Homology lens spaces and $\mathrm{SL}(2,\mathbb{C})$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911183868526592 |
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| author | Ghosh, Sudipta Sivek, Steven Zentner, Raphael |
| author_facet | Ghosh, Sudipta Sivek, Steven Zentner, Raphael |
| contents | We prove that if $Y$ is a closed, oriented 3-manifold with first homology $H_1(Y;\mathbb{Z})$ of order less than $5$, then there is an irreducible representation $π_1(Y) \to \mathrm{SL}(2,\mathbb{C})$ unless $Y$ is homeomorphic to $S^3$, a lens space, or $\mathbb{RP}^3 \# \mathbb{RP}^3$. By previous work it suffices to consider the case $H_1(Y;\mathbb{Z}) \cong \mathbb{Z}/4\mathbb{Z}$, which we accomplish using holonomy perturbation techniques in instanton Floer homology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_25019 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homology lens spaces and $\mathrm{SL}(2,\mathbb{C})$ Ghosh, Sudipta Sivek, Steven Zentner, Raphael Geometric Topology We prove that if $Y$ is a closed, oriented 3-manifold with first homology $H_1(Y;\mathbb{Z})$ of order less than $5$, then there is an irreducible representation $π_1(Y) \to \mathrm{SL}(2,\mathbb{C})$ unless $Y$ is homeomorphic to $S^3$, a lens space, or $\mathbb{RP}^3 \# \mathbb{RP}^3$. By previous work it suffices to consider the case $H_1(Y;\mathbb{Z}) \cong \mathbb{Z}/4\mathbb{Z}$, which we accomplish using holonomy perturbation techniques in instanton Floer homology. |
| title | Homology lens spaces and $\mathrm{SL}(2,\mathbb{C})$ |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2509.25019 |