Morse inequalities for the Koszul complex of multi-persistence

TL;DR

This paper introduces Morse inequalities for multi-parameter persistence modules, providing bounds on homological Morse numbers using spectral sequences and Betti tables, advancing topological data analysis techniques.

Contribution

It defines homological Morse numbers for filtered cell complexes and establishes sharp bounds relating these to Betti tables in multi-parameter persistence.

Findings

Derived strong and weak Morse inequalities for multi-persistence

Established sharp lower bounds for homological Morse numbers

Provided sharp upper bounds in terms of Betti tables

Abstract

In this paper, we define the homological Morse numbers of a filtered cell complex in terms of relative homology of nested filtration pieces, and derive inequalities relating these numbers to the Betti tables of the multi-parameter persistence modules of the considered filtration. Using the Mayer-Vietoris spectral sequence we first obtain strong and weak Morse inequalities involving the above quantities, and then we improve the weak inequalities achieving a sharp lower bound for homological Morse numbers. Furthermore, we prove a sharp upper bound for homological Morse numbers, expressed again in terms of the Betti tables.