A classification of toric, folded-symplectic manifolds

Hockensmith, Daniel Lawrence

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https://hdl.handle.net/2142/88015 Copy

Description

Title

A classification of toric, folded-symplectic manifolds

Author(s)

Hockensmith, Daniel Lawrence

Issue Date

2015-07-15

Director of Research (if dissertation) or Advisor (if thesis)

Lerman, Eugene

Doctoral Committee Chair(s)

Kerman, Ely

Committee Member(s)

• Tolman, Susan
• Watts, Jordan

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• folded-symplectic
• toric
• Delzant
• origami manifolds
• classification
• 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.

Graduation Semester

2015-8

Type of Resource

text

Permalink

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

Copyright and License Information

Copyright 2015 Daniel Hockensmith

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