  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.
