under which the rank invariants of multidimensional persistent homology groups are finite, providing insights into their behavior for spaces embedded in Euclidean space.

Contribution

It introduces a sharp sufficient condition for the finiteness of rank invariants in multidimensional persistent homology, applicable to spaces embeddable in R^n.

Findings

Finiteness condition for rank invariants established

Condition is sharp for spaces embeddable in R^n

Enhances understanding of multidimensional persistent homology

Abstract

Rank invariants are a parametrized version of Betti numbers of a space multi-filtered by a continuous vector-valued function. In this note we give a sufficient condition for their finiteness. This condition is sharp for spaces embeddable in R^n.