#### Vol. 304, No. 2, 2020

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

### Jun Li and Tian-Jun Li

Vol. 304 (2020), No. 2, 561–606
##### Abstract

Let $\left(X,\omega \right)$ be a symplectic rational surface. We study the space of tamed almost complex structures ${\mathsc{𝒥}}_{\omega }$ using a fine decomposition via smooth rational curves and a relative version of the infinite dimensional Alexander–Pontrjagin duality. This decomposition provides new understandings of both the variation and the stability of the symplectomorphism group $Symp\left(X,\omega \right)$ when deforming $\omega$. In particular, we compute the rank of ${\pi }_{1}\left(Symp\left(X,\omega \right)\right)$ with $\chi \left(X\right)\le 7$ in terms of the number ${N}_{\omega }$ of $\left(-2\right)$-symplectic sphere classes.

##### Keywords
symplectomorphism group, almost complex structure, rational symplectic manifold, Lagrangian root system
##### Mathematical Subject Classification 2010
Primary: 32Q65, 53C15, 53D35, 57R17
Secondary: 53D05, 57S05