Bing meets Sobolev

TL;DR

This paper demonstrates the existence of wild involutions on the 3-sphere within specific Sobolev spaces, revealing new complexities in the structure of Sobolev mappings.

Contribution

It establishes the existence of wild involutions in Sobolev class W^{1,p} for 1 ≤ p < 2, a novel result in the study of Sobolev homeomorphisms.

Findings

Existence of wild involutions in W^{1,p} for 1 ≤ p < 2

Extension of known topological phenomena to Sobolev spaces

New insights into the structure of Sobolev mappings on spheres

Abstract

We show that, for each $1\le p < 2$, there exists a wild involution $\mathbb S^3\to \mathbb S^3$ in the Sobolev class $W^{1,p}(\mathbb S^3,\mathbb S^3)$.