Universal realisators for homology classes

Gaifullin, Alexander¹

¹ 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

Geometry & topology, Tome 17 (2013) no. 3, pp. 1745-1772.

Résumé

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

Détail

DOI : 10.2140/gt.2013.17.1745

Classification : 57N65, 53C23, 52B70, 20F55
Keywords: realisation of cycles, hyperbolic manifold, simple polytope, small cover, permutahedron, Coxeter group, negative curvature

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