A differential topology proof that the $SU(2)$ character variety of the genus two surface is homeomorphic to ${\mathbb C} P^3$

TL;DR

This paper proves that the $SU(2)$ character variety of a genus two surface is homeomorphic to ${ m extbf{C} P}^3$ using differential topology, providing new insights into Lagrangian immersions and their relation to 3-manifolds.

Contribution

It offers a differential topology proof of the homeomorphism between the $SU(2)$ character variety of genus two surface and ${ m extbf{C} P}^3$, avoiding the Narasimhan-Seshadri correspondence.

Findings

$ ext{Character variety is a closed compact manifold}$

$ ext{Homeomorphic to } { m extbf{C} P}^3$

$ ext{Examples of Lagrangian immersions and their properties}

Abstract

We provide a proof that the $SU(2)$ character variety of a genus two surface, $\chi(F_2)$, is a closed compact manifold, and a proof of the Narasimhan-Ramanan theorem that $\chi(F_2)$ is homeomorphic to ${\mathbb C} P^3$. This is done entirely in the language of $SU(2)$ representations, differential topology and elementary algebraic topology. It avoids the Narasimhan-Seshadri correspondence, clarifying the nature of Lagrangian immersions into $\chi(F_2)$ induced by 3-manifolds with genus two boundary. We give examples of such Lagrangian immersions and describe a correspondence from multicurves in the pillowcase to Lagrangian immersions in $\chi(F_2)$, induced by a 2-stranded tangle in a punctured genus 2 handlebody. We give an example of a non-transverse pair of smooth Lagrangians in $\chi(F_2)$ induced by a genus 2 Heegaard splitting of $(S^3,W)$ for the ``linked eyeglasses" web $W$, which are made transverse, and hence the corresponding Chern-Simons function Morse, using Goldman flows/holonomy perturbations along embedded curves in the Heegaard surface.