The homological singularities of maps in trace spaces between manifolds

We deal with mappings defined between Riemannian manifolds that belong to a trace space of Sobolev functions. The homological singularities of any such map are represented by a current defined in terms of the boundary of its graph. Under suitable topological assumptions on the domain and target manifolds, we show that the non triviality of the singular current is the only obstruction to the strong density of smooth maps. Moreover, we obtain an upper bound for the minimal integral connection of the singular current that depends on the fractional norm of the mapping.