Sobolev mappings, degree, homotopy classes and rational homology spheres

In the 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 $S^4$.

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