Universal realisators for homology classes

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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 ∈ Hn.(X,Z), there exists a finite-sheeted covering M̂n → M n and a continuous mapping f: M̂ n → X such that f*[M̂n] = 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 languageEnglish
Pages (from-to)1745-1772
Number of pages28
JournalGeometry and Topology
Volume17
Issue number3
DOIs
Publication statusPublished - 2013

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