Fukaya categories of surfaces, spherical objects and mapping class groups
Fukaya categories of surfaces, spherical objects and mapping class groups
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表面、球形物体和映射类组的 Fukaya 类别
DOI：
10.1017/fms.2021.21
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Smith, Ivan
中科院分区：
文献类型：
--
作者：
Auroux, Denis;Smith, Ivan
证明了亏格闭曲面的Fukaya范畴中的每个球对象，如果其Chern特征标表示一个非零的Hochschild同调类，则其拟同构于一条赋有秩局部系的简单闭曲线.(同源假设是必要的。)这在很大程度上回答了海登、卡扎尔科夫和康采维奇的问题。由此可以得出从Fukaya范畴的自等价群到映射类群的自然满射。这些证明吸引并说明了许多最近的发展：包裹范畴的箭图代数模型、Fukaya范畴的映射、有限和连续群作用的等变Floer理论以及同调镜像对称性。给出了它在高维辛映射类群中的应用。
We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank 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.