As with many areas of topology and geometry, the study of 3-dimensional manifolds begins with its codimension-1 submanifolds, namely embedded surfaces. From there, it is very natural to consider foliations of 3-manifolds by surfaces, that is, locally trivial decompositions into (typically noncompact) surface leaves. While every closed orientable 3-manifold has a foliation, the seemingly mild condition of requiring the foliation to be taut restricts both the topology of the ambient manifold (e.g. the universal cover must be R3) and of the leaves, and forces the existences of interesting transverse dynamics.
I will start with an overview of foliations, focusing on examples, and then discuss the basic properties of taut foliations on 3-manifolds. Then I will then turn to the foundational work of Gabai, specifically his tool of sutured manifold hierarchies for constructing foliations. Next, I will discuss connections to contact geometry, namely the work of Eliashberg-Thurston which builds from a taut foliation a pair of tight contact structures. Finally, I will explain how taut foliations fit into the various Floer homology theories of 3-manifolds (the Heegaard, instanton, and monopole Floer homologies).
Prerequisites: Basic knowledge of smooth manifolds and algebraic topology, e.g. Math 518 and Math 525. No prior knowledge of foliations or 3-manifolds will be assumed.
I will not follow any particular text, but recommend the following:
Students registered for the course will need to write a short (2-4 page) paper which will be due on Friday, October 15th. This paper is largely free-form, and can be about any subject related to the content of this course. For instance, it could be a brief account of a result not covered in class, a review of the some related results explaining why they are interesting, a detailed work-out of a proof only sketched in class, or careful solutions to problems from class or taken from one of the texts or other readings.
Notes from each lecture will be posted here.