The authors study Sobolev classes of weakly differentiable mappings \(f:{\mathbb X}\rightarrow {\mathbb Y}\) between compact Riemannian manifolds without boundary. These mappings need not be continuous. They actually possess less regularity than the mappings in \({\mathcal W}^{1,n}({\mathbb X}\, ,\, {\mathbb Y})\,\), \(n=\mbox{dim}\, {\mathbb X}\). The central themes being discussed are:

• smooth approximation of those mappings
• integrability of the Jacobian determinant

The approximation problem in the category of Sobolev spaces between manifolds \({\mathcal W}^{1,p}({\mathbb X}\, ,\, {\mathbb Y})\), \(1\leqslant p \leqslant n\), has been recently settled. However, the point of the results is that the authors make no topological restrictions on manifolds \({\mathbb X}\) and \({\mathbb Y}\). They characterize, essentially all, classes of weakly differentiable mappings which satisfy the approximation property. The novelty of their approach is that they were able to detect tiny sets on which the mappings are continuous. These sets give rise to the so-called web-like structure of \({\mathbb X}\) associated with the given mapping \(f: {\mathbb X}\rightarrow {\mathbb Y}\).

The integrability theory of Jacobians in a manifold setting is really different than one might a priori expect based on the results in the Euclidean space. To the authors' surprise, the case when the target manifold \({\mathbb Y}\) admits only trivial cohomology groups \(H^\ell ({\mathbb Y})\), \(1\leqslant \ell <n= \mbox{dim}\, {\mathbb Y}\), like \(n\)-sphere, is more difficult than the nontrivial case in which \({\mathbb Y}\) has at least one non-zero \(\ell\)-cohomology. The necessity of topological constraints on the target manifold is a new phenomenon in the theory of Jacobians.