## Abstract

An oriented connected closed manifold M^{n} is called a URCmanifold if for any oriented connected closed manifold N^{n} of the same dimension there exists a nonzero-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 cMn of Mn under a continuous mapping f : M^{n} → 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. Bibliography: 39 titles.

Original language | English |
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Pages (from-to) | 1537-1561 |

Number of pages | 25 |

Journal | Sbornik Mathematics |

Volume | 207 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2016 |

## Keywords

- Domination relation
- Graph-associahedron
- Realization of cycles
- Small cover
- URC-manifold