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arXiv · 2006.09689

Fukaya categories of surfaces, spherical objects, and mapping class groups

Denis Auroux·Ivan Smith
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Abstract

We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least two whose Chern character represents a non-zero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank one 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.

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Denis Auroux, Ivan Smith. 2020-06-17. Fukaya categories of surfaces, spherical objects, and mapping class groups. https://arxiv.org/abs/2006.09689

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