Ellis, G. J. Non-abelian exterior products of groups and exact sequences in the homology of groups. (English) Zbl 0631.20040 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 Cited in 2 ReviewsCited in 20 Documents 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 PDF BibTeX XML Cite \textit{G. J. Ellis}, Glasg. Math. J. 29, 13--19 (1987; Zbl 0631.20040) Full Text: DOI OpenURL References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.