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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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