A non-Abelian analogue of Whitney's 2-isomorphism theorem

作者：Katz Eric^*

来源：Journal of Algebraic Combinatorics, 2014, 39(3): 683-690.

DOI：10.1007/s10801-013-0461-x

• Graph theory
• Fundamental group
• Whitney's 2-isomorphism theorem

We give a non-Abelian analogue of Whitney's 2-isomorphism theorem for graphs. Whitney's theorem states that the cycle space determines a graph up to 2-isomorphism. Instead of considering the cycle space of a graph which is an Abelian object, we consider a mildly non-Abelian object, the 2-truncation of the group algebra of the fundamental group of the graph considered as a subalgebra of the 2-truncation of the group algebra of the free group on the edges. The analogue of Whitney's theorem is that this is a complete invariant of 2-edge connected graphs: let G, G' be 2-edge connected finite graphs; if there is a bijective correspondence between the edges of G and G' that induces equality on the 2-truncations of the group algebras of the fundamental groups, then G and G' are isomorphic.

• 出版日期2014-5