Geodesic
A non-abelian analogue of Whitney's 2-isomorphism theorem
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 3, pp. 683-690.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $G$, $G^\prime$ be 2-edge connected finite graphs; if there is a bijective correspondence between the edges of $G$ and $G^\prime$ that induces equality on the 2-truncations of the group algebras of the fundamental groups, then $G$ and $G^\prime$ are isomorphic.
Classification : 05E15, 05C60, 05C25, 14T05
Keywords: fundamental group, Whitney's 2-isomorphism theorem 
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     author = {Katz, Eric},
     title = {A non-abelian analogue of {Whitney's} 2-isomorphism theorem},
     journal = {Journal of Algebraic Combinatorics},
     pages = {683--690},
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     volume = {39},
     number = {3},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2014__39_3_a4/}
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Katz, Eric. A non-abelian analogue of Whitney's 2-isomorphism theorem. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 3, pp. 683-690. http://geodesic.mathdoc.fr/item/JAC_2014__39_3_a4/