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Lerman, Eugene Kerman, Ely Tolman, Susan Watts, Jordan Hockensmith, Daniel Lawrence 2015-09-29T20:38:14Z 2015-09-29T20:38:14Z 2015-08 2015-07-15 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. Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2015-09-29 without embargo terms The student, Daniel Hockensmith, accepted the attached license on 2015-07-09 at 15:47. The student, Daniel Hockensmith, submitted this Dissertation for approval on 2015-07-09 at 15:57. This Dissertation was approved for publication on 2015-07-15 at 09:08. DSpace SAF Submission Ingestion Package generated from Vireo submission #8395 on 2015-09-29 at 13:22:25 Made available in DSpace on 2015-09-29T20:38:14Z (GMT). No. of bitstreams: 2 HOCKENSMITH-DISSERTATION-2015.pdf: 1044445 bytes, checksum: a8d2540daf14f068ac6491b6832b4d45 (MD5) LICENSE.txt: 4215 bytes, checksum: d35f0685529338d0520a6537425e618a (MD5) Previous issue date: 2015-07-15 application/pdf http://hdl.handle.net/2142/88015 en Copyright 2015 Daniel Hockensmith folded-symplectic toric Delzant origami manifolds classification completely integrable system A classification of toric, folded-symplectic manifolds text text 2015-8 Ph.D. Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation