Nontrivial cobordisms with geometrically finite hyperbolic structure
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)].
fundamental group of a closed hyperbolic 3- manifold
scientific article; zbMATH DE number 4020015