Math 221b Spring 1998
Symplectic and Contact
This is an outline of what I hope to cover in the course of this semester; most
of the references may be found on reserve in the library.
- Symplectic Manifolds. 1,2
- Linear algebra and symplectic forms.
- Darboux theorem and Moser stability theorem.
- Compatible almost complex structures.
- Symplectic and Lagrangian submanifolds.
- Cotangent bundle, Kaehler manifolds.
- Kodaira-Thurston examples.
- Gluing constructions.
- J-holomorphic curves.
- Contact manifolds--introduction.
- Basics, especially in dimension 3.3,4
- Darboux theorem and Gray stability theorem.
- Induced foliation on an embedded surface.
- Legendrian knots; Thurson-Bennequin invariant.
- Euler class of a contact structure.
- R3, S3, unit cotangent bundle.
- All 3-manifolds have contact structures.
- Constructions of 3-manifolds: surgery, open books. 5
- Techniques of Lutz, and Thurston-Winkelnkemper. 6
- Relation to symplectic manifolds and Stein manifolds. 7, 8, 9,10
- J-convexity, holomorphic and symplectic fillability.
- Morse theory of Stein manifolds--outline.
- The basic dichotomy in dimension 3: Tight vs. overtwisted contact
- Overtwisted structures.
- Lutz twist.
- Eliashberg's flexibility theorem: homotopic overtwisted structures are
- Sketch of proof for B. 11, 12
- Tight structures.
- The standard structure on S3 is tight, which follows from:
- Holomorphically fillable structures are tight, which follows from:
- Symplectically fillable structures are tight.
- Outline of B. 13
- Bishop's theorem on existence of holomorphic disks.
- Filling by holomorphic disks.
- Pseudoholomorphic disks and part C.
- Eliashberg rigidity theorem: Uniqueness for tight structures on S3.14
- Bennequin's inequality for tight contact manifolds. 15,16
- Gromov/Eliashberg's proof.
- Applications to knot theory.
- Comparison with taut foliations and work of Gabai/Thurston. 17,18
- Sketch of relations to Seiberg-Witten theory.
1 D. McDuff and D. Salamon, Introduction
to Symplectic Topology, Oxford Press (1996), Chapters 1-4 and
2 B. Aebischer et al. Symplectic Geometry, Birkhauser
(1994), Chapters 1,2.
3 Aebischer et al., Chapter 8.
4 Mcduff-Salamon, Chapter 3.4.
5 D. Rolfsen, Knots and Links, Publish or Perish (1976).
6 W. Thurston and H. Winkelnkemper, On the existence of contact
forms, Proc. AMS 52 (1975), 345-347.
7 Y. Eliashberg, Topological characterization of Stein manifolds
of dimension > 2, Int. J. Math. 1 (1990), 29-46.
8 R. Gompf, Handlebody constructions of Stein surfaces.
9 A. Weinstein, Contact surgery and symplectic handlebodies.
Hokkaido Math. J. 20 (1991), 241-251.
10 E. Giroux, Topologie de contact en dimension 3, Seminaire Bourbaki, in Asterisque 216 (1993) 7-33.
11 Y. Eliashberg, Contact 3-manifolds twenty years since J.
Martinet's work. Ann. Inst Fourier 42 (1995), 165-192.
13 Y. Eliashberg, Filling by holomorphic discs and its
applications, in Geometry of Low- Dimensional Manifolds, London
Math. Soc. Lect. Notes 151, Cambridge (1990), 45-67.
16 Y. Eliashberg, Legendrian and transversal knots in tight
contact 3-manifolds, in Topological Methods in Modern Mathematics,
Publish or Perish (1993).
17 D. Gabai, Foliations and the topology of 3-manifolds, J. Diff.
Geom. 18 (1983), 445-503.
18 W. Thurston, A norm for the homology of 3-manifolds, Mem. AMS
59 (1986), no. 339.