Coloring spheres in 3--manifolds
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910451145637888 |
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| author | Bering IV, Edgar A. Haffner, Bennett Ortiz, Estephanie Sanchez, Olivia |
| author_facet | Bering IV, Edgar A. Haffner, Bennett Ortiz, Estephanie Sanchez, Olivia |
| contents | The sphere graph of $M_r$, a connect sum of $r$ copies of $S^1\times S^2$ was introduced by Hatcher as an analog of the curve graph of a surface to study the outer automorphism group of a free group $F_r$. Bestvina, Bromberg, and Fujiwara proved that the chromatic number of the curve graph is finite; bounds were subsequently improved by Gaster, Greene, and Vlamis. Motivated by the analogy, we provide upper and lower bounds for the chromatic number of the sphere graph of $M_r$. As a corollary to the prime decomposition of 3-manifolds, this gives bounds on the chromatic number of the sphere graph for any orientable 3-manifold. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_10932 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coloring spheres in 3--manifolds Bering IV, Edgar A. Haffner, Bennett Ortiz, Estephanie Sanchez, Olivia Geometric Topology Combinatorics The sphere graph of $M_r$, a connect sum of $r$ copies of $S^1\times S^2$ was introduced by Hatcher as an analog of the curve graph of a surface to study the outer automorphism group of a free group $F_r$. Bestvina, Bromberg, and Fujiwara proved that the chromatic number of the curve graph is finite; bounds were subsequently improved by Gaster, Greene, and Vlamis. Motivated by the analogy, we provide upper and lower bounds for the chromatic number of the sphere graph of $M_r$. As a corollary to the prime decomposition of 3-manifolds, this gives bounds on the chromatic number of the sphere graph for any orientable 3-manifold. |
| title | Coloring spheres in 3--manifolds |
| topic | Geometric Topology Combinatorics |
| url | https://arxiv.org/abs/2405.10932 |