A differential topology proof that the $SU(2)$ character variety of the genus two surface is homeomorphic to ${\mathbb C} P^3$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910007544512512 |
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| author | Herald, Christopher M. Kirk, Paul |
| author_facet | Herald, Christopher M. Kirk, Paul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00842 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A differential topology proof that the $SU(2)$ character variety of the genus two surface is homeomorphic to ${\mathbb C} P^3$ Herald, Christopher M. Kirk, Paul Geometric Topology Primary 57K20 57K18, 57K31, 57R17, 53D30, Secondary 14D21 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. |
| title | A differential topology proof that the $SU(2)$ character variety of the genus two surface is homeomorphic to ${\mathbb C} P^3$ |
| topic | Geometric Topology Primary 57K20 57K18, 57K31, 57R17, 53D30, Secondary 14D21 |
| url | https://arxiv.org/abs/2602.00842 |