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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Topology of symplectomorphism groups of rational ruled surfaces
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by Miguel Abreu and Dusa McDuff
J. Amer. Math. Soc. 13 (2000), 971-1009
Published electronically: June 23, 2000


Let $M$ be either $S^2\times S^2$ or the one point blow-up ${\mathbb {C}}P^2\#\overline {{\mathbb {C}}P}^2$ 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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Bibliographic Information
  • Miguel Abreu
  • Affiliation: Department of Mathematics, Instituto Superior Técnico, 1049-001 Lisbon, Portugal
  • Email:
  • Dusa McDuff
  • Affiliation: Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651
  • MR Author ID: 190631
  • Email:
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 13 (2000), 971-1009
  • MSC (2000): Primary 57S05, 57R17; Secondary 53D35
  • DOI:
  • MathSciNet review: 1775741