Geodesic
Non-abelian exterior products of groups and exact sequences in the homology of groups
Glasgow mathematical journal, Tome 29 (1987) no. 1, pp. 13-19

Voir la notice de l'article provenant de la source Cambridge University Press

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). 
Ellis, G. J. Non-abelian exterior products of groups and exact sequences in the homology of groups. Glasgow mathematical journal, Tome 29 (1987) no. 1, pp. 13-19. doi: 10.1017/S0017089500006637
@article{10_1017_S0017089500006637,
     author = {Ellis, G. J.},
     title = {Non-abelian exterior products of groups and exact sequences in the homology of groups},
     journal = {Glasgow mathematical journal},
     pages = {13--19},
     year = {1987},
     volume = {29},
     number = {1},
     doi = {10.1017/S0017089500006637},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500006637/}
}
TY  - JOUR
AU  - Ellis, G. J.
TI  - Non-abelian exterior products of groups and exact sequences in the homology of groups
JO  - Glasgow mathematical journal
PY  - 1987
SP  - 13
EP  - 19
VL  - 29
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500006637/
DO  - 10.1017/S0017089500006637
ID  - 10_1017_S0017089500006637
ER  - 
%0 Journal Article
%A Ellis, G. J.
%T Non-abelian exterior products of groups and exact sequences in the homology of groups
%J Glasgow mathematical journal
%D 1987
%P 13-19
%V 29
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500006637/
%R 10.1017/S0017089500006637
%F 10_1017_S0017089500006637

[1] 1.Brown, R., Coproducts of crossed P-modules: applications to second homotopy groups and to the homology of groups, Topology 23 (1984), 337–345. Google Scholar | DOI

[2] 2.Brown, R. and Loday, J.-L., Excision homotopique en basse dimension, C.R. Acad. Sci. Paris Ser I Math. 298 (1984), 353–356. Google Scholar

[3] 3.Brown, R. and Loday, J.-L., Van Kampen theorems for diagrams of spaces, to appear in Topology. Google Scholar

[4] 4.Ellis, G. J., Crossed modules and their higher dimensional analogues, (University of Wales Ph.D. Thesis, 1984). Google Scholar

[5] 5.Eckmann, B. and Hilton, P. J., On central group extensions and homology, Comment. Math. Helv. 46 (1971), 345–355. Google Scholar | DOI

[6] 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), 102–122. Google Scholar | DOI

[7] 7.Gut, A., A ten term exact sequence in the homology of a group extension, J. Pure Appl. Algebra 8 (1976), 243–260. Google Scholar | DOI

[8] 8.Hilton, P. J. and Stammbach, U., A course in homological algebra, Graduate Texts in Mathematics 4 (Springer-Verlag, 1970). Google Scholar

[9] 9.Loday, J.-L., Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24 (1982), 179–202. Google Scholar | DOI

[10] 10.Lue, A. S.-T., The Ganea map for nilpotent groups, J. London Math. Soc. (2) 14 (1976), 309–312. Google Scholar | DOI

[11] 11.Miller, C., The second homology group of a group, Proc. Amer. Math. Soc. 3 (1952), 588–595. Google Scholar | DOI

Cité par Sources :