Remote Access Journal of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information


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 $M$.

References [Enhancements On Off] (What's this?)

  • 1. M. Abreu, Topology of symplectomorphism groups of $S^2\times S^2$, Inv. Math., 131 (1998), 1-23. MR 99k:57065
  • 2. C. Allday, Rational Whitehead products and a spectral sequence of Quillen, Pac. Journ. Math., 46 (1973), 313-323. MR 48:12519
  • 3. C. Allday, Rational Whitehead products and a spectral sequence of Quillen, II, Houston J. Math., 3 (1977), 301-308. MR 57:13935
  • 4. P. Andrews and M. Arkowitz, Sullivan's minimal models and higher order Whitehead products, Can. J. Math. XXX (1978), 961-982. MR 80b:55008
  • 5. M. Audin, The topology of torus actions on symplectic manifolds, Progress in Math. 93, Birkhäuser, 1991. MR 92m:57046
  • 6. M. Audin and F. Lafontaine (eds.), Holomorphic Curves in Symplectic Geometry, Progress in Math. 117, Birkhäuser, 1994. MR 95i:58005
  • 7. M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Inv. Math., 82 (1985), 307-347. MR 87j:53053
  • 8. P. Iglesias, Les $SO(3)$-variétés symplectiques et leur classification en dimension $4$, Bull. Soc. Math. France, 119 (1991), 371-396. MR 92i:57023
  • 9. P. Kronheimer, Some non-trivial families of symplectic structures, Harvard preprint, 1998.
  • 10. F. Lalonde and D. McDuff, $J$-holomorphic spheres and the classification of rational and ruled symplectic $4$-manifolds, in Contact and Symplectic Geometry, ed. C.Thomas, Cambridge University Press (1996). MR 98d:57045
  • 11. D. McDuff, From symplectic deformation to isotopy, in Topics in Symplectic Topology (Irvine, CA 1996), Internat. Press, Cambridge MA.
  • 12. D. McDuff, Almost complex structures on $S^2\times S^2$, Duke. Math. Journal, 101 (2000), 135-177. CMP 2000:07
  • 13. D. McDuff, Symplectomorphism groups of ruled surfaces, in preparation.
  • 14. D. McDuff and D.A. Salamon, Introduction to Symplectic Topology, 2nd edition, Oxford University Press, 1998. MR 2000g:53098
  • 15. D. McDuff and D.A. Salamon, $J$-holomorphic curves and quantum cohomology, University Lecture Series 6, American Mathematical Society, Providence, RI, 1994. MR 95g:58026
  • 16. D. Salamon, Seiberg-Witten invariants of mapping tori, symplectic fixed points and Lefschetz numbers, Proceedings of 5th Gökova Topology conference, Turkish Journal of Mathematics, (1998), 1-27.
  • 17. P. Seidel, Floer homology and the symplectic isotopy problem, D. Phil. thesis, Oxford (1997).
  • 18. G.W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Math. 61, Springer-Verlag (1978). MR 80b:55001

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 57S05, 57R17, 53D35

Retrieve articles in all journals with MSC (2000): 57S05, 57R17, 53D35

Additional Information

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

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

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