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Title:A classification of toric, folded-symplectic 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
Degree Granting Institution:University of Illinois at Urbana-Champaign
origami manifolds
completely integrable system
Abstract:Given a $G$-toric, folded-symplectic manifold with co-orientable 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, folded-symplectic 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, folded-symplectic 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, folded-symplectic 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 co-orientable. We spend a significant amount of time discussing the fundamentals of equivariant and non-equivariant folded-symplectic geometry. In particular, we characterize folded-symplectic forms in terms of their induced map from the sheaf of vector fields into a distinguished sheaf of one-forms, we relate the existence of an orientation on the folding hypersurface of a fold-form to the intrinsic derivative of the contraction mapping from the tangent bundle to the cotangent bundle, and we show that $G$-toric, folded-symplectic manifolds are stratified by $K$-toric, folded-symplectic 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, folded-symplectic manifold used in the classification. We also give a robust description of folded-symplectic reduction, which we use to construct local models of toric, folded-symplectic manifolds.
Issue Date:2015-07-15
Rights Information:Copyright 2015 Daniel Hockensmith
Date Available in IDEALS:2015-09-29
Date Deposited:August 201

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