Spectral sequences, exact couples and persistent homology of filtrations

Abstract

In this paper we study the relationship between a very classical algebraic object associated to a filtration of topological spaces, namely a spectral sequence introduced by Leray in the 1940s, and a more recently invented object that has found many applications—namely, its persistent homology groups. We show the existence of a long exact sequence of groups linking these two objects and using it derive formulas expressing the dimensions of each individual groups of one object in terms of the dimensions of the groups in the other object. The main tool used to mediate between these objects is the notion of exact couples first introduced by Massey in 1952.