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

  • G. J. Ellis (a1)

Extract

Various authors have obtained an eight term exact sequence in homology

from 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 GNN from a “non-abelian exterior product” of G and N to the group N (the definition of GN, first published in [2], is recalled below). The two short exact sequences

and

where 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).

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References

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1.Brown, R., Coproducts of crossed P-modules: applications to second homotopy groups and to the homology of groups, Topology 23 (1984), 337345.
2.Brown, R. and Loday, J.-L., Excision homotopique en basse dimension, C.R. Acad. Sci. Paris Ser I Math. 298 (1984), 353356.
3.Brown, R. and Loday, J.-L., Van Kampen theorems for diagrams of spaces, to appear in Topology.
4.Ellis, G. J., Crossed modules and their higher dimensional analogues, (University of Wales Ph.D. Thesis, 1984).
5.Eckmann, B. and Hilton, P. J., On central group extensions and homology, Comment. Math. Helv. 46 (1971), 345355.
6.Eckmann, B., Hilton, P. J. and Stammbach, U., On the homology theory of central group extensions: the commutator map and stem extensions, Comment. Math. Helv. 47 (1972), 102122.
7.Gut, A., A ten term exact sequence in the homology of a group extension, J. Pure Appl. Algebra 8 (1976), 243260.
8.Hilton, P. J. and Stammbach, U., A course in homological algebra, Graduate Texts in Mathematics 4 (Springer-Verlag, 1970).
9.Loday, J.-L., Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24 (1982), 179202.
10.Lue, A. S.-T., The Ganea map for nilpotent groups, J. London Math. Soc. (2) 14 (1976), 309312.
11.Miller, C., The second homology group of a group, Proc. Amer. Math. Soc. 3 (1952), 588595.

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

  • G. J. Ellis (a1)

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