Spectral Sequences, Exact Couples and Persistent Homology of filtrations

TL;DR

This paper explores the connection between classical spectral sequences and modern persistent homology groups in filtrations, establishing a long exact sequence linking them and deriving dimension formulas.

Contribution

It introduces a long exact sequence linking spectral sequences and persistent homology, providing formulas to relate their group dimensions using exact couples.

Findings

Established a long exact sequence connecting spectral sequences and persistent homology groups.

Derived formulas expressing the dimensions of groups in one object in terms of the other.

Provided a theoretical framework for understanding the relationship between classical algebraic topology and persistent homology.

Abstract

In this paper we study the relationship between a very classical algebraic object associated to a filtration of spaces, namely a spectral sequence introduced by Leray in the 1940's, 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.