A Continuous Theory of Persistence for Mappings Between Manifolds

TL;DR

This paper develops a continuous persistence theory for mappings between compact manifolds using sheaf theory, ensuring stability under homotopic perturbations, extending persistence concepts to a broader topological context.

Contribution

It introduces a sheaf-based persistence framework for manifold mappings that is stable to homotopic perturbations, expanding the scope of topological data analysis.

Findings

Sheaf construction provides a stable persistence theory for manifold mappings.

The theory applies to orientable manifolds with integer coefficients.

Stability results are analogous to bottleneck stability in persistence.

Abstract

Using sheaf theory, I introduce a continuous theory of persistence for mappings between compact manifolds. In the case both manifolds are orientable, the theory holds for integer coefficients. The sheaf introduced here is stable to homotopic perturbations of the mapping. This stability result has a flavor similar to that of bottleneck stability in persistence.