Summary: Symplectic 4-manifolds, singular
plane curves, and isotopy problems
Massachusetts Inst. of Technology and Ecole Polytechnique
A symplectic structure on a smooth manifold is a 2-form such that d = 0 and
· · · is a volume form.
, 0 = dxi dyi.
(Darboux: every symplectic manifold is locally (R2n
, 0), i.e. there are no local
Example: Riemann surfaces (, vol); CPn
; complex projective manifolds.
The symplectic category is strictly larger (Thurston 1976).
Gompf 1994: G finitely presented group (X4
, ) compact symplectic such that
1(X) = G.
Symplectic manifolds are not always complex, but they are almost-complex, i.e.
there exists J End(TX) such that