Jonny Evans : A First Course in Symplectic Topology

Room 501, Department of Mathematics,

UCL, Gower Street, London, WC1E 6BT

j.d.evans • ucl.ac.uk

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 3.0 Licence. That means you're free to use them as long as you give appropriate attribution and release derivatives under an isomorphic licence.

- Overview and motivation (summary, notes)
- Basics: dynamics; symplectic linear algebra (summary, notes)
- Neighbourhoods (summary, notes)
- Lagrangians I (summary, notes)
- Lagrangians II (summary, notes)
- Projective varieties I (summary, notes)
- Projective varieties II (summary, notes)
- Symplectic blow-up (summary, notes)
- Picard-Lefschetz I (summary, notes)
- Picard-Lefschetz II (summary, notes)
- The non-Kähler world (summary, notes)
- Hamiltonian group actions (summary, notes)
- Pseudoholomorphic curves I (summary, notes)
- Pseudoholomorphic curves II (summary, 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.

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).

Lagrangian submanifolds: zero-sections, graphs of closed forms, Weinstein's neighbourhood theorem (some of its corollaries, e.g. orientable embedded Lagrangians in C^2 are tori); Luttinger surgery, unknottedness of Lagrangian tori in C^2.

Completion of Luttinger's proof of unknottedness; recap of Lagrangian Grassmannian, Maslov class; recap of Chern classes and adjunction.

The Fubini-Study form on CP^n, complex projective varieties as symplectic manifolds, adjunction and Chern classes for projective hypersurfaces; topology of surfaces of low degree in CP^3.

Quadrics, cubic surface; blow-ups, change in first Chern class, rationality of quadric and cubic surfaces, general position requirement for blow-up locus.

Lefschetz hyperplane theorem, sketch via plurisubharmonic Morse theory, holomorphic curves and the maximum principle, Lefschetz pencils (examples).

Parallel transport, vanishing cycles, Dehn twists, Picard-Lefschetz formula.

Kodaira-Thurston manifold, McDuff's example. Symplectic fibre sum; Gompf's theorem on fundamental groups. **Comment on Kähler fundamental groups.**

Symplectic cut along a Hamiltonian circle action, blow-up as an example (connection with fibre sum). Torus actions and the moment polytope. Examples: CP^2, blow-up. Reading off geometry from the moment polytope. Convexity. Delzant's theorem.

Definition. Area and energy. Outline of the analytical setting. Gromov compactness. Good properties in four dimensions. Example existence theorem.

Symplectomorphism group of S^2 x S^2; McDuff's Hopf invariant example.