Title: Generic topological screening and approximation of Sobolev maps Show Chinese title

Title: Sobolev映射的通用拓扑筛选与近似

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 $m - e - 1$. Show Chinese abstract

Abstract: 本文档构建了一个框架，用于紧致流形值的索博列夫映射的强逼近，重点强调局部和全局拓扑性质之间的相互作用。 基于针对可积函数空间 (VMO) 映射调整的拓扑概念（如连续映射的同伦和度），它引入并分析了延拓性质，重点关注了$\ell$-延拓性和其推广，即$(\ell, e)$-延拓性。 我们依赖于富勒德映射，提供了一种处理与索博列夫映射复合问题的强大框架。 多个构造——包括开张、增厚、自适应平滑化和收缩——被精心整合到一个统一的方法中，该方法结合了同伦技术与精确的数量估计。 我们的主要结果表明，定义在维度为$m > kp$的紧致流形上的索博列夫映射$u \in W^{k, p}$可以被光滑映射逼近当且仅当$u$是$(\lfloor kp \rfloor, e)$-延拓，并满足条件$e = m$。 当$e < m$时，逼近仍可使用除了在秩为$m - e - 1$的结构奇点集上不光滑的映射来进行。