Morse inequalities for the Koszul complex of multi-persistence

Abstract

In this paper, we derive inequalities bounding the number of critical cells in a filtered cell complex on the one hand, and the entries of the Betti tables of the multi-parameter persistence modules of such filtrations on the other hand. 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 the number of critical cells.Furthermore, we prove a sharp upper bound for the minimal number of critical cells, expressed again in terms of the entries of Betti tables.