Nontrivial cobordisms with geometrically finite hyperbolic structure
type: Item
instance of: scholarly article
title: Nontrivial cobordisms with geometrically finite hyperbolic structure
zbMATH Open document ID: 0627.57015
author: A. V. Tetenov
author: Boris N. Apanasov
published in: Journal of Differential Geometry
publication date: 1988-01-01T00:00:00Z
review text: In the paper the topology of geometrically finite hyperbolic manifolds with boundary is investigated by means of geometric methods of Kleinian groups in the n-sphere. In dimension 4 a new phenomenon is discovered: the authors construct 4-manifolds M, \(\partial M=N_ 0\cup N_ 1\), such that the cobordism \((M;N_ 0,N_ 1)\) is homologically trivial (i.e. \(H_ *(M,N_ i)=0\), \(i=0,1)\) but it is not h-cobordism. An obstruction here is the fact that the limit set \(L(G)\subset S^ 3=\partial H^ 4\) for the action of \(G=\pi _ 1(M)\) in the hyperbolic space \(H^ 4\) is a wildly embedded 2-dimensional sphere in \(S^ 3\) (even wildly in a dense subset of L(G)). Note, first, that analogous cobordisms are trivial in dimension 3 [see \textit{A. Marden}, Ann. Math., II. Ser. 99, 383-462 (1974; Zbl 0282.30014)] and, second, \(\pi _ 1(M)\) is isomorphic to the fundamental group of a closed hyperbolic 3-manifold and therefore its Whitehead group \(Wh(\pi _ 1(M))\) is trivial and also \(Wh_ 2(\pi _ 1(M))=0\), \~K\({}_ 0({\mathbb{Z}}\pi _ 1(M))=0\), \(K_{-m}({\mathbb{Z}}\pi _ 1(M))=0\) for \(m>0\), \(Wh_ m\pi _ 1(M)\otimes {\mathbb{Q}}=0\) for all m [cf. \textit{F. T. Farrell} and \textit{L. E. Jones}, Bull. Am. Math. Soc., New Ser. 14, 115-119 (1986; Zbl 0619.57016)].
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R75
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57R80
zbMATH DE Number: 4020015
zbMATH Keywords: hyperbolic manifolds with boundary
zbMATH Keywords: Kleinian groups
zbMATH Keywords: 4-manifolds
zbMATH Keywords: h- cobordism
zbMATH Keywords: limit set
zbMATH Keywords: fundamental group of a closed hyperbolic 3- manifold
zbMATH Keywords: Whitehead group
zbMATH Keywords: homologically trivial cobordism
zbMATH Keywords: higher Whitehead groups
Wikidata QID: Q115183502
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214442471
OpenAlex ID: W1508054851
DOI: 10.4310/JDG/1214442471
description: scientific article; zbMATH DE number 4020015