  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 $S^4$.
