Math 221b Spring 1998

Symplectic and Contact Topology

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.

  1. Symplectic Manifolds. 1,2
    1. Basics.
      1. Linear algebra and symplectic forms.
      2. Darboux theorem and Moser stability theorem.
      3. Compatible almost complex structures.
      4. Symplectic and Lagrangian submanifolds.
    2. Examples.
      1. Cotangent bundle, Kaehler manifolds.
      2. Kodaira-Thurston examples.
      3. Gluing constructions.
    3. J-holomorphic curves.
  2. Contact manifolds--introduction.
    1. Basics, especially in dimension 3.3,4
      1. Darboux theorem and Gray stability theorem.
      2. Induced foliation on an embedded surface.
      3. Legendrian knots; Thurson-Bennequin invariant.
      4. Euler class of a contact structure.
    2. Examples.
      1. R3, S3, unit cotangent bundle.
      2. All 3-manifolds have contact structures.
      3. Constructions of 3-manifolds: surgery, open books. 5
      4. Techniques of Lutz, and Thurston-Winkelnkemper. 6
    3. Relation to symplectic manifolds and Stein manifolds. 7, 8, 9,10
      1. J-convexity, holomorphic and symplectic fillability.
      2. Morse theory of Stein manifolds--outline.
    4. The basic dichotomy in dimension 3: Tight vs. overtwisted contact structures.
  3. Overtwisted structures.
    1. Lutz twist.
    2. Eliashberg's flexibility theorem: homotopic overtwisted structures are isotopic.
    3. Sketch of proof for B. 11, 12
  4. Tight structures.
    1. The standard structure on S3 is tight, which follows from:
    2. Holomorphically fillable structures are tight, which follows from:
    3. Symplectically fillable structures are tight.
    4. Outline of B. 13
      1. Bishop's theorem on existence of holomorphic disks.
      2. Filling by holomorphic disks.
      3. Pseudoholomorphic disks and part C.
    5. Eliashberg rigidity theorem: Uniqueness for tight structures on S3.14
  5. Bennequin's inequality for tight contact manifolds. 15,16
    1. Gromov/Eliashberg's proof.
    2. Applications to knot theory.
    3. Comparison with taut foliations and work of Gabai/Thurston. 17,18
  6. Sketch of relations to Seiberg-Witten theory.

References

1 D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Press (1996), Chapters 1-4 and 6.

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. (Preprint, 1997).

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.

12 Giroux.

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.

14 Giroux.

15 Giroux.

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.