W^{1,2} maps u from the 3-ball to S^2 admit a Hopf lift to S^3 precisely when u^*ω_{S^2} is exact, and this condition permits smooth approximations preserving the exactness.
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Some lifting and approximation properties for maps in $W^{1,2}(\mathbb{B}^3;\mathbb{S}^2)$
W^{1,2} maps u from the 3-ball to S^2 admit a Hopf lift to S^3 precisely when u^*ω_{S^2} is exact, and this condition permits smooth approximations preserving the exactness.