#### Volume 17, issue 3 (2013)

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Universal realisators for homology classes

### Alexander Gaifullin

Geometry & Topology 17 (2013) 1745–1772
##### 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 $z\in {H}_{n}\left(X,ℤ\right)$, there exists a finite-sheeted covering ${\stackrel{̂}{M}}^{n}\to {M}^{n}$ and a continuous mapping $f:\phantom{\rule{0.3em}{0ex}}{\stackrel{̂}{M}}^{n}\to X$ such that ${f}_{\ast }\left[{\stackrel{̂}{M}}^{n}\right]=kz$ 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.

##### Keywords
realisation of cycles, hyperbolic manifold, simple polytope, small cover, permutahedron, Coxeter group, negative curvature
##### Mathematical Subject Classification 2010
Primary: 57N65
Secondary: 53C23, 52B70, 20F55
##### Publication
Received: 7 April 2012
Accepted: 4 March 2013
Published: 30 June 2013
Proposed: David Gabai
Seconded: Benson Farb, Ronald Stern
##### Authors
 Alexander Gaifullin Department of Geometry and Topology Steklov Mathematical Institute 8 Gubkina Str Moscow 119991 Russia Lomonosov Moscow State University Leninskie Gory Moscow 119991 Russia Institute for Information Transmission Problems (Kharkevich Institute) 19 Bolshoy Karetny per Moscow 127994 Russia http://www.iitp.ru/en/users/919.htm