Fukaya categories of surfaces, spherical objects and mapping class groups

Fukaya categories of surfaces, spherical objects and mapping class groups

Blog Article

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 Mini Dress 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 Shoe Inserts 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.