2-handle additions producing toroidal and reducible manifolds

Abstract

Let M be a simple 3-manifold and F a boundary component of genus at least two and suppose α,β are two separating slopes on F. It is shown that if both 2-handle attaching M[α] and M[β] are toroidal then Δ(α,β)≤10 or Δ(α,β)=12 or Δ(α,β)=18. Furthermore, in the cases Δ(α,β)=12 or 18, M[α] and M[β] each contains an essential torus which intersects the 2-handle once and F has genus at least 4 or 8, respectively. As a corollary, if g(F)=3 then Δ(α,β)≤10 and if g(F)=2 then Δ(α,β)≤8. Moreover, if F has genus at least two and M[α] is toroidal and M[β] reducible then Δ(α,β)≤4.