Persistent Local Systems

TL;DR

This paper introduces sheaf-theoretic methods to establish persistent lower bounds on the homology of fibers of maps to manifolds, generalizing persistent homology to higher dimensions with stability under perturbations.

Contribution

It develops a new sheaf-theoretic framework for persistent homology, providing stable lower bounds for fiber homology in higher dimensions.

Findings

Lower bounds for fiber homology are stable over open sets.

Persistence of these bounds under map perturbations.

Generalization of persistent homology concepts to higher-dimensional settings.

Abstract

In this paper, we give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds persist over whole open sets of the manifold, and that they are stable under perturbations of the map. This generalizes certain ideas of persistent homology to higher dimensions.