Topology and Sobolev Spaces

Consider the Sobolev class W1, p(M, N) where M and N are compact manifolds. We present some sufficient conditions which guarantee that W1, p(M, N) is path-connected. We also discuss cases where W1, p(M, N) admits more than one component. There are still a number of open problems, especially concerning the values of p where a change in homotopy classes occurs.