Generic topological screening and approximation of Sobolev maps

TL;DR

$The$ $paper$ develops a unified framework for strong approximation of manifold-valued Sobolev maps by smooth maps on domains with $m>kp$, introducing the notions of $\lfloor kp\rfloor$-$\ell$-extendability and $(\ell,e)$-extendability to capture topological obstructions. Central to the approach is the generic composition principle via Fuglede maps: given a Sobolev map $u$ and a suitable summable detector $w$, compositions $u\circ\gamma$ with Lipschitz, lower-dimensional maps $\gamma$ preserve Sobolev or $VMO$ regularity, enabling a topological screening of obstructions. The work then develops $\mathrm{VMO}^\ell$-type detectors and $\ell$-detectors to extend topological notions to $VMO$ settings, proving density results for smooth or bounded uniformly continuous maps and establishing a robust link between $VMO$-based topology and classical $C^0$ topology. Finally, an array of tools—opening, thickening, adaptive smoothing, and shrinking—are adapted to handle higher-order Sobolev spaces and extended to manifolds, yielding a complete density/d approximation theory that couples local geometric/topological data with global extendability criteria.}$

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 $m - e - 1$.

Figures (11)

• Figure 3.1: $u \circ \Phi^{\mathrm{op}}$ is constant in the green region
• Figure 4.1: The polytope of Example \ref{['exampleUglyComplex']}
• Figure 11.1: Opening around vertices (left) and then edges (right)
• Figure 11.2: Thickening in a disc and in a square
• Figure 11.3: Values of $u \circ \Phi^{\mathrm{sh}}$ on the smaller disk (gray) are computed using $u$ mostly from the larger yellow region

Theorems & Definitions (241)

• Example 2.1
• Theorem 2.2
• Theorem 2.3
• Example 2.4
• Example 2.5
• Example 2.6
• Example 2.7
• Example 2.8: Variant of the previous example
• Proposition 2.9
• Theorem 2.10