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Topology of symplectomorphism groups of rational ruled Miguel Abreu

Summary: Topology of symplectomorphism groups of rational ruled
Miguel Abreu 
Instituto Superior Tecnico, Lisbon, Portugal
Dusa McDu y
State University of New York at Stony Brook, USA
October 6, 1999, revised May 13, 2000
1991 Mathematics Subject Classi cation: 53C15, 57S05.
Let M be either S 2  S 2 or the one point blow-up C P 2 # C P 2 of C P 2 . In both
cases M carries a family of symplectic forms ! , where  > 1 determines the co-
homology class [! ]. This paper calculates the rational (co)homology of the group
G of symplectomorphisms of (M; ! ) as well as the rational homotopy type of its
classifying space BG . It turns out that each group G contains a nite collection
Kk ; k = 0; : : : ; ` = `(), of nite dimensional Lie subgroups that generate its homo-
topy. We show that these subgroups \asymptotically commute", i.e. all the higher
Whitehead products that they generate vanish as  ! 1. However, for each xed 
there is essentially one nonvanishing product that gives rise to a \jumping generator"


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


Collections: Mathematics