Strong density results for manifold valued fractional Sobolev maps

We consider fractional Sobolev classes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:math> of maps defined in high dimensional domains and with values into compact smooth manifolds. The problem of strong density of smooth maps for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>s</mml:mi> </mml:math> lower than one is discussed. An equivalent energy convergence defined through extensions in suitable weighted Sobolev spaces is exploited to obtain a new proof of the density of maps with “small” singular set. Moreover, a homotopy-type property is analyzed, yielding to a characterization of approximable maps through topological arguments. We then focus on maps taking values into high dimensional spheres, where homological tools allow to describe the singular set. For suitable values of the product <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> , in fact, strong density of smooth maps is characterized by the triviality of the current of the singularities.