Some lifting and approximation properties for maps in $W^{1,2}(\mathbb{B}^3;\mathbb{S}^2)$

TL;DR

This paper characterizes when $W^{1,2}$ maps from the 3-ball to the 2-sphere can be lifted via the Hopf fibration, and establishes a smooth approximation property that preserves a specific differential form constraint.

Contribution

It provides a characterization of $W^{1,2}$ maps that admit a lift through the Hopf fibration based on the exactness of a pullback form, and proves a related smooth approximation result.

Findings

Maps with exact pullback form can be lifted via the Hopf fibration.

A smooth approximation preserving the form constraint exists.

The characterization links topological and analytical properties of the maps.

Abstract

Smooth maps $u\colon\mathbb B^3\to\mathbb S^2$ can be lifted to $\hat u\colon\mathbb B^3\to\mathbb S^3$ using the Hopf fibration $h\colon \mathbb S^3\to\mathbb S^2$ via the factorization $u=h\circ\hat u$. In this note we characterize the $W^{1,2}$-maps which have this lifting property in terms of exactness of the pullback form $u^*\omega_{\mathbb S^2}$, and deduce a smooth approximation property preserving the constraint $u^*\omega_{\mathbb S^2}=d\eta$.