Sobolev mappings, degree, homotopy classes and rational homology spheres

摘要

In this paper we investigate the degree and the homotopy theory of Orlicz-Sobolev mappings W ~(1,P) (M,N) between manifolds, where the Young function P satisfies a divergence condition and forms a slightly larger space than W ~(1,n), n = dim M. In particular, we prove that if M and N are compact oriented manifolds without boundary and dim M = dim N = n, then the degree is well defined in W ~(1,P) (M,N) if and only if the universal cover of N is not a rational homology sphere, and in the case n = 4, if and only if N is not homeomorphic to S ~4.