Topology of symplectomorphism groups of rational ruled surfaces

Authors:
Miguel Abreu and Dusa McDuff

Journal:
J. Amer. Math. Soc. **13** (2000), 971-1009

MSC (2000):
Primary 57S05, 57R17; Secondary 53D35

Published electronically:
June 23, 2000

MathSciNet review:
1775741

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Abstract | References | Similar Articles | Additional Information

Let be either or the one point blow-up of . In both cases carries a family of symplectic forms , where determines the cohomology class . This paper calculates the rational (co)homology of the group of symplectomorphisms of as well as the rational homotopy type of its classifying space . It turns out that each group contains a finite collection , of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups ``asymptotically commute", i.e. all the higher Whitehead products that they generate vanish as . However, for each fixed there is essentially one nonvanishing product that gives rise to a ``jumping generator" in and to a single relation in the rational cohomology ring . An analog of this generator was also seen by Kronheimer in his study of families of symplectic forms on -manifolds using Seiberg-Witten theory. Our methods involve a close study of the space of -compatible almost complex structures on .

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Additional Information

**Miguel Abreu**

Affiliation:
Department of Mathematics, Instituto Superior Técnico, 1049-001 Lisbon, Portugal

Email:
mabreu@math.ist.utl.pt

**Dusa McDuff**

Affiliation:
Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651

Email:
dusa@math.sunysb.edu

DOI:
https://doi.org/10.1090/S0894-0347-00-00344-1

Received by editor(s):
October 25, 1999

Received by editor(s) in revised form:
May 13, 2000

Published electronically:
June 23, 2000

Additional Notes:
The first author was partially supported by NSF grant DMS 9304580, while at the Institute for Advanced Study (1996/97), and afterwards by FCT grant PCEX/C/MAT/44/96 and PRAXIS XXI through the Research Units Pluriannual Funding Program

The second author was partially supported by NSF grant DMS 9704825.

Article copyright:
© Copyright 2000
American Mathematical Society