Download Symplectic Topology

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.