The distributional k-dimensional Jacobian of a Sobolev map u which takes values in the (k-1)-dimensional sphere can be viewed as a rectifiable current of codimension k located on (a part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator. We show that any boundary M of codimension k can be realized as Jacobian of a Sobolev map; and in case M is polyhedral, the map we construct is smooth outside M plus an additional polyhedral set of lower dimension
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