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In the paper we investigate the degree and the homotopy theory ofOrlicz-Sobolev mappings $W^{1,P}(M,N)$ between manifolds, where the Youngfunction $P$ satisfies a divergence condition and forms a slightly larger spacethan $W^{1,n}$, $n=\dim M$. In particular, we prove that if $M$ and $N$ arecompact oriented manifolds without boundary and $\dim M=\dim N=n$, then thedegree 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 $S^4$.

Sobolev mappings, degree, homotopy classes and rational homology spheres