Persistence of zero sets (Q1689742)

Given a vector valued continuous map \(f:X \to \mathbb{R}^n\) it is natural to be interested in its zero set and in the topological properties of it. Often, however, we only have access to an \(r\) \textit{perturbation} \(g\) of \(f\) (\(r\) a positive real number), i.e. a function such that \(\| f-g \|_\infty <r\). The goal of this paper is the study of the set \(Z_{<r}(f)\) of the zero sets of such perturbations. The paper completes preceding work [\textit{P. Franek} and \textit{M. Krčál}, J. ACM 62, No. 4, Article No. 26, 19 p. (2015; \url{doi:10.1145/2751524})] on the algorithmic decidability of such sets. A related problem had been studied, with varying \(r\), through \textit{well modules} and \textit{zigzag persistence} in [\textit{H. Edelsbrunner} et al., Found. Comput. Math. 11, No. 3, 345--361 (2011; Zbl 1232.55012)]. Surprisingly enough, \(Z_{<r}\) is determined by the set \(A_r\) of points \(x\) on which \(f(x)\geq r\) and by the homotopy class of the normalizing function \(A \to \mathbb{S}^{n-1}\) (Theorem A). On the filtration given by the sets \(A_r\) the authors build a \textit{cohomotopy persistence module} of whose computation they prove polynomial complexity (Theorem B). The research then specializes to manifolds \(X\) and to maps \(f:X \to \mathbb{R}^n\) for which 0 is a regular value. Theorem C characterizes the set, analogous to \(Z_{<r}\) for \textit{regular} \(r\)-perturbations, in terms of cohomotopy (sub)groups when the dimension of the manifold is \(\leq 2n-3\). These are the main results, but the paper has a much wider horizon. All arguments are finely detailed and well commented.