Quantifying Transversality by Measuring the Robustness of Intersections
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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 then we prove that the 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.
KeywordsSmooth mappings Transversality Fixed points Contours Homology Filtrations Zigzag modules Persistence Perturbations Stability
Mathematics Subject Classification (2000)55N99
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