Aspects of the topology of foliations on 3-manifolds

Abstract

In this thesis, we explore two aspects of the topology of codimension 1 foliations on 3-manifolds. In the first part, we study branching in their leaf spaces. We show that for a large class of foliations, the leaf space admits a map to \R such that the action of the fundamental group on the leaf space descends to a Homeo^+(\R) representation. In the second part, we study the holonomy of foliations. We give a new approach to the Eliashberg-Thurston perturbation of foliations into contact structures. Our approach gives control on the Reeb flow of the resulting contact structure, and in particular produces hypertight contact structures.