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**Homology cobordisms, link concordances, and hyperbolic 3-manifolds.**
*(English)*
Zbl 0532.57006

Two closed, oriented 3-manifolds \(M^ 3_ 0\) and \(M^ 3_ 1\) are homology cobordant if there is a compact, oriented 4-manifold \(W^ 4\) such that \(\partial W^ 4=M^ 3_ 0-M^ 3_ 1\) and the inclusion induced homomorphisms \(H_*(M_ i^ 4;{\mathbb{Z}})\to H_*(W^ 4;{\mathbb{Z}})\) are isomorphisms, \(i=0,1\). Ch. Livingston [Pac. J. Math. 94, 193-206 (1981; Zbl 0472.57003)] proved that every closed, oriented 3-manifold is homology cobordant to a Haken manifold. In the paper under review, the author proves the stronger result that every closed, oriented 3-manifold is homology cobordant to a hyperbolic 3- manifold. In addition, the analogous relative result is proven for compact oriented 3-manifolds with boundary. The key idea to prove such a theorem is to show the existence of a disjoint set of properly embedded arcs in a 3-cell having a hyperbolic exterior. The proof is quite long and technical. In addition, the author proves that every knot in \(S^ 3\) is concordant to a knot whose exterior is hyperbolic. This extends the results of R. C. Kirby and W. B. Lickorish [Math. Proc. Cambr. Philos. Soc. 86, 437-441 (1979; Zbl 0426.57001)] and Livingston [loc. cit.] that such knots are concordant to prime knots. To do this, the author modifies his earlier results [Trans. Am. Math. Soc. 273, 75-91 (1982; Zbl 0508.57008)] to show that every compact, oriented 3-manifold whose boundary contains no 2-spheres contains a properly embedded arc whose exterior is hyperbolic. As usual, the author makes use of Thurston’s theorem which equates simple Haken with hyperbolic.

Reviewer: R.Stern

### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

51M10 | Hyperbolic and elliptic geometries (general) and generalizations |

57R65 | Surgery and handlebodies |