Persistence of zero sets

Peter Franek ; Marek Krčál

Homology, homotopy, and applications, Tome 19 (2017) no. 2, pp. 313-342.

Résumé

We study robust properties of zero sets of continuous maps $f : X \to \mathbb{R}^n$. Formally, we analyze the family $Z_{\lt r} (f) := \lbrace g^{-1} (0) : \lVert g - f \rVert \lt r \rbrace$ of all zero sets of all continuous maps $g$ closer to $f$ than $r$ in the max-norm. All of these sets are outside $A := \lbrace x : \lvert f(x) \rvert \geqslant r \rbrace$ and we claim that $Z_{\lt r} (f)$ is fully determined by $A$ and an element of a certain cohomotopy group which (by a recent result) is computable whenever the dimension of $X$ is at most $2n - 3$. By considering all $r \gt 0$ 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).

Détail

DOI : 10.4310/HHA.2017.v19.n2.a16

Classification : 55-04, 65G20, 68Q25
Keywords: system of equations, computational homotopy theory, cohomotopy group