Small covers of graph-associahedra and realization of cycles

Abstract

An oriented connected closed manifold is called a - manifold if for any oriented connected closed manifold of the same dimension there exists a nonzero-degree mapping of a finite-fold covering of onto . This condition is equivalent to the following: for any -dimensional integral homology class of any topological space , a multiple of it can be realized as the image of the fundamental class of a finite-fold covering of under a continuous mapping . 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 -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 -manifolds. In particular, we obtain that the two-fold covering of the small cover of the usual Stasheff associahedron is a -manifold. In dimensions 4 and higher, this manifold is simpler than all the previously known -manifolds. Bibliography: 39 titles.