One of the most famous results in graph theory is that of Kuratowski’s theorem, which states that a graph $G$ is non-planar if and only if it contains one of $K_{3,3}$ or $K_5$ as a topological minor. That is, if some subdivision of either $K_{3,3}$ or $K_5$ appears as a subgraph of $G$. In this case we say that the question of planarity is determined by a finite set of forbidden (topological) minors. A conjecture of Robertson, whose proof was recently announced by Liu and Thomas, characterizes the kinds of graph theoretic properties that can be determined by finitely many forbidden minors. In this extended abstract we will present a categorical version of Robertson’s conjecture, which we have proven in certain cases. We will then illustrate how this categorification, if proven in all cases, would imply many non-trivial statements in the topology of graph configuration spaces.
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Abstract We prove Farber’s conjecture on the stable topological complexity of configuration spaces of graphs. The conjecture follows from a general lower bound derived from recent insights into the topological complexity of aspherical spaces. Our arguments apply equally to higher topological complexity.
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