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

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.

38 pages, 4 figures