Jacobians of $W^{1,p}$ homeomorphisms, case $p=[n/2]$

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 $x$.