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

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


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, K√ľnneth formula
Full Text: DOI


[1] DOI: 10.1016/0040-9383(84)90016-8 · Zbl 0519.55009
[2] DOI: 10.2307/2032593 · Zbl 0047.25703
[3] DOI: 10.1016/0022-4049(82)90014-7 · Zbl 0491.55004
[4] Hilton, A course in homological algebra (1970) · Zbl 0238.18006
[5] Brown, C.R. Acad. Sci. Paris Ser I Math. 298 pp 353– (1984)
[6] DOI: 10.1016/0022-4049(76)90046-3 · Zbl 0361.20057
[7] DOI: 10.1007/BF02566792 · Zbl 0236.20038
[8] DOI: 10.1007/BF02566849 · Zbl 0232.18011
[9] DOI: 10.1112/jlms/s2-14.2.309 · Zbl 0357.20030
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