We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $2$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $1$ local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included.

中文翻译：

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Fukaya 类别的曲面、球形对象和映射类组

我们证明了属闭合面的派生深谷范畴中的每个球形物体至少 $2$ 其 Chern 字符表示非零 Hochschild 同调类与配备秩的简单闭合曲线准同构 $1$ 本地系统。（同调假设是必要的。）这在很大程度上回答了 Haiden、Katzarkov 和 Kontsevich 的问题。由此可见，从深谷范畴的自等价群到映射类群有一个自然的凸出。这些证明吸引并说明了许多最近的发展：包裹类别的颤动代数模型、Fukaya 类别的分层、有限和连续群作用的等变 Floer 理论以及同调镜像对称。包括对高维辛映射类组的应用。