# Spectral Sequences, Exact Couples and Persistent Homology of filtrations

arXiv: https://arxiv.org/abs/1308.0801 · HTML: https://mlsanity.com/paper/1308.0801/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Aug 2013.

## Built on by (most cited first, 1 of 1)

- [Topological data analysis and clustering](https://mlsanity.com/paper/2201.09054/) (2022)

## Abstract (arXiv)

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.