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Geom. Topol., 2013, Volume 17, Issue 3, Pages 1745–1772 (Mi gt3)  

Universal realisators for homology classes

A. A. Gaifullinabc

a Institute for Information Transmission Problems (Kharkevich Institute), 19 Bolshoy Karetny per, Moscow 127994, Russia
b Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia
c Department of Geometry and Topology, Steklov Mathematical Institute, 8 Gubkina Str, Moscow 119991, Russia

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 $z\in H_n(X,\mathbb{Z})$, there exists a finite-sheeted covering $\widehat{M}^n\to M^n$ and a continuous mapping $f:\widehat{M}^n\to X$ such that $f_*[\widehat{M}^n]=kz$ 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.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00694
12-01-31444
Ministry of Education and Science of the Russian Federation MD-4458.2012.1
2010-220-01-077
Dynasty Foundation
The work was partially supported by RFBR (projects 11-01-00694 and 12-01-31444), by a grant of the President of the Russian Federation (project MD-4458.2012.1), by a grant of the Government of the Russian Federation (project 2010-220-01-077), and by a grant from Dmitry Zimin's "Dynasty" foundation.


DOI: https://doi.org/10.2140/gt.2013.17.1745


Bibliographic databases:

UDC: 57N65
MSC: 53C23, 52B70, 20F55
Received: 07.04.2012
Accepted:04.03.2013
Language:

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