Spherical space forms and Dehn filling

Universal bounds for hyperbolic Dehn surgery

“…(See [5] for this argument.) Hence, for one component we obtain the same lower bound for area(T R ), while for the other components the lower bound for area(T R ) is half as large.…”

“…It follows from the work of , see also [5]) that all but a universal number of surgeries on each torus yield 3-manifolds which admit negatively curved metrics. More recent work by Lackenby [33] and, independently, by Agol [2], similarly shows that for all but a universally bounded number of surgeries on each torus the resulting manifolds are irreducible with infinite word hyperbolic fundamental group.…”

Monopoles and lens space surgeries

“…To define an action of Z, we specify that [ξ] + n denotes the homotopy class [ξ] obtained from [ξ] as follows. Let B 3 ⊂ Y be a standard ball, and let ρ : (B 3 , ∂B 3 ) → (SO(3), 1) be a map of degree −2n, regarded as an automorphism of the trivialized tangent bundle of the ball. Outside the ball B 3 , we takeξ = ξ.…”

“…Let K be a knot in S 3 . Given a rational number r, let S 3 r (K) denote the oriented three-manifold obtained from the knot complement by Dehn filling with slope r. The main purpose of this paper is to prove the following conjecture of Gordon (see [18], [19]): Theorem 1.1.…”

Some surface subgroups survive surgery

“…Recall that by construction γ has length more than 2π onÑ , so that using the 2π -theorem [3], we may put negatively curved metrics on both spaces and arrange that the map f γ continues to be a local isometry. We now may prove:…”