The mapping class group action on the odd character variety is faithful

Abstract

The odd character variety of a Riemann surface is a moduli space of SO(3) representations of the fundamental group which can be interpreted as the moduli space of stable holomorphic rank 2 bundles of odd degree and fixed determinant. This is a symplectic manifold, and there is a homomorphism from a finite extension of the mapping class group of the surface to the symplectic mapping class group of this moduli space. When the genus is at least 2, it is shown that this homomomorphism is injective. This answers a question posed by Dostoglou and Salamon and generalizes a theorem of Smith from the genus 2 case to arbitrary genus. A corresponding result on the faithfulness of the action on the Fukaya category of the odd character variety is also proved. The proofs use instanton Floer homology, a version of the Atiyah-Floer Conjecture, and aspects of a strategy used by Clarkson in the Heegaard Floer setting.