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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

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