Symplectic Foliations, Currents, and Local Lie Groupoids

TL;DR

This thesis explores calibrated symplectic foliations and local Lie groupoids, applying currents theory to understand their properties and generalizing key theorems to classify local Lie groupoids and their integrability.

Contribution

It introduces new classifications of local Lie groupoids and extends classical theorems, linking associativity with integrability of algebroids.

Findings

Generalization of Mal'cev's theorem to local Lie groupoids

Classification of local Lie groupoids with integrable algebroids

Relationship between associativity and algebroid integrability

Abstract

This thesis treats two main topics: calibrated symplectic foliations, and local Lie groupoids. Calibrated symplectic foliations are one possible generalization of taut foliations of 3-manifolds to higher dimensions. Their study has been popular in recent years, and we collect several interesting results. We then show how de Rham's theory of currents, and Sullivan's theory of structure currents, can be applied in trying to understand the calibratability of symplectic foliations. Our study of local Lie groupoids begins with their definition and an exploration of some of their basic properties. Next, three main results are obtained. The first is the generalization of a theorem by Mal'cev. The original theorem characterizes the local Lie groups that are part of a (global) Lie group. We give the corresponding result for local Lie groupoids. The second result is the generalization of a theorem by Olver which classifies local Lie groups in terms of Lie groups. Our generalization classifies, in terms of Lie groupoids, those local Lie groupoids that have integrable algebroids. The third and final result demonstrates a relationship between the associativity of a local Lie groupoid, and the integrability of its algebroid. In a certain sense, the monodromy groups of a Lie algebroid manifest themselves combinatorially in a local integration, as a lack of associativity.