A First Course in Symplectic Topology

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This is a summary of the content of a course given at ETH Zürich in the Herbstsemester 2010. It was aimed at giving Masters and PhD students a broad overview of this subject with much emphasis on examples and computations and less on general theory. It owes a lot to this course taught by Ivan Smith at DPMMS in 2006 from which I learned most of the material.

All teaching materials available from this site are released under a CC-BY-SA 4.0 Licence. That means you're free to use them as long as you give appropriate attribution and release derivatives under an isomorphic licence.

Lectures

1. Overview and motivation (notes)
   • Complex and symplectic manifolds, integrability conditions and atlases. Examples from algebraic geometry, dynamics, gauge theory. Symplectomorphism group, rigidity, non-squeezing. Pseudoholomorphic curves. Taubes's theorem and applications to low-dimensional topology.
2. Basics (notes)
   • Hamiltonian dynamics: Review the Hamiltonian formulation of classical dynamics in Euclidean space; understand this formulation from the point of view of symplectic geometry. Generalise this to cotangent bundles to illustrate the passing from linear to nonlinear symplectic manifolds; geodesic flow as an example.
   • Linear algebra: Alternating forms; compatible complex structures; the linear symplectic group; the unitary subgroup as a retract; homogeneous spaces and their topology: compatible complex structures, the Lagrangian Grassmannian and the Maslov class; symplectic manifolds and compatible almost complex structures; contractibility of the space of almost complex structures. First Chern class.
3. Neighbourhoods (notes)
   • Moser's argument, Darboux's theorem, symplectic submanifolds: their normal bundles, symplectic neighbourhood theorem; Banyaga's symplectic isotopy extension theorem (and Auroux's version for symplectic submanifolds).
4. Lagrangians I (notes)
   • Lagrangian submanifolds: zero-sections, graphs of closed forms, Weinstein's neighbourhood theorem (some of its corollaries, e.g. orientable embedded Lagrangians in C² are tori); Luttinger surgery, unknottedness of Lagrangian tori in C².
5. Lagrangians II (notes)
   • Completion of Luttinger's proof of unknottedness; recap of Lagrangian Grassmannian, Maslov class; recap of Chern classes and adjunction.
6. Projective varieties I (notes)
   • The Fubini-Study form on CPⁿ, complex projective varieties as symplectic manifolds, adjunction and Chern classes for projective hypersurfaces; topology of surfaces of low degree in CP³.
7. Projective varieties II (notes)
   • Quadrics, cubic surface; blow-ups, change in first Chern class, rationality of quadric and cubic surfaces, general position requirement for blow-up locus.
8. Symplectic blow-up (notes)
   • Symplectic blow-up of a point, formula for change in cohomology class of the symplectic form. Compatibility. Sketch of Gromov's nonsqueezing theorem.
9. Picard-Lefschetz I (notes)
   • Lefschetz hyperplane theorem, sketch via plurisubharmonic Morse theory, holomorphic curves and the maximum principle, Lefschetz pencils (examples).
10. Picard-Lefschetz II (notes)
    • Parallel transport, vanishing cycles, Dehn twists, Picard-Lefschetz formula.
11. The non-Kähler world (notes)
    • Kodaira-Thurston manifold, McDuff's example. Symplectic fibre sum; Gompf's theorem on fundamental groups. **Comment on Kähler fundamental groups.**
12. Hamiltonian group actions (notes)
    • Symplectic cut along a Hamiltonian circle action, blow-up as an example (connection with fibre sum). Torus actions and the moment polytope. Examples: CP², blow-up. Reading off geometry from the moment polytope. Convexity. Delzant's theorem.
13. Pseudoholomorphic curves I (notes)
    • Definition. Area and energy. Outline of the analytical setting. Gromov compactness. Good properties in four dimensions. Example existence theorem.
14. Pseudoholomorphic curves II (notes)
    • Symplectomorphism group of S² x S²; McDuff's Hopf invariant example.