Crossed modules and homology

Ladra, M.; Grandjean, A.R.

doi:10.1016/0022-4049(94)90117-1

Cited by 26 publications

(9 citation statements)

References 6 publications

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“…We recall from [16] that a crossed module µ : M → P is called abelian if P is an abelian group and the action of P on M is trivial. This implies that M is also abelian.…”

Section: Introductionmentioning

confidence: 99%

N-Fold Čech Derived Functors and Generalised Hopf Type Formulas

Donadze¹,

Inassaridze²,

Porter³

2005

K-Theory

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Abstract. In 1988, Brown and Ellis published [3] a generalised Hopf formula for the higher homology of a group. Although substantially correct, their result lacks one necessary condition. We give here a counterexample to the result without that condition. The main aim of this paper is, however, to generalise this corrected result to derive formulae of Hopf type for the n-foldČech derived functors of the lower central series functors Z k . The paper ends with an application to algebraic K-theory.

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“…We recall from [16] that a crossed module µ : M → P is called abelian if P is an abelian group and the action of P on M is trivial. This implies that M is also abelian.…”

Section: Introductionmentioning

confidence: 99%

N-Fold Čech Derived Functors and Generalised Hopf Type Formulas

Donadze¹,

Inassaridze²,

Porter³

2005

K-Theory

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“…The Set-free crossed module on a function h : X -»• y is the totally free crossed module on /i : A" -»• 7 c F, where F is the free group with basis the set Y. The set-free crossed module can be interpreted by adjoint functors [10]. PROPOSITION …”

Section: Say That a Crossed Module (T G 3) Acts On {S H Pi) If Thmentioning

confidence: 99%

On totally free crossed modules

Grandjean

Ladra

1998

Glasgow Math. J.

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In [10] we associate to a crossed module (T, G, მ) an invariant abelian crossed module H2(T, G, მ). The construction uses presentations by Set-free crossed modules. Now, Set-free crossed modules are special cases of totally free crossed modules, which are algebraic models of 2-dimensional CW complexes used by several authors (see [1] and [6]). The aim of this paper is to show that H2(T, G, მ) can also be constructed from presentations by arbitrary totally free crossed modules.

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“…The algebraic study of the category of crossed modules was initiated by Norrie [18] and has led to a substantial algebraic theory contained essentially in the following papers: [1,7,11,12,13,15,20,21]. In particular, Pirashvili [19] presented the concept of the tensor product of two abelian crossed modules and investigated its relation to the low-dimensional homology of crossed modules.…”

Section: Introductionmentioning

confidence: 99%

The non-abelian tensor product of normal crossed submodules of groups

Salemkar

Taha

2020

CGASA

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In this article, the notions of non-abelian tensor and exterior products of two normal crossed submodules of a given crossed module of groups are introduced and some of their basic properties are established. In particular, we investigate some common properties between normal crossed modules and their tensor products, and present some bounds on the nilpotency class and solvability length of the tensor product, provided such information is given at least on one of the normal crossed submodules.

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Crossed modules and homology

Cited by 26 publications

References 6 publications

N-Fold Čech Derived Functors and Generalised Hopf Type Formulas

N-Fold Čech Derived Functors and Generalised Hopf Type Formulas

On totally free crossed modules

The non-abelian tensor product of normal crossed submodules of groups

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