  We investigate a known problem whether a Sobolev homeomorphism between
domains in $\mathbb{R}^n$ can change sign of the Jacobian. The only case that
remains open is when $f\in W^{1,[n/2]}$, $n\geq 4$. We prove that if $n\geq 4$,
and a sense-preserving homeomorphism $f$ satisfies $f\in W^{1,[n/2]}$,
$f^{-1}\in W^{1,n-[n/2]-1}$ and either $f$ is H\"older continuous on almost all
spheres of dimension $[n/2]$, or $f^{-1}$ is H\"older continuous on almost all
spheres of dimensions $n-[n/2]-1$, then the Jacobian of $f$ is non-negative,
$J_f\geq 0$, almost everywhere. This result is a consequence of a more general
result proved in the paper. Here $[x]$ stands for the greatest integer less
than or equal to $x$.
