A classification of toric, folded-symplectic manifolds

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 H²(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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Hockensmith, Daniel Lawrence

Contributors dc:contributor

• Lerman, Eugene
• Kerman, Ely
• Tolman, Susan
• Watts, Jordan

• folded-symplectic
• toric
• Delzant
• origami manifolds
• classification
• completely integrable system

Statement dc:rights

• Copyright 2015 Daniel Hockensmith

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/88015

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/88015