Coloring spheres in 3--manifolds

Edgar A. Bering IV; Bennett Haffner; Estephanie Ortiz; Olivia Sanchez

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arXiv · 2405.10932

Coloring spheres in 3--manifolds

Edgar A. Bering IV·Bennett Haffner·Estephanie Ortiz·Olivia Sanchez

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Abstract

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.

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Edgar A. Bering IV, Bennett Haffner, Estephanie Ortiz, Olivia Sanchez. 2024-05-17. Coloring spheres in 3--manifolds. https://arxiv.org/abs/2405.10932

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