Persistent Leray's spectral sequence

TL;DR

This paper introduces a persistent version of Leray's spectral sequence to compute persistent cohomology from open covers and their intersections.

Contribution

It develops a novel spectral sequence framework that extends Leray's classical spectral sequence to the persistent cohomology setting.

Findings

Constructed a spectral sequence for persistent cohomology from open covers.

Enabled computation of persistent cohomology via intersections and pre-image coverings.

Provides a new tool for topological data analysis involving persistent invariants.

Abstract

In this work, we construct a persistent version of the well-known Leray spectral sequence. More precisely, we construct a spectral sequence that computes the persistent cohomology of a space from the persistent cohomology in each open set and its intersections with a covering that is the pre-image under a function of a covering of a known space.