Non‐Abelian Homology of Groups

Inassaridze, Hvedri; Inassaridze, Nick

doi:10.1023/a:1007881911578

Cited by 9 publications

(7 citation statements)

References 8 publications

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“…To present the equivariant (co)homology of groups as cotriple (co)homology we will use the free cotriple defined in the category G Γ of Γ -groups given in [21,22] to develop a non-Abelian homology theory of groups. This cotriple corresponds to the tripleability of G Γ over Γ -sets.…”

Section: Equivariant (Co)homology Of Groups As Cotriple (Co)homologymentioning

confidence: 99%

Equivariant homology and cohomology of groups

Inassaridze

2005

Topology and its Applications

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We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ -equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.It is well known that the study of groups with operators has many important applications in algebra and topology. The category of groups enriched with an action by automorphisms of a given group provides a suitable setting for the investigation of an extensive list of subjects with recognized mathematical interest. See, for instance, recent results in equivariant stable homotopy theory [6] and articles devoted to equivariant algebraic K-theory [13,29,24]. The origin of the equivariant investigation in homological algebra, particularly in extension theory of groups, goes back to the article of Whitehead [36]. It should be noted that recently a theory of cohomology of groups with operators was developed [8], motivated by E-mail address: hvedri@rmi.acnet.ge (H. Inassaridze).0166-8641/$ -see front matter 

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Section: Equivariant (Co)homology Of Groups As Cotriple (Co)homologymentioning

confidence: 99%

Equivariant homology and cohomology of groups

Inassaridze

2005

Topology and its Applications

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“…A low-dimensional non-abelian (co)homology of groups and Lie algebras was introduced by Guin (see [7,8]), and has led to a non-abelian (co)homology theory contained essentially in the papers [6,[9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Low-dimensional non-abelian Leibniz cohomology

2011

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We construct the zero and first non-abelian cohomologies of Leibniz algebras with coefficients in crossed modules, which differ from those of Gnedbaye and generalize the zero and first Leibniz cohomologies of Loday and Pirashvili. We also introduce the second non-abelian Leibniz cohomology and describe its relationship with extensions of Leibniz algebras by crossed modules. We obtain a nine-term exact non-abelian cohomology sequence. For Lie algebras we compare the non-abelian Leibniz and Lie cohomologies.

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“…The decomposition of the group of boundaries of the Moore complex of a simplicial group (algebra) as a product of commutator subgroups (sum of product ideals) is of interest in various topological and homological settings. For instance, in calculations of non-abelian homology of groups [12] and non-abelian homology of Lie algebras [13], and to explain the relations among several algebraic models of connected homotopy 3-types: braided regular crossed modules, 2-crossed modules, quadratic modules, crossed squares and simplicial groups with Moore complex of length 2 [1,3,15,16]. It is possible that such decomposition contributes to light complete descriptions of algebraic models of the n-types of specific families of spaces for low values of n, to calculate Samelson and Whitehead products [10] and analogues in homotopy theory of simplicial Lie algebras or to link simplicial groups and weak infinity categoric models.…”

Section: Introductionmentioning

confidence: 99%

Peiffer elements in simplicial groups and algebras

Castiglioni

Ladra

2008

Journal of Pure and Applied Algebra

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The main objective of this paper is to prove in full generality the following two facts:A. For an operad O in Ab, let A be a simplicial O-algebra such that A m is generated as an O-ideal by ( m−1 i=0 s i (A m−1 )), for m > 1, and let NA be the Moore complex of A. Theni∈I p ker d i   where the sum runs over those partitions of [m − 1], I = (I 1 , . . . , I p ), p ≥ 1, and γ is the action of O on A. B. Let G be a simplicial group with Moore complex NG in which G n is generated as a normal subgroup by the degenerate elements in dimension n > 1, then d(N n G) = I,J [ i∈I ker d i , j∈J ker d j ], for I, J ⊆ [n − 1] with I ∪ J = [n − 1]. In both cases, d i is the i-th face of the corresponding simplicial object. The former result completes and generalizes results from Akça and Arvasi [I. Akça, Z. Arvasi, Simplicial and crossed Lie algebras, Homology Homotopy Appl. 4 (1) (2002) 43-57], and Arvasi and Porter [Z. Arvasi, T. Porter, Higher dimensional Peiffer elements in simplicial commutative algebras, Theory Appl. Categ. 3 (1) (1997) 1-23]; the latter completes a result from Mutlu and Porter [A. Mutlu, T. Porter, Applications of Peiffer pairings in the Moore complex of a simplicial group, Theory Appl. Categ. 4 (7) (1998) 148-173].Our approach to the problem is different from that of the cited works. We have first succeeded with a proof for the case of algebras over an operad by introducing a different description of the inverse of the normalization functor N : Ab ∆ op → Ch ≥0 . For the case of simplicial groups, we have then adapted the construction for the inverse equivalence used for algebras to get a simplicial group NG Λ from the Moore complex NG of a simplicial group G. This construction could be of interest in itself.

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Non‐Abelian Homology of Groups

Cited by 9 publications

References 8 publications

Equivariant homology and cohomology of groups

Equivariant homology and cohomology of groups

Low-dimensional non-abelian Leibniz cohomology

Peiffer elements in simplicial groups and algebras

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