## Topology of symplectomorphism groups of rational ruled surfaces

HTML articles powered by AMS MathViewer

- by Miguel Abreu and Dusa McDuff
- J. Amer. Math. Soc.
**13**(2000), 971-1009 - DOI: https://doi.org/10.1090/S0894-0347-00-00344-1
- Published electronically: June 23, 2000
- PDF | Request permission

## Abstract:

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 $M$.## References

- Miguel Abreu,
*Topology of symplectomorphism groups of $S^2\times S^2$*, Invent. Math.**131**(1998), no. 1, 1–23. MR**1489893**, DOI 10.1007/s002220050196 - Christopher Allday,
*Rational Whitehead products and a spectral sequence of Quillen*, Pacific J. Math.**46**(1973), 313–323. MR**334200**, DOI 10.2140/pjm.1973.46.313 - Christopher Allday,
*Rational Whitehead products and a spectral sequence of Quillen. II*, Houston J. Math.**3**(1977), no. 3, 301–308. MR**474288** - Peter Andrews and Martin Arkowitz,
*Sullivan’s minimal models and higher order Whitehead products*, Canadian J. Math.**30**(1978), no. 5, 961–982. MR**506254**, DOI 10.4153/CJM-1978-083-6 - Michèle Audin,
*The topology of torus actions on symplectic manifolds*, Progress in Mathematics, vol. 93, Birkhäuser Verlag, Basel, 1991. Translated from the French by the author. MR**1106194**, DOI 10.1007/978-3-0348-7221-8 - Michèle Audin and Jacques Lafontaine (eds.),
*Holomorphic curves in symplectic geometry*, Progress in Mathematics, vol. 117, Birkhäuser Verlag, Basel, 1994. MR**1274923**, DOI 10.1007/978-3-0348-8508-9 - M. Gromov,
*Pseudo holomorphic curves in symplectic manifolds*, Invent. Math.**82**(1985), no. 2, 307–347. MR**809718**, DOI 10.1007/BF01388806 - Patrick Iglésias,
*Les $\textrm {SO}(3)$-variétés symplectiques et leur classification en dimension $4$*, Bull. Soc. Math. France**119**(1991), no. 3, 371–396 (French, with English summary). MR**1125672**, DOI 10.24033/bsmf.2172
K P. Kronheimer, Some non-trivial families of symplectic structures, Harvard preprint, 1998.
- François Lalonde and Dusa McDuff,
*$J$-curves and the classification of rational and ruled symplectic $4$-manifolds*, Contact and symplectic geometry (Cambridge, 1994) Publ. Newton Inst., vol. 8, Cambridge Univ. Press, Cambridge, 1996, pp. 3–42. MR**1432456**
M2 D. McDuff, From symplectic deformation to isotopy, in - Dusa McDuff and Dietmar Salamon,
*Introduction to symplectic topology*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. MR**1698616** - Dusa McDuff and Dietmar Salamon,
*$J$-holomorphic curves and quantum cohomology*, University Lecture Series, vol. 6, American Mathematical Society, Providence, RI, 1994. MR**1286255**, DOI 10.1090/ulect/006
Sa D. Salamon, Seiberg–Witten invariants of mapping tori, symplectic fixed points and Lefschetz numbers, Proceedings of 5th Gökova Topology conference, - George W. Whitehead,
*Elements of homotopy theory*, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR**516508**, DOI 10.1007/978-1-4612-6318-0

*Topics in Symplectic Topology (Irvine, CA 1996)*, Internat. Press, Cambridge MA. M5 D. McDuff, Almost complex structures on $S^2\times S^2$,

*Duke. Math. Journal*,

**101**(2000), 135–177. Mc D. McDuff, Symplectomorphism groups of ruled surfaces, in preparation.

*Turkish Journal of Mathematics*, (1998), 1–27. Sd P. Seidel, Floer homology and the symplectic isotopy problem, D. Phil. thesis, Oxford (1997).

## Bibliographic 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
- MR Author ID: 190631
- Email: dusa@math.sunysb.edu
- 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: https://doi.org/10.1090/S0894-0347-00-00344-1
- MathSciNet review: 1775741