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Symplectic Topology Ivan Smith Lent 2006 L1: Introduction Riemannian geometry is the geometry of a nondegenerate symmetric bilinear form: we equip each tangent space TxM of a manifold M with such a pairing. Symplectic geometry is the geometry of a non-degenerate skew-symmetric bilinear form, which moreover we insist is “locally constant” as we vary from tangent space to tangent space. Formally, fix the skew form Ω = diag ÃÃ 0 1 −1 0 ! ··· Ã 0 1 −1 0 !! on R 2n, i.e. this is the 2n × 2n matrix which in 2 × 2 blocks just has these equal diagonal entries. The group Sp2n(R ) ≤ GL2n(R ) is the subgroup of matrices which preserve this form: AtΩA = Ω. A symplectic manifold is a 2nmanifold with an atlas of charts for which the transition maps are diffeomorphisms of R 2n whose derivatives belong to Sp2n(R ). Examples: Euclidean space R 2n is trivially symplectic, and is the “local model” for all symplectic manifolds: there are no local invariants, in contrast to curvature in Riemannian geometry. Open subsets and products of symplectic manifolds inherit symplectic structures. Motivation: If a particle moves in R n with force Φ = ∂U/∂q, for a potential U , Newton’s law is: q̈ = −∂U/∂q. Letting H = U + q̇ 2/2 be total energy and p = q̇, Hamilton’s equations: ∂H/∂p = q̇; −∂H/∂q = ṗ for (q, p) ∈ R 2n the position and momentum. We are used to conservation of energy: d ∂H ∂H H(p, q) = q̇ + ṗ = 0 dt ∂q ∂p But the flow also preserves the symplectic structure of R 2n; time evolution in classical mechanics is a map R → Symp(R 2n). A symplectic matrix has determinant +1, so symplectic manifolds have canonically defined volume forms: measure the volume of a subset by covering it in patches and using the measure on R 2n. The choice of charts doesn’t matter since changing charts Uα 7→ φαβ Uβ multiplies the “local integrals” by det(dφαβ ) = 1. Theorem (Moser): (i) two such volume forms on a closed manifold are equivalent if and only if they have the same total volume; (ii) if U , V are open subsets of R k then there is a volumepreserving embedding U ,→ V if and only if vol(U ) ≤ vol(V ). Theorem (Gromov): There is no symplectic embedding B 2n(R) ,→ B 2(r) × R 2n−2 if R > r. This is called the non-squeezing theorem. This amounts to saying symplectic geometry can’t be reduced to volume-preserving geometry, and will come at the end of the course. Motivations from symmetry: suppose we try and classify groups acting locally on R k for which (i) the group acts locally transitively (or we could just reduce dimension to an orbit) (ii) the group has no invariant “foliation”: it’s not of the form (x, y) 7→ φ(x, y) = (f (x), g(x, y)) for R k = R l × R k−l (or simplify by φ 7→ f ). Theorem (Lie): If such a symmetry group is finite-dimensional and compact it’s SO(n), or SU (n) or SP (n) or one of a finite list of exceptions. [Drop compactness: get SO(p, q), SO(n, C ) etc, but still finitely many families.] Theorem (Cartan): If the symmetry group is infinite-dimensional it’s Diff(R k ) or Vol(R k ), or Symp(R 2k ) or Cont(R 2k+1), or a conformal analogue, with no exceptional cases. Symplectic geometry, and it’s odd-dimensional cousin “contact geometry”, are hence very natural, but we won’t come back to this. Motivations from topology: four-dimensional geometry is very special. For instance, R 2n has a unique smooth structure if n 6= 2, but R 4 has uncountably many. In the world of closed simply-connected manifolds, in a given homeomorphism type in dimension k 6= 4 there are at most finitely many diffeomorphism types – and (complicated) homotopy invariants distinguish these – but in dimension 4 there are often infinitely many diffeomorphism types, and no known homotopy interpretation of how to separate them. Conjecture (Donaldson): If X, Y are homeomorphic symplectic four-manifolds, then X is diffeomorphic to Y if and only if X × S 2 and Y × S 2 are deformation equivalent as symplectic manifolds. i.e. symplectic geometry in dimension 6 should – at least, often does – capture exotic smooth geometry in dimension 4. Quantum cohomology: a symplectic manifold has a natural deformation of the ring structure in H ∗(X, R ). Recall the intersection pairing in cohomology is non-degenerate so a cohomology class A is completely determined by giving hA, ci ∈ Z for every c. If now A is itself a cup-product A = a · b then we see that the “three-point function” (a, b, c) 7→ ha · b, ci ∈ Z completely determines the cup-product. Think of this via intersections of cycles: We’re counting intersection points, i.e. maps P1 → X which are constant and have image inside Ca ∩ Cb ∩ Cc. Generalise: count nonconstant “holomorphic” maps P1 → X which take 0 7→ Ca, 1 7→ Cb and ∞ 7→ Cc. We think of this as a deformation: count maps with area N as terms of order εN in an expansion... Plan of the course: Part 1: Background – differential forms on real and complex manifolds, almost complex structures, the first Chern class. Part 2: Moser methods – basic symplectic geometry. What do symplectic manifolds look like locally ? Which manifolds are symplectic What are special symplectic submanifolds? Part 3: Constructions – how to build symplectic manifolds by surgery, and take quotients by group actions, and so forth. We’ll state without proof a “topological characterisation” of symplectic manifolds. Part 4: Holomorphic curves – invariants of symplectic manifolds, by counting certain “minimal surfaces” inside them. We’ll prove the non-squeezing theorem, deduce a “topological characterisation” of symplectic diffeomorphisms, and define quantum cohomology.