Let (X, ω) be a symplectic rational surface. We study the space of tamed almost complex structures Jω 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(X, ω) when deforming ω In particular, we compute the rank of π1(Symp(X, ω)) with X(X) ≥ 7 in terms of the number Nω of (-2)-symplectic sphere classes.
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We appreciate useful discussions with Silvia Anjos, Martin Pinsonnault, Weiwei Wu, Weiyi Zhang. For CP2#4CP2, when [ω] is in the face M4OBC, Anjos and Eden  computed the rational homotopy groups using the toric method (see Remark 4.10). Results in Section 3A overlap with results in Section 4.1 in [Zhang 2017], which are in a slightly different context. We thank the referees for their detailed and constructive comments, which greatly improved the exposition of this article. The research is supported by NSF grant DMS-1611680.
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- Almost complex structure
- Lagrangian root system
- Rational symplectic manifold
- Symplectomorphism group