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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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:

Let $M$ be either $S^2\times S^2$ or the one point blow-up ${\mathbb{C}}P^2\char93 \overline{{\mathbb{C}}P}^2$ of ${\mathbb{C}}P^2$. In both cases $M$ carries a family of symplectic forms $\omega_{\lambda}$, where $\lambda > -1$ determines the cohomology class $[\omega_\lambda]$. This paper calculates the rational (co)homology of the group $G_\lambda$ of symplectomorphisms of $(M,\omega_\lambda)$ as well as the rational homotopy type of its classifying space $BG_\lambda$. It turns out that each group $G_\lambda$ contains a finite collection $K_k, k = 0,\dots,\ell = \ell(\lambda)$, 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 $\lambda\to \infty$. However, for each fixed $\lambda$ there is essentially one nonvanishing product that gives rise to a ``jumping generator" $w_\lambda$ in $H^*(G_\lambda)$ and to a single relation in the rational cohomology ring $H^*(BG_\lambda)$. An analog of this generator $w_\lambda$ was also seen by Kronheimer in his study of families of symplectic forms on $4$-manifolds using Seiberg-Witten theory. Our methods involve a close study of the space of $\omega_\lambda$-compatible almost complex structures on $M$.


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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: http://dx.doi.org/10.1090/S0894-0347-00-00344-1
PII: S 0894-0347(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