AI 中文总结
本文研究带奇数孔曲面的 $SU(2)$ 平坦联络模空间的辛性质，证明映射类群到辛映射类群的同态在维数大于2时是单射，并给出5点爆破射影平面中拉格朗日球面的完全分类及其可移动性判定。
AI 中文摘要
对于具有奇数个孔的曲面，在每个孔周围具有无迹和乐（traceless holonomy）的平坦 $SU(2)$ 联络的模空间是一个辛流形。当模空间非空时，存在一个从带孔曲面的映射类群到该模空间的辛映射类群的自然同态。本文证明了该同态是单射当且仅当模空间的维数大于 $2$。这推广了 Seidel 和 Wehrheim--Woodward 的工作。此外，本文还给出了在射影平面中爆破 $5$ 个点所得的单调辛结构下的拉格朗日球面的完全分类，该射影平面是 $5$ 孔球面的模空间。进一步地，本文确定了两个这样的拉格朗日球面何时可以通过辛同痕（symplectic isotopy）被移动分开。关于 $\mathbb{C}\mathbb{P}^5$ 中两个二次曲面交集内的拉格朗日球面，也获得了相关结果。证明涉及瞬子 Floer 理论和关于 Heegaard 分解的结果。一个主要的技术结果建立了 $SU(2)$ 模空间的任何哈密顿同痕可以通过用于瞬子同调的全纯扰动（holonomy perturbations）来逼近。
英文摘要
For a surface with an odd number of punctures, the moduli space of flat $SU(2)$ connections with traceless holonomy around each puncture is a symplectic manifold. When the moduli space is nonempty, there is a natural homomorphism from the mapping class group of the punctured surface to the symplectic mapping class group of this moduli space. It is shown that this homomorphism is injective if and only if the dimension of the moduli space is greater than $2$. This generalizes work of Seidel and Wehrheim--Woodward. Also given is a complete classification of Lagrangian spheres in the projective plane blown up at $5$ points with its monotone symplectic structure, which is the moduli space for the 5-punctured sphere. Furthermore, it is determined when two such Lagrangian spheres can be displaced by a symplectic isotopy. Results are also obtained regarding Lagrangian spheres in the intersection of two quadrics in $\mathbb{C}\mathbb{P}^5$. The proofs involve instanton Floer theory and results on Heegaard splittings. A main technical result establishes the approximation of any Hamiltonian isotopy of the $SU(2)$ moduli space by holonomy perturbations which are used in instanton homology.
Comments76 pages, 12 figures
