Topology and Sobolev Space

Given two manifolds M, N, the homotopy classes are the connected components of the space C^0(M, N).  When C^0 is replaced by the Sobolev space W^{1,p},  with p greater than or equal to 1 and less than infinity, one may still consider connected components of W^{1,p} (M, N).  If  p is greater than or equal to dim M they have the same structure as the components of C^0.  However, when p is less than dim M, many surprising new phenomena occur.