Coloring spheres in 3--manifolds

TL;DR

This paper investigates the chromatic number of the sphere graph associated with certain 3-manifolds, providing bounds that extend known results from surface curve graphs to 3-manifold topology.

Contribution

It establishes upper and lower bounds for the chromatic number of the sphere graph of connected sums of $S^1\times S^2$, extending the analogy from surface curve graphs to 3-manifolds.

Findings

Bounds for the chromatic number of the sphere graph of $M_r$

Extension of bounds to any orientable 3-manifold via prime decomposition

Connection between sphere graph properties and 3-manifold topology

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.