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Title:  A classification of toric, foldedsymplectic manifolds 
Author(s):  Hockensmith, Daniel Lawrence 
Director of Research:  Lerman, Eugene 
Doctoral Committee Chair(s):  Kerman, Ely 
Doctoral Committee Member(s):  Tolman, Susan; Watts, Jordan 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  foldedsymplectic
toric Delzant origami manifolds classification completely integrable system 
Abstract:  Given a $G$toric, foldedsymplectic manifold with coorientable folding hypersurface, we show that its orbit space is naturally a manifold with corners $W$ equipped with a smooth map $\psi: W \to \frak{g}^*$, where $\frak{g}^*$ is the dual of the Lie algebra of the torus, $G$. The map $\psi$ has fold singularities at points in the image of the folding hypersurface under the quotient map and it is a unimodular local embedding away from these points. Thus, to every $G$toric, foldedsymplectic manifold we can associate its orbit space data $\psi:W \to \fg^*$, a unimodular map with folds. We fix a unimodular map with folds $\psi:W \to \fg^*$ and show that isomorphism classes of $G$toric, foldedsymplectic manifolds whose orbit space data is $\psi:W \to \fg^*$ are in bijection with $H^2(W; \mathbb{Z}_G\times \R)$, where $\mathbb{Z}_G= \ker(\exp:\frak{g} \to G)$ is the integral lattice of $G$. Thus, there is a pair of characteristic classes associated to every $G$toric, foldedsymplectic manifold. This result generalizes a classical theorem of Delzant as well as the classification of toric, origami manifolds, due to Cannas da Silva, Guillemin, and Pires, in the case where the folding hypersurface is coorientable. We spend a significant amount of time discussing the fundamentals of equivariant and nonequivariant foldedsymplectic geometry. In particular, we characterize foldedsymplectic forms in terms of their induced map from the sheaf of vector fields into a distinguished sheaf of oneforms, we relate the existence of an orientation on the folding hypersurface of a foldform to the intrinsic derivative of the contraction mapping from the tangent bundle to the cotangent bundle, and we show that $G$toric, foldedsymplectic manifolds are stratified by $K$toric, foldedsymplectic submanifolds, where $K$ varies over the subtori of $G$ and the action is principal on each stratum. We show how these structures give rise to the rigid orbit space structure of a toric, foldedsymplectic manifold used in the classification. We also give a robust description of foldedsymplectic reduction, which we use to construct local models of toric, foldedsymplectic manifolds. 
Issue Date:  20150715 
Type:  Thesis 
URI:  http://hdl.handle.net/2142/88015 
Rights Information:  Copyright 2015 Daniel Hockensmith 
Date Available in IDEALS:  20150929 
Date Deposited:  August 201 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois