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 $X{\cup }_{F}Y$ 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:

$$g\left(X{\cup }_{F}Y\right)\ge \frac{1}{2}\left(g\left(X\right)+g\left(Y\right)-2g\left(F\right)\right).$$

Both results follow from a new technique to simplify the intersection between an incompressible surface and a strongly irreducible Heegaard splitting.