This semester, as we have sometimes done in the past, the topology seminar will have a common theme. (See the following paragraphs for a few ideas of what this might involve.) Participants in the seminar will sign up to give a talk or two during the semester. The background required is basic algebraic topology (at the level of 121) and the basic knowledge of manifolds (at the level of 110ab), although certain topics will involve more advanced ideas.
The last several years have seen enormous progress in the study of 3-manifolds. The most famous is of course the proof of the Poincare conjecture and Thurston's Geometrization conjecture via Ricci flow. Simultaneously, many other important problems in 3-manifold theory have been solved through a combination of geometric and topological techniques. The most powerful techniques rely on the interaction between, on the one hand, gauge theory and Floer homology (in their many variations: Yang-Mills, Seiberg-Witten, and Heegaard-Floer) and the geometry/topology of 3-manifolds in the form of contact structures and foliations.
The seminar will explore the basics of these geometric structures, starting from the beginning. As it turns out, an important topological notion underlies both contact structures and foliations in dimension 3: the idea of a sutured 3-manifold. This is not much more than a 3-manifold together with a bunch of (oriented) annuli on its boundary, but this simple notion turns out to be extremely useful.
Although we will not get to the multiple applications, it might be of interest to see how it all fits together. A schematic flow-chart of the ideas is drawn below; keep in mind that this encompasses about 10 years of fast-paced development!
List of topics; most of these are more than one lecture's worth of material:
- Basic 3-manifold material: Loop theorem/Dehn's lemma [Rolfsen].
- Incompressible surfaces: Thurston norm and polytope [Thurston, Kapovich].
- Basics of foliations and contact structures [Etnyre].
- Sutured manifolds and foliations [Gabai].
- Sutured manifolds and contact structures [Kazez, Candel-Conlon]
References:
- [Candel-Conlon] Alberto Candel and Lawrence Conlon, Foliations II, AMS Graduate Series in Mathematics 60 (2003).
- [Etnyre]: John Etnyre, Lectures on contact geometry in low-dimensional topology.
- [Gabai]: David Gabai, Foliations and the topology of 3-manifolds, Journal of Differential Geometry 18 (1984), 445-503.
- [Kapovich]: M. Kapovich, Hyperbolic Manifolds and Discrete Groups, Birkhäuser, 2001.
- [Kazez] Kazez, William, A cut-and-paste approach to contact topology, Bol. Soc. Mat. Mexicana 10 (2004), 1--42.
- [Rolfsen] Dale Rolfsen, Knots and Links, Publish or Perish, Inc., Houston, TX, 1990.
- [Thurston] William Thurston, A norm for the homology of 3-manifolds, Memoirs AMS 59 (1986), no. 339.