A combinatorial interpretation of the third integral homology of a group (Q801162)

In this paper the author gives an interpretation of the third integral homology group \(H_ 3(G)\) through the combinatorial aspects of G. The main result is: Theorem: There is a natural isomorphism \(H_ 3(G)\equiv B(X:R)/C(X:R)\) between \(H_ 3(G)\) and the quotient of the group of balanced identities B(X:R) modulo the group of coupled identities C(X:R) for any presentation (X:R) of G. More interesting results are proved in this direction with topological flavor.