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Tensor products and homotopies for ωgroupoids and crossed complexes
, 2007
"... Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed comp ..."
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Cited by 43 (21 self)
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Crossed complexes have longstanding uses, explicit and implicit, in homotopy theory and the cohomology of groups. It is here shown that the category of crossed complexes over groupoids has a symmetric monoidal closed structure in which the internal Hom functor is built from morphisms of crossed complexes, nonabelian chain homotopies between them and similar higher homotopies. The tensor product involves nonabelian constructions related to the commutator calculus and the homotopy addition lemma. This monoidal closed structure is derived from that on the equivalent category of ωgroupoids where the underlying cubical structure gives geometrically natural definitions of tensor products and homotopies.
Computing Crossed Modules Induced By An Inclusion Of A Normal Subgroup, With Applications To Homotopy 2Types
, 1996
"... We obtain some explicit calculations of crossed Qmodules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2types and second homotopy modules of certain homotopy pushouts of maps of classifying ..."
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Cited by 17 (13 self)
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We obtain some explicit calculations of crossed Qmodules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2types and second homotopy modules of certain homotopy pushouts of maps of classifying spaces of discrete groups.
On relative homotopy groups of the product filtration, the James construction, and a formula of Hopf
, 2007
"... ..."
Computing homotopy types using crossed ncubes of groups
 in Adams Memorial Symposium on Algebraic Topology
, 1992
"... Dedicated to the memory of Frank Adams ..."
Nonabelian tensor products of groups: the commutator connection
 In Groups St. Andrews 1997 in
, 1999
"... This is a progress report on some of the developments in nonabelian tensor products of groups since the appearance of the paper “Some Computations of NonAbelian Tensor Products of Groups ” by Brown, Johnson and Robertson, ten years ago. In the spring of 1988 Ronnie Brown came to Binghamton and gave ..."
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Cited by 2 (0 self)
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This is a progress report on some of the developments in nonabelian tensor products of groups since the appearance of the paper “Some Computations of NonAbelian Tensor Products of Groups ” by Brown, Johnson and Robertson, ten years ago. In the spring of 1988 Ronnie Brown came to Binghamton and gave a talk about nonabelian tensor products, in particular about his paper with Johnson and Robertson [7] which had just appeared. I fell in love with tensor products on first sight and started my student Michael Bacon on this topic for his dissertation, and since then others have joined in these investigations. This talk is an invitation for others to join in this research. There are many interesting and accessible problems and it appears likely that there are interesting applications to group theory, in the same way as regular tensors have been applied. All this is provided you do not immediately get thrown off by the notation. In context with nonabelian tensor products left actions are used. Early on I contemplated switching to right action but decided against it. That would be like insisting on driving on the right in a country where everyone else drives on the left. We use the following notation. For elements g, g ′, h, h ′ in a group G we set hg = hgh−1 for the conjugate of g by h, and [h, g] = hgh−1g−1 for the commutator of h and g. The familiar expansion formulas using left action appear as follows: [gg ′ , h] = [ g g ′ , g h][g, h], [g, hh ′ ] = [g, h] [ h g, h h ′]. Now we define the nonabelian tensor product of two groups, preceded by the definition of a compatible action which is intimately connected with nonabelian tensor products. Definition 1. Let G and H be a pair of groups acting upon each other in a compatible way, that is ( g h) g
The Homology of Peiffer Products of Groups
"... Abstract. The Peiffer product of groups first arose in work of J.H.C. Whitehead on the structure of relative homotopy groups, and is closely related to problems of asphericity for twocomplexes. We develop algebraic methods for computing the second integral homology of a Peiffer product. We show tha ..."
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Abstract. The Peiffer product of groups first arose in work of J.H.C. Whitehead on the structure of relative homotopy groups, and is closely related to problems of asphericity for twocomplexes. We develop algebraic methods for computing the second integral homology of a Peiffer product. We show that a Peiffer product of superperfect groups is superperfect, and determine when a Peiffer product of cyclic groups has trivial second homology. We also introduce a double wreath product as a Peiffer product.
AND
, 1985
"... A generalised tensor product G 0 H of groups G, H has been introduced by R. Brown and J.L. Loday in [3,4]. It arises in applications in homotopy theory of a generalised Van Kampen theorem. The reason why G 0 H does not necessarily reduce to GUh Oz Huh, the usual tensor product over Z of the abelian ..."
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A generalised tensor product G 0 H of groups G, H has been introduced by R. Brown and J.L. Loday in [3,4]. It arises in applications in homotopy theory of a generalised Van Kampen theorem. The reason why G 0 H does not necessarily reduce to GUh Oz Huh, the usual tensor product over Z of the abelianisations, is that it is assumed that G acts on H (on the left) and H acts on G (on the left), and these actions are taken into account in the definition of the tensor product. A group G acts on itself by conjugation (“g = hgh ‘) and so the tensor square GO G is always defined. Further, the commutator map G x G + G induces a homomorphism of groups K: G 0 G + G, sending g @ h to [g, h] =ghg‘hl. We write J,(G) for Ker IC; its topological interest is the formula [3,4] qSK(G, l)=J,(G).