Persistence of Zero Sets

TL;DR

This paper introduces a cohomotopy-based framework to analyze the persistent properties of zero sets of continuous maps, offering improved descriptive power and computability over existing methods.

Contribution

It develops a novel cohomotopy group approach to characterize zero set persistence, enhancing both theoretical understanding and practical computability.

Findings

Cohomotopy groups fully determine zero set properties within certain dimensions.

Persistence modules of cohomotopy groups lead to new, more powerful diagrams.

The approach is optimal when considering gradients of zero sets.

Abstract

We study robust properties of zero sets of continuous maps $f:X\to\mathbb{R}^n$. Formally, we analyze the family $Z_r(f)=\{g^{-1}(0):\,\,\|g-f\|<r\}$ of all zero sets of all continuous maps $g$ closer to $f$ than $r$ in the max-norm. The fundamental geometric property of $Z_r(f)$ is that all its zero sets lie outside of $A:=\{x:\,|f(x)|\ge r\}$. We claim that once the space $A$ is fixed, $Z_r(f)$ is \emph{fully} determined by an element of a so-called cohomotopy group which---by a recent result---is computable whenever the dimension of $X$ is at most $2n-3$. More explicitly, the element is a homotopy class of a map from $A$ or $X/A$ into a sphere. By considering all $r>0$ simultaneously, the pointed cohomotopy groups form a persistence module---a structure leading to the persistence diagrams as in the case of \emph{persistent homology} or \emph{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).