Fractional Sobolev spaces and topology

Consider the Sobolev class Ws,p(M,N)Ws,p(M,N) where MM and NN are compact manifolds, and p≥1,s∈(0,1+1/p)p≥1,s∈(0,1+1/p). We present a necessary and sufficient condition for two maps uu and vv in Ws,p(M,N)Ws,p(M,N) to be continuously connected in Ws,p(M,N)Ws,p(M,N). We also discuss the problem of connecting a map u∈Ws,p(M,N)u∈Ws,p(M,N) to a smooth map f∈C∞(M,N)f∈C∞(M,N).