Non-abelian exterior products of groups and exact sequences in the homology of groups. (English) Zbl 0631.20040

Ellis, G. J.

Glasg. Math. J. 29, 13-19 (1987).

Let \(1\to N\to G\to Q\to 1\) be a short exact sequence of groups and let M be a Q-module. It has long been known that there is a five-term homology exact sequence \[ H_ 2(G;M)\to H_ 2(Q;M)\to H_ 1(N)\otimes_ QM\to H_ 1(G;M)\to H_ 1(Q;M)\to 0. \] A variety of extensions of this to eight- and ten-term exact sequences have been obtained by numerous authors, the most general result being due to A. Gut. Recently R. Brown and J.-L. Loday have given an eight-term extension for the case of trivial integral coefficients: \[ H_ 3(G)\to H_ 3(Q)\to V\to H_ 2(G)\to H_ 2(Q)\to N/[G,N]\to H_ 1(G)\to H_ 1(Q)\to 0. \] The advantage of their result is that the term V is given a concrete description involving their concept of “nonabelian exterior product”. (Precisely stated, \(V=\ker (G\wedge N\to N).)\) In the paper under review an infinite-term homology exact sequence is obtained in the case of trivial integral coefficients and it is algebraically shown to extend the Brown-Loday eight-term sequence. In particular this provides an algebraic proof of the Brown-Loday sequence. The argument is quite accessible, essentially based on an interesting description of \(H_ 2(G)\) derived by C. Miller which is reformulated here in terms of nonabelian exterior products. A key step in Ellis’ proof (and a by-product of the Brown-Loday sequence) is the derivation of an analogue for \(H_ 3(G)\) of the Hopf formula for \(H_ 2(G)\).

Reviewer: A.Miller

MSC:

20J05	Homological methods in group theory
55U99	Applied homological algebra and category theory in algebraic topology
20E22	Extensions, wreath products, and other compositions of groups
55U25	Homology of a product, Künneth formula

Keywords:

homology of groups; exact sequences; nonabelian exterior product; homology exact sequence; nonabelian exterior products; Hopf formula

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