Spectral sequences, exact couples and persistent homology of filtrations (Q2407570)

This paper considers the relation between the Leray spectral sequence of a filtration \(\mathcal F\) on a space \(X\) and the persistent homology of \(X\) associated to the same filtration. The authors show that there is a long exact sequence of groups linking the two objects. Using the notion of exact couples the main results show that \(E^{(r)}_{p,q}({\mathcal F})\) is equal to an expression in the persistent Betti numbers \(b^{s,t}_{n}({\mathcal F})\). Further the persistent Betti numbers can be expressed as a sum of the dimensions of suitable terms in the spectral sequence. Thus each concept carries the same information.