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).