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arXiv · 1911.11073

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

Jun Li·Tian-Jun Li·Weiwei Wu
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Abstract

For $(\mathbb{C} P^2 \# 5{\overline {\mathbb{C} P^2}},ω)$, let $N_ω$ 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 $N_ω>8$; it is $π_0(Diff^+(S^2,5))$ if $N_ω=0$ (by Paul Seidel and Jonathan Evans); it is $π_0(Diff^+(S^2,4))$ in the remaining case. Further, we completely determine the rank of $π_1(Symp(\mathbb{C} P^2 \# 5{\overline {\mathbb{C} P^2}}, ω)$ 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.

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Jun Li, Tian-Jun Li, Weiwei Wu. 2019-11-25. Symplectic $(-2)$-spheres and the symplectomorphism group of small rational 4-manifolds, II. https://arxiv.org/abs/1911.11073

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