Dehn Fillings Producing Reducible Manifolds and Toroidal Manifolds

This paper studies one of the problems concerning Dehn fillings producing reducible or toroidal 3-manifolds. Let M be an orientable, irreducible, atoroidal, anannular 3-manifold with T as a torus boundary component. Let γ be an essential simple loop on T . Denote by M(γ) the manifold obtained by Dehn filling along the curve γ, i.e. M(γ) = M ∪φ J , where J is a solid torus, and φ : T ∼= ∂J is a gluing map sending γ to a meridian curve of J . For two curves γ1, γ2 on T , denote by ∆ = ∆(γ1, γ2) the minimal geometric intersection number between the isotopy classes of γ1 and γ2. In [GLu] Gordon and Luecke proved that if both M(γi) are reducible then ∆ ≤ 1. Gordon [Go] proved that if both M(γi) are toroidal then ∆ ≤ 5 except for a few manifolds M , for which the maximal ∆ is 6, 7 or 8. When M(γ1) is reducible andM(γ2) is toroidal, Gordon and Litherland [GLi] showed that ∆ ≤ 5. This was improved by Boyer and Zhang [BZ] to ∆ ≤ 4. The main theorem of this paper is: