  An oriented connected closed manifold $M^n$ is called a URC-manifold if for
any oriented connected closed manifold $N^n$ of the same dimension there exists
a nonzero degree mapping of a finite-fold covering $\widehat{M}^n$ of $M^n$
onto $N^n$. This condition is equivalent to the following: For any
$n$-dimensional integral homology class of any topological space $X$, a
multiple of it can be realized as the image of the fundamental class of a
finite-fold covering $\widehat{M}^n$ of $M^n$ under a continuous mapping
$f\colon \widehat{M}^n\to X$. In 2007 the author gave a constructive proof of
the classical result by Thom that a multiple of any integral homology class can
be realized as an image of the fundamental class of an oriented smooth
manifold. This construction yields the existence of URC-manifolds of all
dimensions. For an important class of manifolds, the so-called small covers of
graph-associahedra corresponding to connected graphs, we prove that either they
or their two-fold orientation coverings are URC-manifolds. In particular, we
obtain that the two-fold covering of the small cover of the usual Stasheff
associahedron is a URC-manifold. In dimensions 4 and higher, this manifold is
simpler than all previously known URC-manifolds.
