M.Sc Student | Hagay Shacham |
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Subject | Combinatorial Conditions which Affect the Heegaard Distance |

Department | Department of Mathematics |

Supervisor | Full Professor Moriyah Yoav |

Full Thesis text |

Every closed orientable 3-dimensional
manifold M admits a Heegaard splitting, i.e. a decomposition into two handlebodies which meet along their boundary.
This common boundary is called a Heegaard surface in M, and is usually
considered only up to isotopy in M. The genus g of the Heegaard surface is said
to be the genus of the handlebodies. A Heegaard splitting gives us the Heegaard
distance, which is defined using the curve complex. The fact that a Heegaard
splitting is high distance has important consequences for the geometry of the
3-manifold determined by it. In this research thesis we discuss
two previously introduced combinatorial conditions on the Heegaard
distance - the rectangle condition, first introduced be Casson and Gordon, and
the double rectangle condition, first introduced by Lustig and Moriah - and
their effect on the Heegaard distance, and hence on the geometry of the
3-manifold. Both of these conditions mean high complexity of the Heegaard
splitting, and so it seems reasonable to ask whether they also imply a large
Heegaard distance, namely Heegaard distance strictly larger than 2. We will
show, using induction on the genus *g * of the handlebodies which define
the Heegaard splitting, that for each of these combinatorial conditions, and
for every genus* g*, there exists a 3-manifold M_{g} with Heegaard
splitting of genus g with two complete decomposing systems which satisfy the
relevant condition and implies Heegaard distance of exactly 2.