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Topology of Symplectomorphism Groups of Miguel Abreu #

Summary: Topology of Symplectomorphism Groups of
S 2
× S 2
Miguel Abreu #
Stanford University
final version
December 4, 1996
0 Introduction
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 [5]. 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
dx 1
# dy 1 + dx 2
# dy 2 .


Source: Abreu, Miguel - Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa


Collections: Mathematics