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 perturbation 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. Communicated by Gunnar Carlsson AMS Subject Classification - 55N99 Keywords. Smooth mappings, transversality, fixed points, contours, homology, filtrations, zigzag modules, persistence, perturbations, stability.
Herbert Edelsbrunner, Dmitriy Morozov, Amit K. Pat