Small covers of graph-associahedra and realization of cycles

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 ${\hat{M}}^n$ 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 ${\hat{M}}^n$ of $M^n$ under a continuous mapping $f:{\hat{M}}^n\to X$. In 2007 the author gave a constructive proof of Thom's classical result 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 the previously known $URC$-manifolds.

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