Leray Spectral Sequence for Simplicial Maps

TL;DR

This paper explores the Leray spectral sequence for simplicial maps, detailing its relation to Leray cosheaves and applying it to level set persistent homology and Reeb spaces.

Contribution

It provides a comprehensive exposition linking the Leray spectral sequence to cosheaves for simplicial maps and demonstrates applications in topological data analysis.

Findings

Detailed relation between Leray spectral sequence and cosheaves.

Applications to level set persistent homology.

Insights into Reeb spaces.

Abstract

The Leray spectral sequence of a map $f$ computes the homology of the domain of $f$ from the fibers of $f$. In this expository paper, we relate in full detail the Leray spectral sequence associated to a simplicial map $f$ to the Leray cosheaves of $f$. We then give applications to level set persistent homology and Reeb spaces.