Symplectic $(-2)$-spheres and the symplectomorphism group of small rational 4-manifolds, II

Abstract

For $(\mathbb{C}{P}^2\#5\overline{\mathbb{C}{P}^2},\omega )$, let $N_{\omega }$ be the number of $(-2)$-symplectic spherical homology classes.We completely determine the Torelli symplectic mapping class group (Torelli SMCG): the Torelli SMCG is trivial if $N_{\omega }>8$; it is ${\pi }_0(Dif{f}^{+}(S^2,5))$ if $N_{\omega }=0$ (by Paul Seidel and Jonathan Evans); it is ${\pi }_0(Dif{f}^{+}(S^2,4))$ in the remaining case. Further, we completely determine the rank of ${\pi }_1(Symp(\mathbb{C}{P}^2\#5\overline{\mathbb{C}{P}^2},\omega )$ for any given symplectic form. Our results can be uniformly presented regarding Dynkin diagrams of type $\mathbb{A}$ and type $\mathbb{D}$ Lie algebras. We also provide a solution to the smooth isotopy problem of rational $4$-manifolds.

arXiv comment44 pages, 5 tables, 8 figures. Comments welcome. This paper is a split from the original version of the arXiv paper arXiv:1611.07436 (replaced accordingly), containing and strengthening the results for 5-point blowup of $\mathbb{C}{P}^2$, adding a new proof the the smooth isotopy theorem

References in corpus (8)

• Growth of maps, distortion in groups and symplectic geometry
• The topology of the space of symplectic balls in rational 4-manifolds
• Lagrangian spheres, symplectic surfaces and the symplectic mapping class group
• Dehn twists and free subgroups of symplectic mapping class groups
• Symplectomorphism groups and almost complex structures
• Secondary Stiefel-Whitney class and diffeomorphisms of rational and ruled symplectic 4-manifolds
• Stability of the symplectomorphism group of rational surfaces
• Topology of symplectomorphism groups and ball-swappings

Cited by in corpus (4)

• Stability of the symplectomorphism group of rational surfaces
• Topology of symplectomorphism groups and ball-swappings
• Centralizers of Hamiltonian circle actions on rational ruled surfaces
• Symplectic isotopy on non-minimal ruled surfaces