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 $f \in C(\mathbb{R}^n, M) \cap W^{1,n}_{\mathrm{loc}}(\mathbb{R}^n, M)$ satisfying $\lvert Df(x) \rvert^n \le K J_f(x) + \operatorname{dist}^n(f(x), f(x_0)) \Sigma(x)$ a.e. in $\mathbb{R}^n$ with $K \ge 1$, $x_0 \in \mathbb{R}^n$, and $\Sigma \in L^1(\mathbb{R}^n) \cap L^{1+\varepsilon}_{\mathrm{loc}}(\mathbb{R}^n)$ for some $\varepsilon > 0$, then the real singular cohomology ring $H^*(M; \mathbb{R})$ of $M$ embeds into the exterior algebra $\wedge^* \mathbb{R}^n$ 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.