Summary: Topology of Symplectomorphism Groups of
× S 2
Miguel Abreu #
December 4, 1996
In dimension 4, due to nonexistence of adequate tools, very little is known
about the topology of groups of di#eomorphisms. For example, it is unknown
if the group of compactly supported di#eomorphisms of R 4 is connected.
The situation is much better if one wants to study groups of symplectomor
phisms. This is due to the existence of powerful tools, going by the name of
``pseudoholomorphic curve techniques'' and introduced in symplectic geometry
by M.Gromov in his seminal paper of 1985 . Gromov proved in that paper,
among several other remarkable results, the contractibility of the group of com
pactly supported symplectomorphisms of R 4 with its standard symplectic form
# dy 1 + dx 2
# dy 2 .