Topology of symplectomorphism groups of rational ruled surfaces
Authors:
Miguel Abreu and Dusa McDuff
Journal:
J. Amer. Math. Soc. 13 (2000), 9711009
MSC (2000):
Primary 57S05, 57R17; Secondary 53D35
Published electronically:
June 23, 2000
MathSciNet review:
1775741
Fulltext PDF Free Access
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Abstract: Let be either or the one point blowup 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 SeibergWitten 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, 1049001 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 117943651
Email:
dusa@math.sunysb.edu
DOI:
http://dx.doi.org/10.1090/S0894034700003441
PII:
S 08940347(00)003441
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.
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© Copyright 2000 American Mathematical Society
