Heegaard surfaces and the distance of amalgamation

Abstract

Let $M_1$ and $M_2$ be orientable irreducible 3--manifolds with connected boundary and suppose $\partial M_1\cong \partial M_2$. Let $M$ be a closed 3--manifold obtained by gluing $M_1$ to $M_2$ along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then $M$ is not homeomorphic to $S^3$ and all small-genus Heegaard splittings of $M$ are standard in a certain sense. In particular, $g(M)=g(M_1)+g(M_2)-g(\partial M_i)$, where $g(M)$ denotes the Heegaard genus of $M$. This theorem is also true for certain manifolds with multiple boundary components.

38 pages, 2 figures

References in corpus (6)

• Geometry of the Complex of Curves I: Hyperbolicity
• Alternate Heegaard genus bounds distance
• Sweepouts of amalgamated 3-manifolds
• On the Heegaard splittings of amalgamated 3-manifolds
• Saddle tangencies and the distance of Heegaard splittings
• The Heegaard structure of Dehn filled manifolds

Cited by in corpus (4)

• Bridge distance and plat projections
• The Heegaard distances cover all non-negative integers
• Small 3-manifolds with large Heegaard distance
• On the width of complicated JSJ decompositions