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Crossed Modules and Symmetric Cohomology of Groups Homology, Homotopy and Applications, vol. 22(2), 2020, pp.123–134 CROSSED MODULES AND SYMMETRIC COHOMOLOGY OF GROUPS MARIAM PIRASHVILI (communicated by Graham Ellis) Abstract This paper links the third symmetric cohomology (intro- duced by Staic [10]andZarelua[12]) to crossed modules with certain properties. The equivalent result in the language of 2- groups states that an extension of 2-groups corresponds to an element of HS3 iff it possesses a section which preserves inverses in the 2-categorical sense. This ties in with Staic’s (and Zare- lua’s) result regarding HS2 and abelian extensions of groups. 1. Introduction Let G be a group and M a G-module. Symmetric cohomology HS∗(G, M)was introduced by Staic [9] as a variant of classical group cohomology H∗(G, M). There has been interest in the mathematical community in exploring symmetric cohomol- ogy. Several papers have already been published on this topic, e.g. [2], [3], [8], [11]. ∗ Zarelua’s prior definition [12] of exterior cohomology Hλ(G, M) is very closely related to this, as shown in [6]. Namely, we proved in [6] that symmetric cohomology has a functorial decomposition ∗ ∗ ⊕ ∗ HS (G, M)=Hλ(G, M) Hδ (G, M), i where Hδ(G, M)=0when0 i 4orM hasnoelementsofordertwo. n n → n n n There are natural homomorphisms α : HS (G, M) H (G, M) (and β : Hλ (G, M) → Hn(G, M)), which are isomorphisms for n =0, 1 and a monomorphism for n = 2. This was shown by Staic in [10]. According to our results in [6], the map αn is an isomorphism for n =2ifG has no elements of order two. In this case, α3 is a monomorphism. More generally, αn is an isomorphism if G is torsion free. Classical H2(G, M) classifies extensions of G by M. Staic showed in [10]that HS2(G, M) classifies a subclass of extensions of G by M, namely those that have a section-preserving inversion. It is also well known that H3(G, M) classifies the so-called crossed extensions of 3 G by M [4] or, equivalently, H (G, M) classifies 2-groups Γ, with π0(Γ) = G and π1(Γ) = M. Received April 1, 2019, revised July 25, 2019, August 14, 2019; published on April 15, 2020. 2010 Mathematics Subject Classification: 20J06, 18D05. Key words and phrases: group cohomology, crossed modules, symmetric cohomology. Article available at http://dx.doi.org/10.4310/HHA.2020.v22.n2.a7 Copyright c 2020, Mariam Pirashvili. Permission to copy for private use granted. 124 MARIAM PIRASHVILI The main result of this paper is that for groups G with no elements of order 2 the group HS3(G, M) classifies crossed extensions of G by M with a certain condition on the section. 2. Preliminaries 2.1. Preliminaries on symmetric cohomology of groups Let G be a group and M a G-module. Recall that the group cohomology H∗(G, M) is defined as the cohomology of the cochain complex C∗(G, M), where the group of n-cochains of G with coefficients in M is the set of functions from Gn to M: Cn(G, M)={φ: Gn → M} and the coboundary map dn : Cn(G, M) → Cn+1(G, M) is defined by n n i d (φ)(g0,g1,...,gn)=g0 · φ(g1,...,gn)+ (−1) φ(g0,...,gi−2,gi−1gi,gi+1,...,gn) i=1 n+1 +(−1) φ(g0,...,gn−1). Acochainφ is called normalised if φ(g1,...,gn) = 0 whenever some gj =1.Thecol- ∗ lection of normalised cochains is denoted by CN (G, M) and the classical normalisation ∗ → ∗ theorem claims that the inclusion CN (G, M) C (G, M) induces an isomorphism on cohomology. In [9] Staic introduced a subcomplex CS∗(G, M) ⊂ C∗(G, M), whose homology is known as the symmetric cohomology of G with coefficients in M and is denoted ∗ n by HS (G, M). The definition is based on an action of Σn+1 on C (G, M)(for every n) compatible with the differential. In order to define this action, it is enough n to define how the transpositions τi =(i, i +1),1 i n act. For φ ∈ C (G, M)one defines: ⎧ −1 ⎨⎪−g1φ(g1 ,g1g2,g3,...,gn), if i =1, −1 (τiφ)(g1,g2,g3,...,gn)= −φ(g1,...,g −2,g −1g ,g ,g g +1,...,g ), 1 <i<n, ⎩⎪ i i i i i i n − −1 φ(g1,g2,g3,...,gn−1gn,gn ), if i = n. Denote by CSn(G, M) the subgroup of the invariants of this action. That is, CSn(G, M)=Cn(G, M)Σn+1 . Staic proved that CS∗(G, M) is a subcomplex of C∗(G, M)[9], [10] and hence the groups HS∗(G, M) are well defined. There is an obvious natural transformation αn : HSn(G, M) → Hn(G, M),n 0. Accordingto[9], [10], αn is an isomorphism if n =0, 1 and is a monomorphism for n = 2. For extensive study of the homomorphism αn for n 2 we refer to [6]. ∗ ∗ ∩ ∗ Denote by CSN (G, M) the intersection CS (G, M) CN (G, M). Unlike classical ∗ → ∗ cohomology the inclusion CSN (G, M) CS (G, M) does not always induce an iso- ∗ ∗ ∗ morphism on cohomology. The groups Hλ(G, M)=H (CSN (G, M)) are isomorphic to the so called exterior cohomology of groups introduced by Zarelua in [12]. Accord- ingto[6] the canonical map n → n γn : Hλ (G, M) HS (G, M) ∗ → ∗ induced by the inclusion CSN (G, M) CS (G, M), is an isomorphism if n 4, or M hasnoelementsofordertwo. CROSSED MODULES AND SYMMETRIC COHOMOLOGY OF GROUPS 125 2.2. Symmetric extensions and HS2 It is a classical fact that H2(G, M) classifies the extension of G by M and H3(G, M) classifies the crossed extensions of G by M. One can ask what objects classify the symmetric cohomology groups HS2(G, M)andHS3(G, M). The answer to this ques- tion in dimension two was given in [10]. The aim of this work is to prove a similar result in dimension three. Let G be a group and M a G-module. Recall that an extension of G by M is a short exact sequence of groups p 0 → M −−i→ K −−→ G → 0 such that for any k ∈ K and m ∈ M, one has ki(m)k−1 = i(p(k)m). An s-section to this extension is a map s: G → K such that p ◦s(x)=x for all x ∈ G.LetExtgr(G, M) be the category whose objects are extensions of G and M and morphisms are com- mutative diagrams p 0 /M i /K /G /0 id id p 0 /M i /K /G /0. The set of connected components of the category Extgr(G, M) is denoted by Extgr(G,M). It is well known that there exists a natural map Extgr(G,M)→H2(G, M), which is a bijection. To construct this map, one needs to choose an s-section s and then define f ∈ C2(G, M)by s(x)s(y)=i(f(x, y))s(xy). One checks that f is a 2-cocycle and its class in H2 is independent of the chosen s-section s. p Let 0 → M −−i→ K −−→ G → 0 be an extension and s an s-section. Then s is called symmetric if s(x−1)=s(x)−1 holds for all x ∈ G. An extension is called symmetric if it possesses a symmetric s-section. The symmetric extensions form a full subcat- egory ExS(G, M) of the category Extgr(G, M). The set of connected components of ExS(G, M) is denoted by ExS(G, M). The main result of [10] claims that the restriction of the bijection Extgr(G, M) → H2(G, M)onExS(G, M) yields a bijection ExS(G, M) → HS2(G, M). 2.3. Crossed modules Recall the classical relationship between third cohomology of groups and crossed modules. A crossed module is a group homomorphism ∂ : T → R together with an action of R on T satisfying: ∂(rt)=r∂(t)r−1 and ∂ts = tst−1,r∈ R, t, s ∈ T. It follows from the definition that the image Im(∂) is a normal subgroup of R,and the kernel M =Ker(∂) is in the centre of T . Moreover, the action of R on T induces an action of G on Ker(∂), where G = Coker(∂). A morphism from a crossed module ∂ : T → R to a crossed module ∂ : T → R is 126 MARIAM PIRASHVILI a pair of group homomorphisms (φ: T → T ,ψ: R → R) such that ψ ◦ ∂ = ∂ ◦ φ, φ(rt)= ψ(r)φ(t),r∈ R, t ∈ T. For a group G and for a G-module M one denotes by Xext(G, M) the category of exact sequences 0 → M → T −−∂→ R → G → 0, where ∂ : T → R is a crossed module and the action of G on M induced from the crossed module structure coincides with the prescribed one. The morphisms in Xext(G, M) are commutative diagrams p 0 /M /T ∂ /R /G /0 id φ ψ id p 0 /M /T ∂ /R /G /0, where (φ, f) is a morphism of crossed modules (T,G,∂) → (T ,G,∂). We let Xext(G, M) be the class of the connected components of the category Xext(G, M). Objects of the category Xext(G, M) are called crossed extensions of G by M. It is a classical fact (see for example [4]) that there is a canonical bijection ξ : Xext(G, M) → H3(G, M).
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