Fibers of monotone maps of finite distortion

We study topologically monotone surjective $W^{1,n}$-maps of finite distortion $f\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$, where $\mathrm{\Omega },{\mathrm{\Omega }}^{\mathrm{\prime }}$ are domains in ${\mathbb{R}}^n$, $n\ge 2$. If the outer distortion function $K_f \in L_{$\mathrm{l}\mathrm{o}\mathrm{c}$}^{p}(\Omega)$with$p \geq n-1$,thenanysuchmap$f$ is known to be homeomorphic, and hence the fibers $f^{-1}\{y\}$ are singletons. We show that as the exponent of integrability $p$ of the distortion function $K_f$ increases in the range $1\mathrm{/}(n-1)\le p<n-1$, then the fibers $f^{-1}\{y\}$ of $f$ start satisfying increasingly strong homological limitations. We also give a Sobolev realization of a topological example by Bing of a monotone $f \colon $\mathbb{R}$^3 \to $\mathbb{R}$^3$ with homologically nontrivial fibers, and show that this example has $K_f \in L^{1/2 - \varepsilon}_{$\mathrm{l}\mathrm{o}\mathrm{c}$}($\mathbb{R}$^3)$forall$\varepsilon > 0$.