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 is an element of H-n(X, Z), there exists a finite-sheeted covering (M) over cap (n) -> M-n and a continuous mapping f: (M) over cap (n) -> X such that f(*)[(M) over cap (n)] = 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 Loh.