Small covers of graph-associahedra and realization of cycles

Abstract

Abstract. 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 non-zero degree mapping of a finite-fold covering 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 M n of M n under a continuous mapping f : M n → 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.

Polyhedral products and commutator subgroups of right-angled Artin and Coxeter groups

“…These groups have attracted much attention recently in geometric group theory and lowdimensional topology. The manifolds R K corresponding to (the dual complexes of) higher-dimensional permutahedra and graph-associahedra also feature as the universal realisators in the works of Gaifullin [Ga1], [Ga2] on the problem of realisation of homology classes by manifolds.…”