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

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.