Representation varieties and genus-three Torelli maps

TL;DR

The paper analyzes SU(2) representation and character varieties for mapping tori of genus‑three Torelli maps, focusing on powers of a bounding-pair map. Through an explicit action of the Torelli-induced automorphism on the genus‑3 fundamental group and a reduction to a tractable parameter space governed by commutator data, it derives precise counts of connected components for both the representation and character varieties of the mapping tori and for the fixed-point sets under pullback. The main results show that ${\mathcal{R}}(\Sigma_{\Phi^n})$ and ${\mathfrak{X}}(\Sigma_{\Phi^n})$ have $n^2{+}1$ components when $n$ is even and $n^2$ components when $n$ is odd, while the fixed-point sets ${\mathrm{Fix}_{\mathcal{R}}}((\Phi^n)^*)$ and ${\mathrm{Fix}_{\mathfrak{X}}}((\Phi^n)^*)$ have $\left\lfloor \frac{n^2}{2} \right\rfloor{+}1$ (even) or $\frac{n^2{+}1}{2}$ (odd) components. A key device is the connectedness of commutator fibers in $\mathrm{SU}(2)$ and the decomposition into auxiliary spaces whose fibers are controlled by the commutator map. The results provide a constructive description of representations across components and establish a bridge between representation-theoretic fixed points and torus topology, with potential extension to other Lie groups sharing the same centralizer properties.

Abstract

We consider the family of Torelli homeomorphisms on a genus-three surface given by powers of a fixed bounding pair map. For each such homeomorphism $φ$ we determine the number of connected components of the fixed point set of the induced map on the representation variety of the surface, as well as the number of connected components of the representation variety of the mapping torus of $φ$.

Figures (2)

• Figure 1: Illustrated here are the (unoriented) curves $\gamma_1, \gamma_2$ relative to which the bounding pair map $\Phi = T_{\gamma_1} \circ T_{\gamma_2}^{-1}$ is formed. Note that the basepoint $x_0$ appears here to the left of $\gamma_2$ (this is the same basepoint from Figure \ref{['fig:2']}, and appears on the "$\alpha_2$-$\beta_2$" side of $\gamma_2$). Since $x_0$ does not lie on either of $\gamma_1$ or $\gamma_2$, we can define the Dehn twists $T_{\gamma_i}$ to have support near $x_0$, and thus $\Phi(x_0) = x_0$.
• Figure 2: Illustrated here is our generating set $\alpha_i, \beta_i$ for the fundamental group of our genus-three surface $\Sigma$. These all have basepoint $x_0$ (in the middle of the figure) and, with the orientations as indicated, these generators satisfy the relation $\prod_{i = 1}^3 [\alpha_i, \beta_i ] = 1$. As such, these give a standard presentation for $\pi_1(\Sigma, x_0)$.

Theorems & Definitions (20)

• Remark
• Lemma 1
• proof
• Lemma 2
• Proposition 3
• Proposition 4
• Proposition 5
• proof : Proof Sketch
• Corollary 6
• Lemma 7