Quantifying Transversality by Measuring the Robustness of Intersections

TL;DR

This paper introduces a measure of transversality using persistent homology, quantifying the stability of intersections and fixed points under perturbations, with proven stability properties.

Contribution

It extends the concept of transversality to a quantitative measure via persistent homology and proves its stability under perturbations.

Findings

Robustness of intersections can be quantified using persistent homology.

Robustness measure is stable under perturbations.

Application to fixed points and contours demonstrates practical utility.

Abstract

By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbations in this space necessary to kill it, and prove that robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings.