## Geometry & Topology

### Heegaard surfaces and the distance of amalgamation

Tao Li

#### Abstract

Let $M1$ and $M2$ be orientable irreducible $3$–manifolds with connected boundary and suppose $∂M1≅∂M2$. Let $M$ be a closed $3$–manifold obtained by gluing $M1$ to $M2$ along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then $M$ is not homeomorphic to $S3$ and all small-genus Heegaard splittings of $M$ are standard in a certain sense. In particular, $g(M)=g(M1)+g(M2)−g(∂Mi)$, where $g(M)$ denotes the Heegaard genus of $M$. This theorem is also true for certain manifolds with multiple boundary components.

#### Article information

Source
Geom. Topol., Volume 14, Number 4 (2010), 1871-1919.

Dates
Revised: 9 March 2010
Accepted: 7 June 2010
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883509

Digital Object Identifier
doi:10.2140/gt.2010.14.1871

Mathematical Reviews number (MathSciNet)
MR2680206

Zentralblatt MATH identifier
1207.57031

#### Citation

Li, Tao. Heegaard surfaces and the distance of amalgamation. Geom. Topol. 14 (2010), no. 4, 1871--1919. doi:10.2140/gt.2010.14.1871. https://projecteuclid.org/euclid.gt/1513883509

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