Sweepouts of amalgamated 3–manifolds

Abstract

We show that if two 3-manifolds with toroidal boundary are glued via a "sufficiently complicated" map then every Heegaard splitting of the resulting 3-manifold is weakly reducible. Additionally, suppose X [ F Y is a manifold obtained by gluing X and Y , two connected small manifolds with incompressible boundary, along a closed surface F . Then the following inequality on genera is obtained: