A non-Abelian analogue of Whitneys 2-isomorphism theorem
We give a non-Abelian analogue of Whitneys 2-isomorphism theorem for graphs. Whitneys 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 Whitneys 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.
whitneys 2-isomorphism theorem fundamental group graph theory