## Abstract

We study oriented closed manifolds M ^{n} possessing the following Universal Realisation of Cycles (URC) Property: For each topological space X and each homology class z ∈ H_{n}.(X,Z), there exists a finite-sheeted covering M̂^{n} → M ^{n} and a continuous mapping f: M̂ ^{n} → X such that f_{*}[M̂^{n}] = kz for a non-zero integer k. We find a wide class of examples of such manifolds M ^{n} among so-called small covers of simple polytopes. In particular, we find 4-dimensional hyperbolic manifolds possessing the URC property. As a consequence, we obtain that for each 4-dimensional oriented closed manifold N ^{4}, there exists a mapping of non-zero degree of a hyperbolic manifold M ^{4} to N ^{4}. This was earlier conjectured by Kotschick and Löh.

Original language | English |
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Pages (from-to) | 1745-1772 |

Number of pages | 28 |

Journal | Geometry and Topology |

Volume | 17 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 |