## Homology cobordisms, link concordances, and hyperbolic $3$-manifolds

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- by Robert Myers PDF
- Trans. Amer. Math. Soc.
**278**(1983), 271-288 Request permission

## Abstract:

Let $M_0^3$ and $M_1^3$ be compact, oriented $3$-manifolds. They are*homology cobordant*(respectively

*relative homology cobordant*) if $\partial M_1^3 = \emptyset \;({\text {resp.}}\;\partial M_1^3 \ne \emptyset )$ and there is a smooth, compact oriented $4$-manifold ${W^4}$ such that $\partial {W^4} = M_0^3 - M_1^3$ (resp. $\partial {W^4} = M_0^3 - M_1^3) \cup (M_i^3 \times [0,1])$ and ${H_{\ast }}(M_i^3;{\mathbf {Z}}) \to {H_{\ast }}({W^4};{\mathbf {Z}})$ are isomorphisms, $i = 0,1$. Theorem.

*Every closed, oriented*$3$-

*manifold is homology cobordant to a hyperbolic*$3$-

*manifold*. Theorem.

*Every compact, oriented*$3$-

*manifold whose boundary is nonempty and contains no*$2$-

*spheres is relative homology cobordant to a hyperbolic*$3$-

*manifold*. Two oriented links ${L_0}$ and ${L_1}$ in a $3$-manifold ${M^3}$ are

*concordant*if there is a set ${A^2}$ of smooth, disjoint, oriented annuli in $M \times [0,1]$ such that $\partial {A^2} = {L_0} - {L_1}$, where ${L_{i}} \subseteq \;{M^3} \times \{ i\} ,i = 0,1$. Theorem.

*Every link in a compact, oriented*$3$-

*manifold*${M^3}$

*whose boundary contains no*$2$-

*spheres is concordant to a link whose exterior is hyperbolic*. Corollary.

*Every knot in*${S^3}$

*is concordant to a knot whose exterior is hyperbolic*.

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## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**278**(1983), 271-288 - MSC: Primary 57N10; Secondary 57M40, 57N70
- DOI: https://doi.org/10.1090/S0002-9947-1983-0697074-4
- MathSciNet review: 697074