Examples of bireducible Dehn fillings

Abstract

If an irreducible manifold M admits two Dehn fillings along distinct slopes each filling resulting in a reducible manifold, then we call these bireducible Dehn fillings. The first example of bireducible Dehn fillings is due to Gordon and Litherland. More recently, Eudave-Muñoz and Wu presented the first infinite family of manifolds which admit bireducible Dehn fillings. We present another infinite family of hyperbolic manifolds which admit bireducible Dehn fillings. The manifolds obtained by the fillings are always the connect sum of two lens spaces.

A census of exceptional Dehn fillings

“…Specifically, the summands all have finite π 1 or are S 2 ×S 1 ; there are only two summands in all but three cases: the filling o9 39343 (1, 0) is R P 3 # R P 3 # R P 3 and both o9 41447 (1, 0) and o9 43255 (1, 0) are the manifold L(3, 1) # R P 3 # R P 3 . While there are infinite families with two connected sum fillings [EMW], it is an open question whether there is a manifold with three such fillings, see [HM,§4]. In C 9 there are only 14 manifolds with two distinct Dehn fillings that are connected sums, and none with more than two.…”

“…In C 9 there are only 14 manifolds with two distinct Dehn fillings that are connected sums, and none with more than two. Another question from [HM,§4] is when there are two such fillings, must both have at least one summand that is R P 3 = L(2, 1), L(3, 1), or L(4, 1)?…”

Chern–Simons theory, surface separability, and volumes of 3-manifolds

“…There is a natural section-fiber basis (s, h) of M 1 , where h is parallel to S 1 and s is parallel to ∂F . On the other hand, it follows from [16] that there are infinitely many one cusped, complete, finite volume hyperbolic manifolds M 2 endowed with a basis (μ, λ) ⊂ ∂M 2 such that both M 2 (λ) and M 2 (μ) have zero simplicial volume (because they are actually connected sums of lens spaces). Denote by ϕ : ∂M 1 → ∂M 2 the homeomorphism defined by ϕ(s) = μ and ϕ(h) = λ −1 .…”

“…To prove Theorem 1.7, some arguments in [11], an example of Motegi [28] and a result of Hoffman-Matignon [16] are also applied.…”

Exceptional surgeries in 3-manifolds

“…While non-trivial reducible surgeries on hyperbolic knots in reducible manifolds do exist, see e.g. [HM03], we suspect that manifolds whose prime decompositions have at least 3 summands are candidates.…”