Let M³₀ and M³₁ be compact, oriented 3-manifolds. They are homology cobordant (respectively relative homology cobordant) if ∂M³ᵢ = ø (resp. ∂M³ᵢ ≠ ø) and there is a smooth, compact oriented 4-manifold W⁴ such that ∂W⁴ = M³₀ - M³₁ (resp. ∂W⁴ = M³₀ - M³₁) U (M³₁;Z) → H*(W⁴;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 L₀ and L₁ in a 3-manifold M³ are concordant if there is a set A² of smooth, disjoint, oriented annuli in M X [0,1] such that ∂A² = L₀ - L₁, where Lᵢ ⊆ M³ X {i}, i = 0, 1.

Theorem. Every link in a compact, oriented 3-manifold M³ whose boundary contains no 2-spheres is concordant to a link whose exterior is hyperbolic.

Corollary. Every knot in S³ is concordant to a knot whose exterior is hyperbolic.