Non-abelian exterior products of groups and exact sequences in the homology of groups

Abstract

Various authors have obtained an eight term exact sequence in homologyfrom a short exact sequence of groups,the term V varying from author to author (see [7] and [2]; see also [5] for the simpler case where N is central in G, and [6] for the case where N is central and N ⊂ [G, G]). The most satisfying version of the sequence is obtained by Brown and Loday [2] (full details of [2] are in [3]) as a corollary to their van Kampen type theorem for squares of spaces: they give the term V as the kernel of a map G ∧ N → N from a “non-abelian exterior product” of G and N to the group N (the definition of G ∧ N, first published in [2], is recalled below). The two short exact sequencesandwhere F is free, together with the fact that H2(F) = 0 and H3(F) = 0, imply isomorphisms..The isomorphism (2) is essentially the description of H2(G) proved algebraically in [11]. As noted in [2], the isomorphism (3) is the analogue for H3(G) of the Hopf formula for H2(G).