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 | References | Similar Articles | Additional Information

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

Keywords:
<IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$3$">-manifold,
hyperbolic <IMG WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$3$">-manifold,
knot,
link,
tangle,
homology cobordism,
knot concordance,
link concordance

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© Copyright 1983
American Mathematical Society