TY - JOUR
T1 - Robertson’s Conjecture in Algebraic Topology
AU - Knudsen, Ben
AU - Ramos, Eric
N1 - Publisher Copyright:
© 2023, Universitat Wien. All rights reserved.
PY - 2023
Y1 - 2023
N2 - 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 K3,3 or K5 as a topological minor. That is, if some subdivision of either K3,3 or K5 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.
AB - 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 K3,3 or K5 as a topological minor. That is, if some subdivision of either K3,3 or K5 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.
KW - Configuration Spaces
KW - Graph Well-Quasi-Orders
KW - Representation Stability
KW - Representations of Categories
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M3 - Article
AN - SCOPUS:85181936494
JO - Seminaire Lotharingien de Combinatoire
JF - Seminaire Lotharingien de Combinatoire
IS - 89
M1 - #49
ER -