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

TL;DR

This paper introduces a non-abelian analogue of Whitney's 2-isomorphism theorem, using 2-truncations of group algebras of fundamental groups to uniquely identify 2-edge connected graphs, extending classical graph invariants.

Contribution

It develops a non-abelian invariant based on 2-truncations of group algebras that fully characterizes 2-edge connected graphs, generalizing Whitney's cycle space concept.

Findings

The 2-truncation of the group algebra uniquely determines 2-edge connected graphs.

The invariant is complete: graphs with the same 2-truncation are isomorphic.

Extends classical abelian invariants to a non-abelian setting.

Abstract

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.