Small Heegaard genus and SU(2)
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866912542247354368 |
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| author | Baldwin, John A. Sivek, Steven |
| author_facet | Baldwin, John A. Sivek, Steven |
| contents | Let $Y$ be a closed, orientable 3-manifold with Heegaard genus 2. We prove that if $H_1(Y;\mathbb{Z})$ has order $1$, $3$, or $5$, then there is a representation $π_1(Y) \to \mathrm{SU}(2)$ with non-abelian image. Similarly, if $H_1(Y;\mathbb{Z})$ has order $2$ then we find a non-abelian representation $π_1(Y) \to \mathrm{SO}(3)$. We also prove that a knot $K$ in $S^3$ is a trefoil if and only if there is a unique conjugacy class of irreducible representations $π_1(S^3\setminus K) \to \mathrm{SU}(2)$ sending a fixed meridian to $\left(\begin{smallmatrix}i&0\\0&-i\end{smallmatrix}\right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_09780 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Small Heegaard genus and SU(2) Baldwin, John A. Sivek, Steven Geometric Topology Let $Y$ be a closed, orientable 3-manifold with Heegaard genus 2. We prove that if $H_1(Y;\mathbb{Z})$ has order $1$, $3$, or $5$, then there is a representation $π_1(Y) \to \mathrm{SU}(2)$ with non-abelian image. Similarly, if $H_1(Y;\mathbb{Z})$ has order $2$ then we find a non-abelian representation $π_1(Y) \to \mathrm{SO}(3)$. We also prove that a knot $K$ in $S^3$ is a trefoil if and only if there is a unique conjugacy class of irreducible representations $π_1(S^3\setminus K) \to \mathrm{SU}(2)$ sending a fixed meridian to $\left(\begin{smallmatrix}i&0\\0&-i\end{smallmatrix}\right)$. |
| title | Small Heegaard genus and SU(2) |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2309.09780 |