The restriction of $f$ to the boundary has degree zero. It is true also in higher dimensions.
Theorem. If $f\in W^{1,n}(B^n,S^{n-1})$ and $f|_{\partial B^n}\in C^0$, then $f|_{\partial B^n}:S^{n-1}\to S^{n-1}$ has degree zero.
Here the restriction to the boundary $f|_{\partial B^n}$ is defined as a trace of a $W^{1,n}$ function.
When I find time I will sketch a proof. I hope to do it tomorrow.