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

TL;DR

This paper investigates the topology of the symplectomorphism group of small rational 4-manifolds by analyzing the space of almost complex structures and the role of symplectic -2 spheres, revealing new stability and variation properties.

Contribution

It introduces a detailed decomposition of the space of almost complex structures and computes the fundamental group rank of the symplectomorphism group based on -2 sphere classes.

Findings

Computed the rank of π₁(Symp(X,ω)) for χ(X) ≤ 7.

Developed a new decomposition approach for the space of almost complex structures.

Provided insights into the stability of the symplectomorphism group under deformations.

Abstract

Let $(X,\omega)$ be a symplectic rational 4 manifold. We study the space of tamed almost complex structures $\mathcal{J}_{\omega}$ using a fine decomposition via smooth rational curves and a relative version of the infinite-dimensional Alexander duality. This decomposition provides new understandings of both the variation and stability of the symplectomorphism group $Symp(X,\omega)$ when deforming $\omega$. In particular, we compute the rank of $\pi_1(Symp(X,\omega))$ with Euler number $\chi(X)\leq 7$ in terms of the number $N$ of -2 symplectic sphere classes.