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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2310.17965 |
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| _version_ | 1866911531023728640 |
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| author | Ghosh, Sudipta Sivek, Steven Zentner, Raphael |
| author_facet | Ghosh, Sudipta Sivek, Steven Zentner, Raphael |
| contents | We use instanton gauge theory to prove that if $Y$ is a closed, orientable $3$-manifold such that $H_1(Y;\mathbb{Z})$ is nontrivial and either $2$-torsion or $3$-torsion, and if $Y$ is neither $\#^r \mathbb{RP}^3$ for some $r\geq 1$ nor $\pm L(3,1)$, then there is an irreducible representation $π_1(Y) \to \mathrm{SL}(2,\mathbb{C})$. We apply this to show that the Kauffman bracket skein module of a non-prime 3-manifold has nontrivial torsion whenever two of the prime summands are different from $\mathbb{RP}^3$, answering a conjecture of Przytycki (Kirby problem 1.92(F)) unless every summand but one is $\mathbb{RP}^3$. As part of the proof in the $2$-torsion case, we also show that if $M$ is a compact, orientable $3$-manifold with torus boundary whose rational longitude has order 2 in $H_1(M)$, then $M$ admits a degree-1 map onto the twisted $I$-bundle over the Klein bottle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_17965 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rational homology 3-spheres and SL(2,$\mathbb{C}$) representations Ghosh, Sudipta Sivek, Steven Zentner, Raphael Geometric Topology We use instanton gauge theory to prove that if $Y$ is a closed, orientable $3$-manifold such that $H_1(Y;\mathbb{Z})$ is nontrivial and either $2$-torsion or $3$-torsion, and if $Y$ is neither $\#^r \mathbb{RP}^3$ for some $r\geq 1$ nor $\pm L(3,1)$, then there is an irreducible representation $π_1(Y) \to \mathrm{SL}(2,\mathbb{C})$. We apply this to show that the Kauffman bracket skein module of a non-prime 3-manifold has nontrivial torsion whenever two of the prime summands are different from $\mathbb{RP}^3$, answering a conjecture of Przytycki (Kirby problem 1.92(F)) unless every summand but one is $\mathbb{RP}^3$. As part of the proof in the $2$-torsion case, we also show that if $M$ is a compact, orientable $3$-manifold with torus boundary whose rational longitude has order 2 in $H_1(M)$, then $M$ admits a degree-1 map onto the twisted $I$-bundle over the Klein bottle. |
| title | Rational homology 3-spheres and SL(2,$\mathbb{C}$) representations |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2310.17965 |