Editing Symplectic geometry (section)

==Examples and structures==
Every [[Kähler manifold]] is also a symplectic manifold. Well into the 1970s, symplectic experts were unsure whether any compact non-Kähler symplectic manifolds existed, but since then many examples have been constructed (the first was due to [[William Thurston]]); in particular, [[Robert Gompf]] has shown that every [[finitely presented group]] occurs as the [[fundamental group]] of some symplectic 4-manifold, in marked contrast with the Kähler case.

Most symplectic manifolds, one can say, are not Kähler; and so do not have an integrable [[Linear complex structure|complex structure]]  compatible with the symplectic form. [[Mikhail Gromov (mathematician)|Mikhail Gromov]], however, made the important observation that symplectic manifolds do admit an abundance of compatible [[almost complex structure]]s, so that they satisfy all the axioms for a Kähler manifold ''except'' the requirement that the [[transition map]]s be [[Holomorphic function|holomorphic]].

Gromov used the existence of almost complex structures on symplectic manifolds to develop a theory of [[pseudoholomorphic curve]]s,<ref>Gromov, Mikhael. "Pseudo holomorphic curves in symplectic manifolds." Inventiones mathematicae 82.2 (1985): 307–347.</ref> which has led to a number of advancements in symplectic topology, including a class of symplectic invariants now known as [[Gromov–Witten invariant]]s. Later, using the pseudoholomorphic curve technique [[Andreas Floer]] invented another important tool in symplectic geometry known as the [[Floer homology]].<ref>Floer, Andreas. "Morse theory for Lagrangian intersections." Journal of differential geometry 28.3 (1988): 513–547.</ref>