Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces
Piotr Hajlasz and Armin Schikorra
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Submission date: 16. Aug. 2013
published in: Annales Academiae Scientiarum Fennicae / Mathematica, 39 (2014) 2, p. 593-604
DOI number (of the published article): 10.5186/aasfm.2014.3932
MSC-Numbers: 46E35, 55Q70
We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip(Sˆn,Y) are not dense in the Sobolev space Wˆ1,n(Sˆn,Y). On the other hand we show that if a metric space Y is Lipschitz (n-1)-connected, then Lipschitz mappings Lip(X,Y) are dense in Nˆ1,p(X,Y) whenever the Nagata dimension of X is bounded by n and the space X supports the p-Poincare inequality.