Topology of symplectomorphism groups of S 2 × S 2

Abstract

In dimension 4, due to non-existence of adequate tools, very little is known about the topology of groups of diffeomorphisms. For example, it is unknown if the group of compactly supported diffeomorphisms of R is connected. The situation is much better if one wants to study groups of symplectomorphisms. This is due to the existence of powerful tools, going by the name of “pseudo-holomorphic 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 compactly supported symplectomorphisms of R with its standard symplectic form dx ∧ dy + dx ∧ dy. Gromov also studied the following example. LetMλ be the symplectic manifold (S×S, ωλ = (1+λ)σ0⊕σ0), where 0 ≤ λ ∈ R and σ0 is a standard area form on S with total area equal to 1. Denote by Gλ the group of symplectomorphisms of Mλ that act as the identity on H2(S × S;Z). Gromov proved in [5] that G0 is homotopy equivalent to its subgroup of standard isometries SO(3)×SO(3). He also showed why that would not be true for Gλ with λ > 0, and in 1987 D.McDuff [10] constructed explicitly an element of infinite order in H1(Gλ), λ > 0. In this paper we will give a more detailed description of Gλ, for 0 < λ ≤ 1. In particular we will prove the following: ∗On leave from Instituto Superior Tecnico, Lisbon, Portugal. Partially supported by JNICT Programas Ciencia and Praxis, Fundacao Luso-Americana para o Desenvolvimento and a Sloan Doctoral Dissertation Fellowship. Current address: School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA. E-mail: abreu@math.ias.edu