Small Covers of Graph-associahedra and Realization of Cycles

An oriented connected closed manifold Mn is called a URC-manifold if for any oriented connected closed manifold Nn of the same dimension there exists a non-zero degree mapping of a finite-fold covering M̂n of Mn onto Nn. 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 M̂n of Mn under a continuous mapping f : M̂n → 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.