#### Volume 14, issue 4 (2010)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
Heegaard surfaces and the distance of amalgamation

### Tao Li

Geometry & Topology 14 (2010) 1871–1919
##### 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, $g\left(M\right)=g\left({M}_{1}\right)+g\left({M}_{2}\right)-g\left(\partial {M}_{i}\right)$, where $g\left(M\right)$ denotes the Heegaard genus of $M$. This theorem is also true for certain manifolds with multiple boundary components.

##### Keywords
Heegaard splitting, amalgamation, curve complex
Primary: 57N10
Secondary: 57M50