M 392C (57685) Symplectic Topology

Time: MWF 12:00 – 1:00 pm

Room: RLM 10.176

Instructor: Robert Gompf

Course Description:

Symplectic structures (closed, nondegenerate 2-forms) arose from classical physics and algebraic geometry, but in recent years they have developed deep connections with other areas such as low-dimensional topology. These structures can be thought of as skew-symmetric analogs of constant curvature Riemannian metrics, or as generalizations of (Kaehler) complex structures. However, they are more flexible than these other structures, and so are more amenable to topological (cut and paste) constructions. On 4-manifolds, the existence of symplectic structures is a delicate question perhaps analogous to hyperbolization of 3-manifolds. The answer depends not just on the topology, but also on the smooth structure of the underlying 4-manifold. While symplectic manifolds are necessarily even dimensional, their natural boundaries are odd-dimensional contact manifolds. Contact structures are closely related to foliations, and so are intimately related to the topology of the underlying manifold - particularly in dimension 3.We will explore the topology of symplectic manifolds, with emphasis on explicit constructions, especially for 4-manifolds and their contact 3-manifold boundaries. Topics will include linear and nonlinear symplectic geometry, symplectic fiber bundles and Lefschetz pencils, symplectic normal connected sums, realizing preassigned fundamental groups and characteristic numbers, symplectic fillings of 3-manifolds, and other topics as time permits.

Prerequisites:

Basic algebraic topology (fundamental group and homology), some familiarity with smooth manifolds, differential forms. 



Textbook(s):

None required. Some of this material can be found in the following texts:

·       Aebischer et. al., Symplectic Geometry, Progress in Math 124, Birkhauser 1994.

·       McDuff and Salamon, Introduction to Symplectic Topology, Oxford Univ. Press 1995

·       Gompf and Stipsicz, 4-manifolds and Kirby Calculus, Grad. Studies in Math. 20, Amer. Math. Soc. 1999.