Generic topological screening and approximation of Sobolev maps

TL;DR

This paper presents a topological framework for approximating Sobolev maps into manifolds, linking local extendability properties to the possibility of smooth approximation, and introduces new techniques for handling singularities.

Contribution

It develops a unified approach combining topological and quantitative methods to characterize when Sobolev maps can be approximated by smooth maps based on extendability properties.

Findings

Sobolev maps are approximable by smooth maps iff they are $(loor{kp}, e)$-extendable with $e = m$.

Approximation is possible with maps smooth outside structured singular sets when $e < m$.

The framework integrates topological concepts with quantitative estimates for Sobolev maps.

Abstract

This manuscript develops a framework for the strong approximation of Sobolev maps with values in compact manifolds, emphasizing the interplay between local and global topological properties. Building on topological concepts adapted to VMO maps, such as homotopy and the degree of continuous maps, it introduces and analyzes extendability properties, focusing on the notions of $\ell$-extendability and its generalization, $(\ell, e)$-extendability. We rely on Fuglede maps, providing a robust setting for handling compositions with Sobolev maps. Several constructions -- including opening, thickening, adaptive smoothing, and shrinking -- are carefully integrated into a unified approach that combines homotopical techniques with precise quantitative estimates. Our main results establish that a Sobolev map $u \in W^{k, p}$ defined on a compact manifold of dimension $m > kp$ can be approximated by smooth maps if and only if $u$ is $(\lfloor kp \rfloor, e)$-extendable with $e = m$. When $e < m$, the approximation can still be carried out using maps that are smooth except on structured singular sets of rank $m - e - 1$.