Document Type

Dissertation

Date of Degree

Spring 2015

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Keiko Kawamuro

Abstract

Contact structures on 3-manifolds are 2-plane fields satisfying a set of conditions. The study of contact structures can be traced back for over two-hundred years, and has been of interest to mathematicians such as Hamilton, Jacobi, Cartan, and Darboux. In the late 1900's, the study of these structures gained momentum as the work of Eliashberg and Bennequin described subtleties in these structures that could be used to find new invariants. In particular, it was discovered that contact structures fell into two classes: tight and overtwisted. While overtwisted contact structures are relatively well understood, tight contact structures remain an area of active research. One area of active study, in particular, is the classification of tight contact structures on 3-manifolds. This began with Eliashberg, who showed that the standard contact structure in real three-dimensional space is unique, and it has been expanded on since. Some major advancements and new techniques were introduced by Kanda, Honda, Etnyre, Kazez, Matić, and others. Convex decomposition theory was one product of these explorations. This technique involves cutting a manifold along convex surfaces (i.e. surfaces arranged in a particular way in relation to the contact structure) and investigating a particular set on these cutting surfaces to say something about the original contact structure. In the cases where the cutting surfaces are fairly nice, in some sense, Honda established a correspondence between information on the cutting surfaces and the tight contact structures supported by the original manifold.

In this thesis, convex surface theory is applied to the case of handlebodies with a restricted class of dividing sets. For some cases, classification is achieved, and for others, some interesting patterns arise and are investigated.

Public Abstract

Three manifolds are simply spaces that, when viewed by an observer living inside them, look like the three dimensional space with which we are familiar. These spaces might have global properties that make them different from real three dimensional space, but up close they are generally the same.

A contact structure on a three manifold is an arrangement of planes called contact planes. Think of a room where hundreds of sheets of paper are flying around. If you freeze time, you can think of each of those pieces of paper as representing one of the contact planes. It is impossible to visualize all of the contact planes because they will fill the space and intersect one another. To be a contact structure, these planes must follow a strict set of rules.

The rules which determine whether an arrangement of planes makes up a contact structure are very specific, but they are not so rigid that there is always only one way to do it. That is, sometimes there are two or more arrangements that follow the rules of contact structures. Sometimes these arrangements only look different, but are the same up to some notion of equivalence. So, the challenge becomes listing all of the fundamentally different ways that these planes can be arranged to follow the rules.

Here, this is achieved by cutting a three manifold into simple pieces and investigating how the contact planes along these cuts are arranged.

Keywords

publicabstract, convex decomposition, cotact topology, topology

Pages

x, 65 pages

Bibliography

Includes bibliographical references (pages 64-65).

Copyright

Copyright 2015 Marcos Arthur Ortiz

Included in

Mathematics Commons

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