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

Christopher M. Herald, Paul Kirk

TL;DR

This work provides a self-contained, representation-theoretic proof that the SU(2) character variety χ(F2) of the genus two surface is a closed 6-manifold and is homeomorphic to ${\mathbb C}P^3$, avoiding the Narasimhan–Seshadri correspondence. The strategy hinges on analyzing an involution on χ(S^2,6) whose quotient is χ(F2), establishing a double branched-cover framework, and applying Wall’s classification to identify a CP^3-structure via a controlled decomposition along a separating curve. A central technical feat is matching the κ-level decomposition of χ(F2) with a Morse-Bott decomposition on CP^3 using the real locus ${\mathbb R}P^3$ and a smooth quadric, yielding a diffeomorphism after gluing two tubular neighborhoods. The latter part develops a rich Lagrangian-immersion picture: holonomy perturbations and Goldman flows realize explicit Lagrangian correspondences from 3-manifolds with genus two boundary into χ(F2), illustrated by a suite of tangles, perturbations, and multicurves that produce embedded submanifolds representing homology classes and explicit intersection phenomena. The results connect to Floer-type frameworks by providing concrete Lagrangian data in χ(F2) and show how 3-manifold topology can be encoded inside the CP^3-model of χ(F2), with potential implications for bordered Floer theories and instanton- Floer-type constructions.

Abstract

We provide a proof that the $SU(2)$ character variety of a genus two surface, $χ(F_2)$, is a closed compact manifold, and a proof of the Narasimhan-Ramanan theorem that $χ(F_2)$ 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 $χ(F_2)$ 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 $χ(F_2)$, 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 $χ(F_2)$ 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.

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 work provides a self-contained, representation-theoretic proof that the SU(2) character variety χ(F2) of the genus two surface is a closed 6-manifold and is homeomorphic to , avoiding the Narasimhan–Seshadri correspondence. The strategy hinges on analyzing an involution on χ(S^2,6) whose quotient is χ(F2), establishing a double branched-cover framework, and applying Wall’s classification to identify a CP^3-structure via a controlled decomposition along a separating curve. A central technical feat is matching the κ-level decomposition of χ(F2) with a Morse-Bott decomposition on CP^3 using the real locus and a smooth quadric, yielding a diffeomorphism after gluing two tubular neighborhoods. The latter part develops a rich Lagrangian-immersion picture: holonomy perturbations and Goldman flows realize explicit Lagrangian correspondences from 3-manifolds with genus two boundary into χ(F2), illustrated by a suite of tangles, perturbations, and multicurves that produce embedded submanifolds representing homology classes and explicit intersection phenomena. The results connect to Floer-type frameworks by providing concrete Lagrangian data in χ(F2) and show how 3-manifold topology can be encoded inside the CP^3-model of χ(F2), with potential implications for bordered Floer theories and instanton- Floer-type constructions.

Abstract

We provide a proof that the character variety of a genus two surface, , is a closed compact manifold, and a proof of the Narasimhan-Ramanan theorem that is homeomorphic to . This is done entirely in the language of representations, differential topology and elementary algebraic topology. It avoids the Narasimhan-Seshadri correspondence, clarifying the nature of Lagrangian immersions into 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 , 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 induced by a genus 2 Heegaard splitting of for the ``linked eyeglasses" web , which are made transverse, and hence the corresponding Chern-Simons function Morse, using Goldman flows/holonomy perturbations along embedded curves in the Heegaard surface.
Paper Structure (57 sections, 31 theorems, 185 equations, 4 figures)

This paper contains 57 sections, 31 theorems, 185 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Discussion
  3. Relationship to previous work
  4. Contents
  5. Acknowledgments.
  6. Preliminaries
  7. Notation
  8. Double branched covers
  9. $SU(2)$ character varieties of finitely presented groups
  10. Smooth points and tangent spaces of $SU(2)$ character varieties.
  11. Surfaces and the Atiyah-Bott-Goldman form
  12. Symplectic structure
  13. The link of a point in a character variety
  14. The genus two surface
  15. Six-punctured 2-sphere
  16. ...and 42 more sections

Key Result

Theorem 1

Let $C$ be an embedded curve which separates $F_2$ into two punctured tori, and let $\kappa\colon\chi(F_2)\to [-1,1]$ send a conjugacy class $[\rho]$ to $\tfrac{1}{2} {\rm Trace}(\rho(C)).$ Then the decomposition satisfies The abelian locus $\chi^{\hbox{\scriptsize \sl ab}}(F_2)$ is contained in $\{\kappa=1\}$.

Figures (4)

  • Figure 1: The genus two surface $F_2$
  • Figure 2: A tangle cobordism from $(S^2,4)$ to $F_2$
  • Figure 3: Four-ended tangles in the ball and a modified product tangle in $S^2\times I$, enhanced with perturbation curves and $w_2$ arcs
  • Figure 4: A genus two splitting of a linked pair of eyeglasses

Theorems & Definitions (50)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Proposition 2.2
  • Theorem 3.1: K6
  • Corollary 3.2
  • proof
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • ...and 40 more