Volume 17, issue 3 (2013)

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Universal realisators for homology classes

Alexander Gaifullin

Geometry & Topology 17 (2013) 1745–1772

We study oriented closed manifolds Mn possessing the following universal realisation of cycles (URC) property: For each topological space X and each homology class z Hn(X, ), there exists a finite-sheeted covering M̂n Mn and a continuous mapping f : M̂n X such that f[M̂n] = kz for a non-zero integer k. We find a wide class of examples of such manifolds Mn 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 N4, there exists a mapping of non-zero degree of a hyperbolic manifold M4 to N4. This was earlier conjectured by Kotschick and Löh.

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