Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces

TL;DR

This paper constructs a specific manifold with trivial Lipschitz homotopy groups where Lipschitz maps are not dense in Sobolev spaces, and also establishes conditions for density of Lipschitz maps in Sobolev spaces based on Lipschitz connectivity and Nagata dimension.

Contribution

It provides a counterexample to the density of Lipschitz maps in Sobolev spaces and identifies conditions ensuring density based on Lipschitz connectivity and geometric properties.

Findings

Constructed a manifold with trivial Lipschitz homotopy groups but non-dense Lipschitz maps in Sobolev space.

Proved density of Lipschitz maps under Lipschitz (n-1)-connectivity and geometric conditions.

Established links between Lipschitz homotopy, Sobolev spaces, and geometric properties of metric spaces.

Abstract

We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip(S^n,Y) are not dense in the Sobolev space W^{1,n}(S^n,Y). On the other hand we show that if a metric space Y is Lipschitz (n-1)-connected, then Lipschitz mappings Lip(X,Y) are dense in N^{1,p}(X,Y) whenever the Nagata dimension of X is bounded by n and the space X supports the p-Poincare inequality.