DOI

10.17077/etd.d4bf73qk

Document Type

Dissertation

Date of Degree

Spring 2017

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Kawamuro, Keiko

First Committee Member

Kawamuro, Keiko

Second Committee Member

Cooper, Benjamin

Third Committee Member

Fang, Hao

Fourth Committee Member

Frohman, Charles

Fifth Committee Member

Tomova, Maggy

Abstract

A contact structure is a maximally non-integrable hyperplane field $\xi$ on an odd-dimensional manifold $M$. In $3$-dimensional contact geometry, there is a fundamental dichotomy, where a contact structure is either tight or overtwisted. Making use of Giroux's correspondence between contact structures and open books for $3$-dimensional manifolds, Honda, Kazez, and Mat\'{i}c proved that verifying whether a mapping class is right-veering or not gives a way of detecting tightness of the compatible contact structure. As a counter-part to right-veering mapping classes, right-veering closed braids have been studied by Baldwin and others. Ito and Kawamuro have shown how various results on open books can be translated to results on closed braids; introducing the notion of quasi right-veering closed braids to provide a sufficient condition which guarantees tightness.

We use the related concept of $P$-bigon right-veeringness for closed braids to show that given a $3$-dimensional contact manifold $(M, \xi)$ supported by an open book $(S, \phi)$, if $L \subset (M, \xi)$ is a non-$P$-bigon right-veering transverse link in pure braid position with respect to $(S, \phi)$, performing $0$-surgery along $L$ produces an overtwisted contact manifold $(M', \xi')$. Furthermore, if we suppose $L \subset (M, \xi)$ is a pure and non-quasi right-veering braid with respect to $(S, \phi)$, performing $p$-surgery along $L$, for $p \geq 0$, gives rise to an open book $(S', \phi')$ which supports an overtwisted contact manifold $(M', \xi')$.

Keywords

contact geometry, contact structures, contact topology, overtwisted contact structures, p-bigon right-veeringness

Pages

xiii, 94 pages

Bibliography

Includes bibliographical references (pages 91-94).

Copyright

Copyright © 2017 Camila Alexandra Ramirez

Included in

Mathematics Commons

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