Persistence of zero sets
Peter Franek, Marek Krčál · Homology Homotopy and Applications · 2017
We study robust properties of zero sets of continuous maps f : X → R n .Formally, we analyze the family Z 0 simultaneously, the pointed cohomotopy groups form a persistence module-a structure leading to persistence diagrams as in the case of persistent homology or well groups.Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams.Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C).