Non abelian tensor products of groups and related constructions, and applications
Last updated, December 15, 2015
Some remarks on history
The earliest paper which uses a version of the nonabelian tensor square, and so
a replacement of the commutatot map by a morphism, is surely
[1] by Claire Miller. The next publication [2] defined a tensor product for a crossed module, and Abe Lue was annoyed he had not thought
of the more symmetric definition given in [5-6].
It should be emphasised that a start for the work with Loday was a seminar I gave in Strasbourg in 1981 on the joint work with Philip Higgins;
Jean-Louis immediately saw its relevance. Indeed, he had a conjecture which I saw as a triadic Hurewicz Theorem. But Higgins and I had deduced
the Relative Hurewicz Theorem from a van Kampen type theorem. So we conjectured a van Kampen type theorem
for his n-cat-groups. In 1982 it was realised that such a theorem would give rise to applications of a nonabelian tensor product, and this was part of papers
[5-7]. This work was improved with the input of the preprints [3-4]. The problem of calculation arose out of writing the paper [6], once we had seen that the tensor square of finite groups was finite.
Calculations for dihedral and quaternionic groups appeared in [6], and the general problem of calculation was put to David Johnson, leading to the publication [8].
A seminar I gave in Binghampton attracted the interest of L.-C. Kappe, and much work from her group.
Graham Ellis was a PhD student at Bangor, 1984-7, hence his interest in this area.
This version has been revised to be in chronological order, at least in terms
of years, in order to show better the development of the area.
Among `related constructions' we include non abelian tensor products of other
algebraic structures (see [17]), such as Lie algebras, since these were motivated
by the construction for groups. Also included (see [21,74]) is the Peiffer
product of groups which act on each other.
Suggestions for further entries or other comments are welcomed. See also
the survey by L.-C. Kappe, listed as [68].
On my preprint page is a link to presentation
on this topic for a Colloquium in Goettingen, May 5, 2011. See also on trhe same page a presentation in June 2015 at CT2015 on "A philosophy of modelling and computing homotopy types".
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