Quasiregular values and cohomology

We prove that the recently shown cohomological obstruction for quasiregular ellipticity has a generalization in the theory of quasiregular values. More specifically, if $M$ is a closed, connected, and oriented Riemannian $n$-manifold, and there exists a map $f\in C({\mathbb{R}}^n,M)\cap W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,n}({\mathbb{R}}^n,M)$ satisfying $\mid Df(x){\mid }^n\le K{J}_f(x)+{\mathrm{dist}}^n(f(x),f(x_0))\mathrm{\Sigma }(x)$ a.e. in ${\mathbb{R}}^n$ with $K\ge 1$, $x_0\in {\mathbb{R}}^n$, and $\mathrm{\Sigma }\in L^1({\mathbb{R}}^n)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1+\epsilon }({\mathbb{R}}^n)$ for some $\epsilon >0$, then the real singular cohomology ring $H^{\ast }(M;\mathbb{R})$ of $M$ embeds into the exterior algebra ${\wedge }^{\ast }{\mathbb{R}}^n$ in a graded manner. We also show a partial version of our result for $M$ with dimension greater than $n$, by using a class of maps that combines properties of quasiregular values and quasiregular curves.