Topology of symplectomorphism groups of rational ruled surfaces

Abreu, Miguel¹ ; McDuff, Dusa²

¹ Department of Mathematics, Instituto Superior Técnico, 1049-001 Lisbon, Portugal
² Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651

Journal of the American Mathematical Society, Tome 13 (2000) no. 4, pp. 971-1009

Résumé

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 $M$.

Détail

DOI : 10.1090/S0894-0347-00-00344-1

Bibliographie

[1] Abreu, Miguel Topology of symplectomorphism groups of 𝑆²×𝑆² Invent. Math. 1998 1 23

[2] Allday, Christopher Rational Whitehead products and a spectral sequence of Quillen Pacific J. Math. 1973 313 323

[3] Allday, Christopher Rational Whitehead products and a spectral sequence of Quillen. II Houston J. Math. 1977 301 308

[4] Andrews, Peter, Arkowitz, Martin Sullivan’s minimal models and higher order Whitehead products Canadian J. Math. 1978 961 982

[5] Audin, Michèle The topology of torus actions on symplectic manifolds 1991 181

[6] Holomorphic curves in symplectic geometry 1994

[7] Gromov, M. Pseudo holomorphic curves in symplectic manifolds Invent. Math. 1985 307 347

[8] Iglésias, Patrick Les 𝑆𝑂(3)-variétés symplectiques et leur classification en dimension 4 Bull. Soc. Math. France 1991 371 396

[9] Lalonde, François, Mcduff, Dusa 𝐽-curves and the classification of rational and ruled symplectic 4-manifolds 1996 3 42

[10] Mcduff, Dusa, Salamon, Dietmar Introduction to symplectic topology 1998

[11] Mcduff, Dusa, Salamon, Dietmar 𝐽-holomorphic curves and quantum cohomology 1994

[12] Whitehead, George W. Elements of homotopy theory 1978