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May 6, 2018 at 18:41 comment added Piotr Hajlasz Since you were interested in the question you might look at my answer. I do not wee right away why the fact that $W^{1,2}(D^2,S^1)$ is path connected implies that the degree of the map on the boundary has to be zero.
Feb 21, 2017 at 9:15 history answered Denis Serre CC BY-SA 3.0