Universal realisators for homology classes

We study oriented closed manifolds $M^n$ possessing the following universal realisation of cycles (URC) property: For each topological space $X$ and each homology class $z\in H_n(X,\mathbb{Z})$, there exists a finite-sheeted covering $\widehat{M}^n\to M^n$ and a continuous mapping $f:\widehat{M}^n\to X$ such that $f_*\left[\widehat{M}^n\right]=kz$ for a non-zero integer $k$. We find a wide class of examples of such manifolds $M^n$ among so-called small covers of simple polytopes. In particular, we find $4$–dimensional hyperbolic manifolds possessing the URC property. As a consequence, we obtain that for each $4$–dimensional oriented closed manifold $N^4$, there exists a mapping of non-zero degree of a hyperbolic manifold $M^4$ to $N^4$. This was earlier conjectured by Kotschick and Löh.