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

Smooth maps $u\text{\hspace{0.17em}⁣}:{\mathbb{B}}^3\to {\mathbb{S}}^2$ can be lifted to $\widehat{u}\text{\hspace{0.17em}⁣}:{\mathbb{B}}^3\to {\mathbb{S}}^3$ using the Hopf fibration $h\text{\hspace{0.17em}⁣}:{\mathbb{S}}^3\to {\mathbb{S}}^2$ via the factorization $u=h\circ \widehat{u}$. In this note we characterize the $W^{1,2}$-maps which have this lifting property in terms of exactness of the pullback form $u^{\ast }{\omega }_{{\mathbb{S}}^2}$, and deduce a smooth approximation property preserving the constraint $u^{\ast }{\omega }_{{\mathbb{S}}^2}=d\eta$.