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## Triangulations and Heegaard splittings

Liu, Zhenyi

Liu, Zhenyi

##### Abstract

Recently William Jaco, J. Hyam Rubinstein and Stephan Tillmann together proved that the generalized quaternion spaces S3 /Q4k, k ≥ 2, which are small Seifert fibered spaces Mk= (S2 : (2, 1),(2, 1),(k, -k + 1)), have complexity k, which is the minimal number of tetrahedra in a triangulation of Mk. The techniques used can be expanded to show that the layered chain pair triangulations of Seifert fibered spaces (S2: (2, -1),(r + 1, 1),(s + 1, 1)), r, s ≥ 1 are minimal.

My thesis is to closely study the minimal, 0-efficient triangulations of the above two infinite families of Seifert fiberd spaces. One family is called the twisted layered loop triangulation, and the other family is called layered chain pair triangulations. They were named by Ben Burton. We classify all normal and almost normal surfaces by identifying one-sided incompressible surfaces, orientable incompressible surfaces and Heegaard splitting surfaces. We also use combinatorial methods to study and classify irreducible Heegaard splitting surfaces, up to isotopy, in these two infinite families of Seifert fibered manifolds.

In the twisted layered loop triangulations of the Seifert fibered space Mk, k ≥ 2. We prove that a properly embedded surface S is a Heegaard splitting surface if and only if it is an almost normal tubed surface with the almost normal tube at the same level of a thin edge-linking tube. Furthermore, any genus two Heegaard splitting surface is vertical. A combinatorial proof is given that there is a unique irreducible genus 2 Heegaard splitting surface, up to isotopy, in Mk, k ≥ 2.

In the layered chain pair triangulation of Seifert fibered spaces Mr,s=( S2: (2, -1),(r+ 1, 1),(s+1, 1)), r, s ≥ 1, we prove that an almost normal tubed surface with the almost normal tube at the same level of a thin edge-linking tube is a Heegaard splitting surface. Moreover, if the genus of it is 2, then it is not only an irreducible Heegaard splitting but also a vertical one. We give a combinatorial proof that there is a unique irreducible vertical Heegaard splitting surface, up to isotopy, in Mr,s, r, s ≥ 1.

Our work follows the methods used by Jaco and Rubinstein in studying layered triangulations of the solid torus and their classification of normal surfaces in these triangulations.

My thesis is to closely study the minimal, 0-efficient triangulations of the above two infinite families of Seifert fiberd spaces. One family is called the twisted layered loop triangulation, and the other family is called layered chain pair triangulations. They were named by Ben Burton. We classify all normal and almost normal surfaces by identifying one-sided incompressible surfaces, orientable incompressible surfaces and Heegaard splitting surfaces. We also use combinatorial methods to study and classify irreducible Heegaard splitting surfaces, up to isotopy, in these two infinite families of Seifert fibered manifolds.

In the twisted layered loop triangulations of the Seifert fibered space Mk, k ≥ 2. We prove that a properly embedded surface S is a Heegaard splitting surface if and only if it is an almost normal tubed surface with the almost normal tube at the same level of a thin edge-linking tube. Furthermore, any genus two Heegaard splitting surface is vertical. A combinatorial proof is given that there is a unique irreducible genus 2 Heegaard splitting surface, up to isotopy, in Mk, k ≥ 2.

In the layered chain pair triangulation of Seifert fibered spaces Mr,s=( S2: (2, -1),(r+ 1, 1),(s+1, 1)), r, s ≥ 1, we prove that an almost normal tubed surface with the almost normal tube at the same level of a thin edge-linking tube is a Heegaard splitting surface. Moreover, if the genus of it is 2, then it is not only an irreducible Heegaard splitting but also a vertical one. We give a combinatorial proof that there is a unique irreducible vertical Heegaard splitting surface, up to isotopy, in Mr,s, r, s ≥ 1.

Our work follows the methods used by Jaco and Rubinstein in studying layered triangulations of the solid torus and their classification of normal surfaces in these triangulations.

##### Date

2010-05