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


Author: Robert Myers
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
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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

DOI: https://doi.org/10.1090/S0002-9947-1983-0697074-4
Keywords: $ 3$-manifold, hyperbolic $ 3$-manifold, knot, link, tangle, homology cobordism, knot concordance, link concordance
Article copyright: © Copyright 1983 American Mathematical Society

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