Topology of Symplectomorphism Groups of S 2 × S 2

Content Provider	Semantic Scholar
Author	Abreu, Miguel
Copyright Year	1998
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. Let Mλ 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 Técnico, Lisbon, Portugal. Partially supported by JNICT Programas Ciência and Praxis, Fundação 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
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Alternate Webpage(s)	http://www.math.ist.utl.pt/~mabreu/preprints/symp.ps
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Language	English
Access Restriction	Open
Subject Keyword	Altran Praxis Anatomy, Regional Pseudo brand of pseudoephedrine Subgroup A Nepoviruses SxS Symplectic integrator manifold
Content Type	Text
Resource Type	Article