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

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 $u^*\omega_{\mathbb S^2}=d\eta$.