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

TL;DR

The paper establishes that the extended mapping class group action on the odd character variety $M_g$ (the moduli space of flat $SO(3)$ connections with fixed odd determinant) is faithful for genus $g\ge 2$, via the injectivity of $\widehat{\rho}:\widehat{\Gamma}_g\to\pi_0\text{Symp}(M_g)$; for $g\ge 3$ this action is also faithful on the monotone Fukaya category. The genus‑2 case is shown to have kernel contained in an order‑2 subgroup generated by a hyperelliptic involution, extending Smith’s $g=2$ result to all genera. The proof blends instanton Floer homology, a version of the Atiyah–Floer conjecture, and Clarkson’s Heegaard–Floer strategy to connect 3‑manifold topology with symplectic and categorical actions. The work strengthens the bridge between mapping class group actions on character varieties and their Fukaya-categorical avatars, and points to generalizations to higher rank and parabolic/extended moduli spaces.

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.

Figures (2)

• Figure 1: The compression body $H^\circ$ is a genus $g$ handlebody (here $g=3$) with an open neighborhood of a core $1$-handle removed. The boundary created by this removal is the $2$-torus $T$, which in the picture sits interior to the outer boundary $\Sigma_g$. The circle $w'$ is another core, and $w"$ is an arc from $T$ to $\Sigma_g$.
• Figure 2: Schematic diagrams of $Y_\phi$, $Y_{\phi}^\circ$ and $Y_\phi'$. First, $Y_{\phi}$ is obtained by gluing $H_1^\circ$ to $H_2^\circ$ along $\Sigma_g$ using $\phi$ and along $T$ using the identity. $Y_\phi^\circ$ is obtained from $Y_\phi$ by cutting along $T$. $Y_\phi'$ is obtained from $Y_\phi^\circ$ by filling in the resulting $T$-components; it is the Heegaard splitting formed by the two handlebodies $H_1$ and $H_2$ glued along $\Sigma_g$ via $\phi$. The curves $w_1=w_1'\cup w_1"\subset H_1^\circ$ and $w_2=w_2"\subset H_2^\circ$ are also indicated.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Remark 2.2
• Theorem 2.3: dfl
• Lemma 2.4
• proof
• Proposition 2.5
• proof