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The non-abelian tensor product of groups and related constructions

Published online by Cambridge University Press:  18 May 2009

N. D. Gilbert  and
P. J. Higgins
N. D. Gilbert
Affiliation:
Department of Pure Mathematics, University College of North Wales, Bangor, Gwynedd LL57 2UW
P. J. Higgins
Affiliation:
Department of Mathematical Sciences, University of Durham, Science Laboratories, South Road, Durham DH1 3LE
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The tensor product of two arbitrary groups acting on each other was introduced by R. Brown and J.-L. Loday in [5, 6]. It arose from consideration of the pushout of crossed squares in connection with applications of a van Kampen theorem for crossed squares. Special cases of the product had previously been studied by A. S.-T. Lue [10] and R. K. Dennis [7]. The tensor product of crossed complexes was introduced by R. Brown and the second author [3] in connection with the fundamental crossed complex π(X) of a filtered space X, which also satisfies a van Kampen theorem. This tensor product provides an algebraic description of the crossed complex π(X ⊗ Y) and gives a symmetric monoidal closed structure to the category of crossed complexes (over groupoids). Both constructions involve non-abelian bilinearity conditions which are versions of standard identities between group commutators. Since any group can be viewed as a crossed complex of rank 1, a close relationship might be expected between the two products. One purpose of this paper is to display the direct connections that exist between them and to clarify their differences.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 1 , January 1989 , pp. 17 - 29
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

References

1.Brown, R., Coproducts of crossed P-modules: applications to second homotopy groups and to the homology of groups, Topology 23 (1984), 337–345.CrossRefGoogle Scholar
2.Brown, R. and Higgins, P. J., On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc. (3) 36 (1978), 193–212.CrossRefGoogle Scholar
3.Brown, R. and Higgins, P. J., Tensor products and homotopies for w-groupoids and crossed complexes, J. Pure Appl. Algebra 47 (1987), 1–33.CrossRefGoogle Scholar
4.Brown, R., Johnson, D. L. and Robertson, E. F., Some computations of non-abelian tensor products of groups, J. Algebra 111 (1987), 177–202.CrossRefGoogle Scholar
5.Brown, R. and Loday, J.-L., Excision homotopique en basse dimension, C.R. Acad. Sci. Paris Sér. I Math. 298 (1984), 353–356.Google Scholar
6.Brown, R. and Loday, J.-L., Van Kampen theorems for diagrams of spaces, Topology 26 (1987), 311–335.CrossRefGoogle Scholar
7.Dennis, R. K., In search of new homology functors having a close relationship to K-theory preprint (Cornell Univ., 1976).Google Scholar
8.Ellis, G. J., The non-abelian tensor product of finite groups is finite, J. Algebra 111 (1987), 203–205.CrossRefGoogle Scholar
9.Levi, F. W., The commutator group of a free product, J. Indian Math. Soc. (N.S.) 4 (1940), 136–144.Google Scholar
10.Lue, A. S.-T., The Ganea map for nilpotent groups, J. London Math. Soc. (2) 14 (1976), 309–312.CrossRefGoogle Scholar
11.Whitehead, J. H. C., On adding relations to homotopy groups, Ann. of Math. (2) 42 (1941), 409–428.CrossRefGoogle Scholar