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.
| Identifer | oai:union.ndltd.org:uiowa.edu/oai:ir.uiowa.edu:etd-5765 |
| Date | 01 May 2015 |
| Creators | Ortiz, Marcos A |
| Contributors | Kawamuro, Keiko |
| Publisher | University of Iowa |
| Source Sets | University of Iowa |
| Language | English |
| Detected Language | English |
| Type | dissertation |
| Format | application/pdf |
| Source | Theses and Dissertations |
| Rights | Copyright 2015 Marcos Arthur Ortiz |
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