Memoirs of the American Mathematical Society 2008; 72 pp; softcover Volume: 192 ISBN10: 0821840797 ISBN13: 9780821840795 List Price: US$65 Individual Members: US$39 Institutional Members: US$52 Order Code: MEMO/192/899
 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 socalled weblike 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 nonzero \(\ell\)cohomology. The necessity of topological constraints on the target manifold is a new phenomenon in the theory of Jacobians. Table of Contents  Introduction
 Preliminaries concerning manifolds
 Examples
 Some classes of functions
 Smooth approximation
 \({\mathcal L}^1\)Estimates of the Jacobian
 \({\mathcal H}^1\)Estimates
 Degree theory
 Mappings of finite distortion
 Bibliography
