Symplectic topology
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Symplectic topology (also called symplectic geometry) is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with closed, nondegenerate, 2-forms. Symplectic topology has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.
Symplectic topology has a number of similarities and differences with Riemannian geometry, which is the study of differentiable manifolds equipped with nondegenerate, symmetric 2-tensors (called metric tensors). Unlike the Riemannian case, symplectic manifolds have no local invariants such as curvature. This is a consequence of Darboux's theorem which states that every pair of symplectic manifolds are locally isomorphic. Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions. For starters, the manifold must be even-dimensional. Much work in symplectic topology has centered around investigating which manifolds admit symplectic 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 complex structure compatible with the symplectic form. Mikhail Gromov made, however, the important observation that symplectic manifolds do admit an abundance of compatible almost complex structures, so that they satisfy all the axioms for a complex manifold except the requirement that the transition functions be holomorphic.
A Riemann surface mapped into a symplectic manifold compatibly with the almost complex structure is called a pseudoholomorphic curve, and Gromov proved a compactness theorem for such curves; this result has led to the development of a fairly large subdiscipline of symplectic topology. Results arising from Gromov's theory include Gromov's nonsqueezing theorem concerning symplectic embeddings of spheres into cylinders, and also a proof of a conjecture of Vladimir Arnol'd concerning the number of fixed points of Hamiltonian flows. This was proven (in increasing generality) by several researchers beginning with Andreas Floer, who introduced what is now known as Floer homology using Gromov's methods.
Pseudoholomorphic curves are also a source of symplectic invariants, known as Gromov-Witten invariants, by which two different symplectic manifolds could in principle be distinguished.
See also
References
- Dusa McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford University Press, 1998. ISBN 0-198-50451-9.
- A. T. Fomenko, Symplectic Geometry (2nd edition) (1995) Gordon and Breach Publishers, ISBN 2-88124-901-9. (Provides an undergrad level introduction.)