GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 4, 1997, 313-331

NON-ABELIAN COHOMOLOGY OF GROUPS

H. INASSARIDZE

Abstract. Following Guin’s approach to non-abelian cohomology [4] and, using the notion of a crossed bimodule, a second pointed set of cohomology is defined with coefficients in a crossed module, and Guin’s six-term exact cohomology sequence is extended to a nine-term exact sequence of cohomology up to dimension 2.

Introduction

In this and forthcoming papers [1] we discuss the cohomology H^{∗}(G, A) of a group G with coefficients in a G-group A.When A is abelian this cohomology is the well-known classical cohomology of groups which can be defined as derived functors either of the functor Hom_{Z[G]}(−, A) in the category of Z[G]-modules or of the functor Der(−, A) in the category of groups acting on A. When A is non-abelian, a functorial pointed set of cohomology H^{1}(G, A) not equipped with a group structure was defined in a natural way in [2]. Guin defined, in [3]–[4], a first cohomology group when the coefficient group is a crossed G-module and obtained a six-term exact sequence of cohomology for any short exact coefficient sequence of crossed G-modules.

Our approach to a non-abelian cohomology of groups follows Guin’s cohomology theory of groups [3]–[4] which differs from the classical first nonabelian cohomology pointed set [2] and from the setting of various papers on non-abelian cohomology [5]–[7] extending the classical exact non-abelian cohomology sequence from lower dimensions [2] to higher dimensions.

Let G and R be groups and let (A, μ) be a crossed R-module. We introduce the notion of a crossed G − R-bimodule signifying an action of G on the crossed R-module (A, μ) and generalizing the notion of a crossed G-module.The group of derivations Der(G, (A, μ)) from G to (A, μ) is defined to obtain a pointed set of cohomology H^{2}(G, A) when A is a crossed G-module. The group Der(G, (A, μ)) and the pointed set H^{2}(G, A) coincide

313 respectively with the group Der_{G}(G, A) of Guin [4] when (A, μ) is a crossed G-module and with the usual cohomology group when A is abelian. A coefficient short exact sequence of crossed G-modules gives rise to a nine-term exact sequence of cohomology which extends the six-term exact cohomology sequence of Guin [4]. In [1] these results are generalized when the coefficients are crossed bimodules; in that case H^{1}(G, (A, μ)) is equipped with a partial product, and, finally, in [1] the definition of a pointed set of cohomology H^{n}(G, (A, μ)) of a group G with coefficients in a crossed G − R-bimodule (A, μ) for all n ≥ 1 is given.

1991 Mathematics Subject Classification. 18G50, 18G55.

Key words and phrases. Crossed module, derivation, simplicial kernel, crossed bimodule.

1072-947X/97/0700-0313$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

All considered groups will be arbitrary (not necessarily commutative). An action of a group G on a group A means an action on the left of G on A by automorphisms and will be denoted by^{g}a, g ∈ G, a ∈ A. We assume that G acts on itself by conjugation. The center of a group G will be denoted by Z(G). If the groups G and R act on a group A then the notation ^{gr}a means ^{g}(^{r}a), g ∈ G, r ∈ R, a ∈ A.

1. Crossed Bimodules

A precrossed G-module (A, μ) consists of a group G acting on a group A and a homomorphism μ : A ⟶ G such that

μ(^{g}a) = gμ(a)g^{−1}, g ∈ G, a ∈ A.

If in addition we have

μ(a)a′ = aa′a^{−1}

for a, a′ ∈ A, then (A, μ) is a crossed G-module.

Definition 1. Let G, R, and A be groups. It will be said that (A, μ) is a precrossed G − R-bimodule if

(1) (A, μ) is a precrossed R-module,

(2) G acts on R and A,

(3) the homomorphism μ : A ⟶ R is a homomorphism of G-groups,

(4) ^{(^{g}r)}a =^{grg^{−1}} a (compatibility condition) for g ∈ G, r ∈ R, a ∈ A. If in addition (A, μ) is a crossed R-module then (A, μ) will be called a crossed G − R-bimodule. If conditions (1)–(3) hold it will be said that the group G acts on the precrossed (resp. crossed) R-module (A, μ).

It is easy to see that any precrossed (resp. crossed) G-module (A, μ) is in a natural way a precrossed (resp. crossed) G − G-bimodule. It is also clear that if (A, μ) is a crossed G − R-bimodule and f : G′ ⟶ G is a homomorphism of groups then (A, μ) is a crossed G′ − R-bimodule induced by f, G′ acting on A and R via f.

A homomorphism f : (A, μ) ⟶ (B, λ) of precrossed (crossed) G − Rbimodules is a homomorphism of groups f : A ⟶ B such that

(1) f(^{r}a) =^{r} f(a), r ∈ R, a ∈ A, (2) f(^{g}a) =^{g} f(a), g ∈ G, a ∈ A,

(3) μ = λf.

2. The Group Der (G, (A, μ))

Consider a crossed G − R-bimodule (A, μ).

Definition 2. Denote by Der(G, (A, μ)) the set of pairs (α, r) where α is a crossed homomorphism from G to A, i.e.,

α(xy) = α(x)^{x}α(y), x, y ∈ G,

and r is an element of R such that

μα(x) = r ^{x}r^{−1}, x ∈ G.

This set will be called the set of derivations from G to (A, μ).

We define in Der(G, (A, μ)) a product by

(α, r)(β, s) = (α ∗ β, rs),

where (α ∗ β)(x) = ^{r}β(x)α(x), x ∈ G.

Proposition 3. Under the aforementioned product Der(G, (A, μ)) becomes a group which coincides with the group Der_{G}(G, A) of Guin when (A, μ) is a crossed G-module viewed as a crossed G − G-bimodule.

Proof. We have to show that (α ∗ β, rs) ∈ Der(G, (A, μ)). Put γ = α ∗ β. At first we prove that γ is a crossed homomorphism. In effect, we have

γ(xy) = ^{r}β(xy)α(xy) = ^{r}(β(x)^{x} β(x))α(x)^{x}α(y) =
= ^{r}β(x)^{rx}β(y)α(x)^{x}α(y).

On the other hand,

γ(x)^{x}γ(y) = ^{r}β(x)α(x)^{x}(^{r}β(y)α(y)) =
= ^{r}β(x)α(x)^{xr}β(y)^{x}α(y).

For any a ∈ A and (α, r) ∈ Der(G, (A, μ)) the equality

α(x)^{xr}a = ^{rx}aα(x), x ∈ G,  (1)

holds, since α(x)^{xr}aα(x)^{−1} = ^{μα(x)·xr}a = ^{r·^{x}r^{−1}}(^{xr}a) = ^{rxr^{−1}·x^{−1}·xr}a = rxa.

It follows that γ(xy) = γ(x)^{x}γ(y). Further,we have

μγ(x) = μ(^{r}β(x)α(x)) = ^{r}μβ(x)μα(x) =
= ^{r}(s^{x}s^{−1})r^{x}r^{−1} = ^{r}s^{r}(^{x}s^{−1})r^{x}r^{−1} = ^{r}sr^{x}s^{−1} ^{x}r^{−1} =
= rs^{x}(rs)^{−1}.

Therefore (α ∗ β, rs) ∈ Der(G, (A, μ)). It is evident that this product is associative. It is also obvious that

(α_{0}, 1) ∈ Der(G, (A, μ)), where α_{0}(x) = 1 for all x ∈ G, and (α_{0}, 1) is the unit of Der(G, (A, μ)).

Now we will show that for (α, r) ∈ Der(G, (A, μ)) we have

r^{−1}·xa^{r^{−1}}α(x)^{−1} = ^{r^{−1}}α(x)^{−1} ^{xr^{−1}}a, x ∈ G, a ∈ A.  (2)

Since μ(^{r^{−1}}α(x)^{−1}) = r^{−1} · μα(x)^{−1} · r = r^{−1} ^{x}r, this implies

μ(^{r^{−1}} α(x)^{−1})(^{xr^{−1}}a) = ^{r^{−1}^{x}r}(^{xr^{−1}}a) =^{r^{−1}xrx^{−1}xr^{−1}} a =^{r^{−1}x} a.

On the other hand,

μ(^{r^{−1}} α(x)^{−1})(^{xr^{−1}}a) = r^{−1}α(x) ·^{xr^{−1}} a ·^{r^{−1}} α(x)

and equality (2) is proved.

For (α, r)∈Der(G, (A, μ)) take the pair (\overline{α}, r^{−1}) where \overline{α}(x)=^{r^{−1}} α(x)^{−1}, x ∈ G. It will be shown that (\overline{α}, r^{−1}) ∈ Der(G, (A, μ)). We have

\overline{α}(xy) = ^{r^{−1}}α(xy)^{−1} = ^{r^{−1}}(^{x}α(y)^{−1} · α(x)^{−1}) =
= ^{r^{−1}x}α(y)^{−1} ^{r^{−1}}α(x)^{−1}

and \overline{α}(x) ·^{x} \overline{α}(y) = ^{r^{−1}}α(x)^{−1} ^{xr^{−1}}α(y)^{−1}.

By (2) one gets \overline{α}(xy) = \overline{α}(x)^{x}\overline{α}(y), i.e., \overline{α} is a crossed homomorphism. We also have

μ\overline{α}(x) = μ(^{r^{−1}}α(x)^{−1}) = r^{−1}μα(x)^{−1}r = r^{−1}^{x}r · r^{−1} · r = r^{−1}^{x}r.

Therefore (\overline{α}, r^{−1}) ∈ Der(G, (A, μ)).

It is easy to check that

(α, r)(\overline{α}, r^{−1}) = (\overline{α}, r^{−1})(α, r) = (α_{0}, 1).

We conclude that Der(G, (A, μ)) is a group. If (A, μ) is a crossed Gmodule and (α, g) ∈ Der(G, (A, μ)) then μα(x) = g^{x}g^{−1} = gxg^{−1}x^{−1}.

In Der_{G}(G, A) this product was defined by Guin [4] and it follows that the group Der(G, (A, μ)) coincides with Der_{G}(G, A) when (A, μ) is a crossed G-module.

If (A, μ) is a precrossed R-module and (B, λ) is a crossed R-module then (B, λ) is a crossed A − R-bimodule induced by μ and the group Der_{G}(A, B) of Guin [4] is the group Der(A, (B, λ)).

It is clear that a homomorphism of G−R-bimodules f : (A, μ) ⟶ (B, λ) induces a homomorphism

f^{∗} : Der(G, (A, μ)) ⟶ Der(G, (B, λ))

given by (α, r) ⟼ (fα, r).

There is an action of G on Der(G, (A, μ)) defined by

g(α, r) = (α̃,^{g} r), g ∈ G, r ∈ R,

with α̃(x) = ^{g}α(^{g^{−1}}x), x ∈ G.

In effect,we have

α̃(xy) = ^{g}α(^{g^{−1}}(xy)) = ^{g}α(^{g^{−1}}x^{g^{−1}}y) = ^{g}α(^{g^{−1}}x)^{xg}α(^{g^{−1}}y) =

= α̃(x)^{x}α̃(y)

and μα̃(x) = μ(^{g}α(^{g^{−1}}x)) = ^{g}μα(^{g^{−1}}x) = ^{g}(r ^{(^{g^{−1}} x)}r^{−1}) = ^{g}r^{xg}r^{−1}, whence (α̃,^{g} r) ∈ Der(G, (A, μ)). It is easy to verify that one gets an action of G on the group Der(G, (A, μ)). In effect,

g((α, r)(β, s)) =^{g} (α ∗ β, rs) = ({ \atop α ∗ β},^{g} (rs)),

where ({ \atop α × β})(x) = ^{g}(α ∗ β)(^{g^{−1}}x) = ^{g}(^{r}β(^{g^{−1}}x)) · α(^{g^{−1}}x) = ^{gr}β(^{g^{−1}}x)) ·

gα(^{g^{−1}}x) and ^{g}(α, r)^{g}(β, s) = (α̃,^{g} r)(β̃,^{g} s) = ({ \atop α ∗ β},^{g} (rs)) where (α̃ ∗

β̃)(x) = ^{^{g}r}(^{g}β(^{g^{−1}}x))^{g}α(^{g^{−1}}x) = ^{grg^{−1}}(^{g}β(^{g^{−1}}x))^{g}α(^{g^{−1}}x) = ^{gr}β(^{g^{−1}}x) gα(^{g^{−1}}x).

Thus, ^{g}((α, r)(β, s)) = ^{g}(α, r)^{g}(β, s) and it is clear that ^{gg′} (α, r) = g(^{g′} (α, r)). This action on the group Der_{G}(A, B) is defined in [4].

Let (A, μ) be a crossed G − R-bimodule. If R acts on G and the compatibility condition

(^{r}g)a =^{rgr^{−1}} a, ^{(^{r}g)}r′ = ^{rgr^{−1}}r′ for r, r′ ∈ R, g ∈ G, a ∈ A, (3)

holds, then there is also an action of R on Der(G, (A, μ)) given by

r(α, s) = (α̃,^{r} s),

where α̃(x) = ^{r}α(^{r^{−1}}x), x ∈ G.

A calculation similar to the case of the action of G on Der(G, (A, μ)) shows that (α̃,^{r} s) is an element of Der(G, (A, μ)).

Let G and R be groups acting on each other and on themselves by conjugation. It is known [8] that these actions are said to be compatible if

(^{g}r)g′ = ^{grg^{−1}}g′, ^{(^{r}g)}r′ = ^{rgr^{−1}}r′

for g, g′ ∈ G and r, r′ ∈ R.

Definition 4. It will be said that the groups G and R act on a group A compatibly if

(^{g}r)a = ^{grg^{−1}}a, ^{(^{r}g)}a = ^{rgr^{−1}}a

for g ∈ G, r ∈ R, a ∈ A.

Proposition 5. Let (A, μ) be a crossed G − R-bimodule. Let the groups G and R act on each other and on A compatibly. Under the aforementioned actions of G and R on Der(G, (A, μ)) and the homomorphism γ : Der(G, (A, μ)) ⟶ R given by (α, r) ⟼ r, the pair (Der(G, (A, μ)), γ) is a precrossed G − R-bimodule.

Proof. We have only to show that

(^{g}r)(α, s) = ^{grg^{−1}}(α, s),

for g ∈ G, r ∈ R.

In effect,

(^{g}r)(α, s) = (β,^{(^{g}r)} s),

where β(x) = ^{(^{g}r)}α(^{(^{g}r^{−1})}x) = ^{grg^{−1}}α(^{gr^{−1}g^{−1}}x), x ∈ G.

On the other hand,

grg^{−1}(α, s) = (γ,^{grg^{−1}} s),

where γ(x) = ^{grg^{−1}}α(^{gr^{−1}g^{−1}}x) and

grg^{−1}s = ^{g}(r^{g^{−1}}sr^{−1}) = ^{g}rs^{g}r^{−1} = ^{(^{g}r)}s.

Therefore ^{(^{g}r)}(α, s) = ^{grg^{−1}}(α, s).

3. The Pointed Set H^{2}(G, A)

We will use the group of derivations in a crossed bimodule to define H^{2}(G, A) when A is a crossed G-module.

We start by the following characterization of H^{2}(G, A) when A is a Z[G]- module.

Consider the diagram

M →_{l_{1}}^{⟶^{l_{0}}} F ⟶^{τ} G  (4)

where F is a free group, τ is a surjective homomorphism, M is the set of pairs (x, y), x, y ∈ F, such that τ(x) = τ(y) and l_{0}, l_{1}are canonical projections, l_{0}(x, y) = x, l_{1}(x, y) = y. Thus, (M, l_{0}, l_{1}) is the simplicial kernel of τ. Put Δ = {(x, x), x ∈ F } ⊂ M.

Let f be a map from an arbitrary group C to a group D. Then in what follows by f^{−1} : C ⟶ D will always be denoted a map with f^{−1}(c) = f(c)^{−1}, c ∈ C.

Let A be a Z[G]-module. It is clear that A is a M-module via τl_{0} and a F -module via τ. Denote by Z^{1}(M, A) (resp. Z^{1}(F, A)) the abelian group of crossed homomorphims from M to A (resp. from F to A). Let Z̃^{1}(M, A) be a subgroup of Z^{1}(M, A) consisting of all elements α such that α(Δ) = 1. There is a homomorphism

κ : Z^{1}(F, A) ⟶ Z̃^{1}(M, A)

defined by β ⟼ βl_{0}βl_{1}^{−1}.

Proposition 6. H^{2}(G, A) is canonically isomorphic to Coker κ.

Proof. It is sufficient to show that Coker κ is isomorphic to Opext(G, A, φ) where φ : G ⟶ Aut(A) denotes the action of G on A.

Let α ∈ Z̃^{1}(M, A) and introduce in the semi-direct product A ▷◁ F the relation

(a, x) ∼ (a′, x′) ⟺ τ(x) = τ(x′)

and a · α(x, x′) = a′.

It is easy to see that this relation is an equivalence; use the fact that if (x, x′, x′′) is a triple of elements of F such that τ(x) = τ(x′) = τ(x′′) then α(x, x′′) = α(x, x′)α(x′, x′′). Denote this equivalence by ρ and take the quotient set (A ▷◁ F)/ρ. We will show that ρ is in fact a congruence and therefore C = (A ▷◁ F)/ρ is a group.

Let (a, x) ∼ (a′, x′) and (b, y) ∼ (b′, y′). Then τ(x) = τ(x′), τ(y) = τ(y′), aα(x, x′) = a′, bα(y, y′) = b′.

Further, (a, x)(b, y) = (a^{x}b, xy), (a′, x′)(b′, y′) = (a′ ^{x′} b′, x′y′).

We have

xb^{x}α(y, y′) = ^{x}b′ = ^{x′} b′,

whence a·α(x, x′)^{x}b^{x}α(y, y′)=a′ ^{x′} b′. Since α(xy, x′y′)=α(x, x′)^{x}α(y, y′), it follows that

a^{x}bα(xy, x′y′) = a′ ^{x′} b′ .

One gets a commutative diagram

M ⟶^{⟶}^{l_{0}}_{l_{1}} F ⟶^{τ} G
↓ α  ↓ β  ∥
A ⟶^{σ} C ⟶^{ψ} G

where σ(a) = [(a, 1)], ψ[(a, x)] = τ(x), β(x) = [(1, x)]. Denote by E the exact sequence

0 ⟶ A ⟶^{σ} C ⟶^{ψ} G ⟶ 1

which gives an element of Opext(G, A, φ).

Define a map

ϑ : Coker κ ⟶ Opext(g, A, φ)

given by [α] ⟼ [E].

By standard calculations it can be easily proved that ϑ is a correctly defined homomorphism which is bijective.

Let (A, μ) be a crossed G-module. Then (A, μ) is a crossed M − Gbimodule induced by τl_{0} ( or by τl_{1} ) and a crossed F −G-bimodule induced by τ (see diagram (4)).

Consider the group Der(M, (A, μ)) and let D̃er(M, (A, μ)) be the subgroup of Der(M, (A, μ)) consisting of elements (α, g) such that α(Δ) = 1. If (α, g) ∈ D̃er(M, (A, μ)) this implies g ∈ Z(G). Then we have μα(m) = 1 for any m ∈ M and α(M) ⊂ Z(A). Denote by Z̃^{1}(M, (A, μ)) a subset of D̃er(M, (A, μ)) consisting of all elements of the form (α, 1).

Define, on the set Z̃^{1}(M, (A, μ)), a relation

(α′, 1) ∼ (α, 1) ⇔ ∃(β, h) ∈ Der(F, (A, μ))

such that

(α′, 1) = (βl_{0}, h)(α, 1)(βl_{1}, h)^{−1}

in the group Der(M, (A, μ)).

We see that if (α′, 1) ∼ (α, 1) one has

α′(x) = βl_{1}(x)^{−1}^{h}α(x)βl_{0}(x), x ∈ M,

for some (β, h) ∈ Der(F, (A, μ)).

Proposition 7. The relation ∼ defined on Z̃^{1}(M, (A, μ)) is an equivalence.

Proof. The reflexivity is clear. If (α′, 1) ∼ (α, 1), i.e., (α′, 1) = (βl_{0}, h)(α, 1) (βl_{1}, h)^{−1} where (β, h) ∈ Der(F, (A, μ)), then (α, 1) = (βl_{0}, h)^{−1}(α′, 1) (βl_{1}, h) where (βl_{0}, h)^{−1} = (β̃l_{0}, h^{−1}) and (βl_{1}, h) = (β̃l_{1}, h^{−1})^{−1} with (β̃, h^{−1}) = (β, h)^{−1} ∈ Der(F, (A, μ)). Thus the relation ∼ is symmetric.

Let (α′, 1) ∼ (α, 1) and (α′′, 1) ∼ (α′, 1); then one has

(α′, 1) = (βl_{0}, h)(α, 1)(βl_{1}, h)^{−1},
(α′′, 1) = (β′l_{0}, h′)(α′, 1)(β′l_{1}, h′)^{−1},

where (β, h), (β′, h′) ∈ Der(F, (A, μ)).

It follows that

(α′′, 1) = (β′l_{0}, h′)(βl_{0}, h)(α, 1)(βl_{1}, h)^{−1}(β′l_{1}, h′)^{−1} =
= ((β′ ∗ β)l_{0}, h′h)(α, 1)((β′ ∗ β)l_{1}, h′h)^{−1},

where (β′ ∗ β, h′h) = (β′, h′)(β, h) ∈ Der(F, (A, μ)). This means that (α′′, 1) ∼ (α, 1) and the relation ∼ is an equivalence.

Proposition 8. Let (A, μ) be a crossed G-module. Then the quotient set Z̃^{1}(M, (A, μ))/ ∼ is independent of the diagram (4) and is unique up to bijection.

We need the

Lemma 9. Let A be a G-group and let α : M ⟶ A be a crossed homomorphism such that α(Δ) = 1. Then there exists a map q : F ⟶ A such that

α(y) = ql_{1}(y)^{−1}ql_{0}(y), y ∈ M.

Proof. Observe that if (x, x′′), (x′, x′′) ∈ M, then α(x, x′′) = α(x′, x′′) α(x, x′). In effect,the equality (x, x′′) = (1, x′′x^{′−1})(x, x′) implies α(x, x′′) = α(1, x′′x^{′−1}) α(x, x′). But (x′, x′′) = (1, x′′x^{′−1})(x′, x′). Thus α(x′, x′′) = α(1, x′′x^{′−1}) α(x′, x′) = α(1, x′′x^{′−1}) and we get the desired equality.

In particular, applying this equality one gets α(x, x) = α(x′, x) · α(x, x′) for (x, x), (x′, x) ∈ M. Therefore α(x′, x) = α(x, x′)^{−1} for any (x, x′) ∈ M.

Take a section η : G ⟶ F, τη = 1_{G} and define a map q : F ⟶ A by

q(x) = α(x, ητ(x)), x ∈ F.

For (x, x′) ∈ M one has

ql_{1}(x, x′)^{−1}ql_{0}(x, x′) = q(x′)^{−1}q(x) = (α(x′, ητ(x′))^{−1}α(x, ητ(x)) =
= α(ητ(x′), x′)α(x, ητ(x)).

On the other hand, since α(x, x′) = α(1, x′x^{−1}) for all (x, x′) ∈ M, one has α(ητ(x′), x′) = α(1, x′ητ(x′)^{−1}) and α(x, ητ(x)) = α(1, ητ(x)x^{−1}).

But (1, x′ητ(x′)^{−1})(1, ητ(x)x^{−1}) = (1, x′x^{−1}). Therefore, α(x, x′) = α(1, x′ητ(x′)^{−1})α(1, ητ(x)x^{−1}) = ql_{1}(x, x′)^{−1}ql_{0}(x, x′).

Proof of Proposition 8. Consider a commutative diagram

M′  →_{l′_{1}}^{⟶^{l{′ \atop 0}}} F′  ⟶^{τ′} G
\overline{γ_{1}} ↓ \overline{γ_{2}}  γ_{1} ↓ γ_{2}  ∥
M  →_{l_{1}}^{⟶^{l_{0}}}  F  ⟶^{τ} G

where (M, l_{0}, l_{1}) and (M′, l0^{′}, l1^{′}) are the simplicial kernels of τ_{1}and τ_{2} res-

pectively, l_{i}\overline{γ_{1}} = γ_{1}li^{′}, l_{i}\overline{γ_{2}} = γ_{2}li^{′}, i = 0, 1, τγ_{1} = τγ_{2} = τ′.

The pair (γ_{i}, \overline{γ_{i}}) induces a homomorphism

Der(M, (A, μ)) ⟶ Der(M′, (A, μ))

given by (α, g) ⟼ (α\overline{γ_{i}}, g), i = 1, 2.

If (α′, 1) ∼ (α, 1), i.e.,

(α′, 1) = (βl_{0}, h)(α, 1)(βl_{1}, h)^{−1}

with (β, h) ∈ Der(F, (A, μ)), then

α′ \overline{γ_{i}}(y) = βγ_{i}l1^{′}(y)^{−1}^{h}α\overline{γ_{i}}(y)βγ_{i}l0^{′}(y), y ∈ M′.

Thus (α′ \overline{γ_{i}}, 1) ∼ (α\overline{γ_{i}}, 1), i = 1, 2, and one gets a natural map

ϵ_{i} : Z̃^{1}(M, (A, μ))/ ∼⟶ Z̃^{1}(M′, (A, μ))/ ∼

induced by the pair (γ_{i}, \overline{γ_{i}}) and given by [(α, 1)] ⟼ [(α\overline{γ_{i}}, 1)], i = 1, 2.

We will show that ϵ_{1} = ϵ_{2}. By Lemma 9 there is a map q : F ⟶ A such that

α(y) = ql_{1}(y)^{−1}ql_{0}(y), y ∈ M.

Consider the homomorphism s : F′ ⟶ M given by

s(x′) = (γ_{1}(x′), γ_{2}(x′)), x′ ∈ F′.

It is clear that (αs, 1) ∈ Der(F′, (A, μ)).

Further we have

((αsl_{1})^{−1}α\overline{γ_{2}}αsl0^{′})(x0^{′}, x1^{′}) = αs(x1^{′})^{−1} α\overline{γ_{2}}(x0^{′}, x1^{′})αs(x0^{′}) =

= α(γ_{1}(x1^{′}), γ_{2}(x1^{′}))^{−1}α\overline{γ_{2}}(x0^{′}, x1^{′}) α(γ_{1}(x0^{′}), γ_{2}(x0^{′})) = qγ_{1}(x1^{′})^{−1}qγ_{2}(x1^{′}) =

= qγ_{2}(x1^{′})^{−1}qγ_{2}(x0^{′})qγ_{2}(x0^{′})^{−1}qγ_{1}(x0^{′}) = qγ_{1}(x1^{′})^{−1}qγ_{1}(x0^{′}) = α\overline{γ_{1}}(x0^{′}, x1^{′})

for (x0^{′}, x1^{′}) ∈ M′.

Therefore (α\overline{γ_{1}}, 1) ∼ (α\overline{γ_{2}}, 1) with (αs, 1) ∈ Der(F′, (A, μ)) and one gets ϵ_{1} = ϵ_{2}.

The rest of the proof of the uniqueness is standard.

Let (A, μ) be a crossed G − R-bimodule. Denote by I Der(G, (A, μ)) a subgroup of Der(G, (A, μ)) consisting of elements of the form (α, r) with r ∈ H^{0}(G, R). If (A, μ) is a crossed G-module viewed as a crossed G − Gbimodule then

I Der(G, (A, μ)) = {(α, r), g ∈ Z(G)}.

Consider the diagram

M_{G} ⟶^{⟶}_{l_{1}}^{l_{0}} F_{G} ⟶^{τ_{G}} G  (5)

where F_{G} is the free group generated by G, τ_{G} is the canonical homomorphism and (M_{G}, l_{0}, l_{1}) is the simplicial kernel of τ_{G}.

Proposition 10. Let (A, μ) be a crossed G-module. Then:

(i) there is a canonical surjective map

ϑ′ : H^{2}(G, Kerμ) ⟶ Z̃^{1}(M_{G}, (A, μ))/ ∼

given by the composite map [E] ⟼^{ϑ^{−1}} [α] ⟼ [(α, 1)];

(ii) if we assume Der(F_{G}, (A, μ)) = I Der(F_{G}, (A, μ)) (in particular, it is so if either μ is the trivial map or G is abelian) we can introduce, in the pointed set Z̃^{1}(M_{G}, (A, μ))/ ∼, an abelian group structure defined by

[(α, 1)][(β, 1)] = [(α ∗ β, 1)]

where (A, μ) is viewed as a crossed F_{G} − G-bimodule induced by τ_{G}and [(α, 1)] denotes the equivalence class containing (α, 1). Under this product the map ϑ′ becomes an isomorphism.

Proof. To prove (i) we have only to show the correctness of [α] ⟼ [(α, 1)] where α : M_{G} ⟶ A is a crossed homomorphism with α(Δ) = 1 and α(M_{G}) ⊂ Ker μ.

Let α′ ∈ [α], i.e.,

α′(x) = βl_{1}^{−1}(x)α(x)βl_{0}(x), x ∈ M_{G},

where β : F_{G} ⟶ Ker μ is a crossed homomorphism. Then (β, 1) ∈ Der(F_{G}, (A, μ)) and we have

(α′, 1) = (βl_{0}, 1)(α, 1)(βl_{1}, 1)^{−1}.

The surjectivity of ϑ′ is clear.

(ii) Let (α′, 1) ∼ (α, 1). Then

α′(x) = ηl_{1}(x)^{−1}^{g}α(x)ηl_{0}(x), x ∈ M_{G},

for some (η, g) ∈ Der(F_{G}, (A, μ)). By assumption, g ∈ Z(G). Thus μη(x) = gxg^{−1}x^{−1} = 1, x ∈ M_{G}. It follows that [α′] = [^{g}α] with g ∈ Z(G). But it is known that Z(G) acts trivially on H^{2}(G, Ker μ). Therefore we have [α′] = [α]. Hence there is a crossed homomorphism γ : F_{G} ⟶ Ker μ such that

α′(x) = γl_{1}^{−1}(x)α(x)γl_{0}(x), x ∈ M_{G}.

It follows that if (α, 1) ∼ (α′, 1) and (β, 1) ∼ (β′, 1) then (α ∗ β, 1) ∼ (α′ ∗ β′, 1). We conclude that the product is correctly defined and the map ϑ′ is an isomorphism when Der(F_{G}, (A, μ)) = I Der(F_{G}, (A, μ)).

Proposition10 motivates the following definition of the second cohomology of groups with coefficients in crossed modules.

Definition 11. Let (A, μ) be a crossed G-module. One denotes by H^{2}(G, A) the quotient set Z̃^{1}(M, (A, μ)) / ∼ which will be called the second set of cohomology of G with coefficients in the crossed G-module (A, μ).

Remark. Using diagram (4) it is possible to define the second cohomology of G with coefficients in a crossed G-module (A, μ) by a different “less abelian” way. Consider the set Z̃^{1}(M, A) of all crossed homomorphims α : M ⟶ A with α(Δ) = 1 (the equality μα = 1 is not required and therefore α(M) is not necessarily contained in Z(A)). Introduce in Z̃^{1}(M, A) a relation ∼ of equivalence as follows:

α′ ∼ α if ∃ (β, h) Der(F, (A, μ)) such that α′(x) = βl_{1}(x)^{−1} ^{h}α(x) βl_{0}(x), x ∈ M.

Define \overline{H^{2}}(G, A) = Z̃^{1}(M, A) / ∼. It is obvious that H^{2}(G, A) ⊂ \overline{H^{2}}(G, A). But it seems the exact cohomology sequence (Theorem 13) does not hold for \overline{H^{2}}(G, A).

It is clear H^{2}(G, A) is a pointed set with [(α_{0}, 1)] as a distinguished element where α_{0}(y) = 1 for all y ∈ M.

A homomorphism of crossed G-modules f : (A, μ) ⟶ (B, λ) induces a map of pointed sets

f^{2} : H^{2}(G, A) ⟶ H^{2}(G, B), f^{2}([(α, 1)]) = [(fα, 1)].

There is an action of G on F_{G} (see diagram (5)) defined as follows:

g(| g_{1} |^{ϵ} ⋯ | g_{n} |^{ϵ}) =|^{g} g_{1} |^{ϵ} ⋯ |^{g} g_{n} |^{ϵ}, g, g_{1}, …, g_{n} ∈ G,

where ϵ = ±1.

This action induces an action of G on M_{G} by

g(x, x′) = (^{g}x,^{g} x′), g ∈ G, (x, x′) ∈ M_{G}.

Let (A, μ) be a crossed G-module. Then we have an action of G on Der(M_{G}, (A, μ)) given by

g(α, h) = (α̃,^{g} h)

where α̃(m) = ^{g}α(^{g^{−1}}m), g ∈ G, m ∈ M_{G}.

Proposition 12. Let (A, μ) be a crossed G-module. There is an action of G on H^{2}(G,A) induced by the above-defined action of G on Der(M_{G}, (A,μ)) under which the center Z(G) acts trivially.

Proof. Obviously, the action of G on Der(M_{G}, (A, μ)) induces an action of G on D̃er(M_{G}, (A, μ)). Thus one gets an action of G on Z̃^{1}(M_{G}, (A, μ)).

If (α′, 1) ∼ (α, 1), where (α, 1), (α′, 1) ∈ Z̃^{1}(M_{G}, (A, μ)), we have

α′(y) = βl_{1}(y)^{−1} ^{h}α(y)βl_{0}(y), y ∈ M_{G},

for some (β, h) ∈ Der(F_{G}, (A, μ)).

This implies

gα′(^{g^{−1}}y) = ^{g}βl_{1}(^{g^{−1}}y)^{−1} ^{gh}α(^{g^{−1}}y)^{g}βl_{0}(^{g^{−1}}y), y ∈ M_{G}.

Hence α̃′(y) = ^{g}β(^{g^{−1}}l_{1}(y))^{−1} ^{ghg^{−1}} α̃(^{g^{−1}}y)^{g}β(^{g^{−1}}l_{0}(y)) with (β̃,^{g} h) ∈ Der(F_{G}, (A, μ)). Therefore (α̃^{1}, 1) ∼ (α̃, 1). It is obvious that the abovedefined map ϑ′ : H^{2}(G, ker μ) ⟶ H^{2}(G, A) is a G-map. Since ϑ′ is surjective (see Proposition 11) and Z(G) acts trivially on H^{2}(G, ker μ), it follows that Z(G) acts trivially on H^{2}(G, A) too.

4. An Exact Cohomology Sequence

For any G-group A denote by H^{0}(G, A) a subgroup of A consisting of all invariant elements under the action of G on A.

Theorem 13. Let

1 ⟶ (A, 1) ⟶^{φ} (B, μ) ⟶^{ψ} (C, λ) ⟶ 1  (6)

be an exact sequence of crossed G-modules. Then there is an exact sequence

1 ⟶ H^{0}(G, A) ⟶^{φ^{0}} H^{0}(G, B) ⟶^{ψ^{0}} H^{0}(G, C) ⟶^{δ^{0}} H^{1}(G, A) ⟶^{φ^{1}}

⟶^{φ^{1}} H^{1}(G, B) ⟶^{ψ^{1}} H^{1}(G, C) ⟶^{δ^{1}} H^{2}(G, A) ⟶^{φ^{2}} H^{2}(G, B) ⟶^{ψ^{2}} H^{2}(G, C)

where φ^{0}, ψ^{0}, δ^{0}, φ^{1}, ψ^{1} are group homomorphisms, δ^{1} is a crossed homomorphism under the action of H^{1}(G, C) on H^{2}(G, A) induced by the action of G on A, and φ^{2}, ψ^{2} are maps of pointed G-sets.

Proof. The exactness of

1 ⟶ H^{0}(G, A) ⟶^{φ^{0}} H^{0}(G, B) ⟶^{ψ^{0}} H^{0}(G, C) ⟶^{δ^{0}} H^{1}(G, A) ⟶^{φ^{1}}

⟶^{φ^{1}} H^{1}(G, B) ⟶^{ψ^{1}} H^{1}(G, C) ⟶^{δ^{1}} H^{2}(G, A)

is proved in [4].

We have only to show the exactness of

H^{1}(G, C) ⟶^{δ^{1}} H^{2}(G, A) ⟶^{φ^{2}} H^{2}(G, B) ⟶^{ψ^{2}} H^{2}(G, C).

Let [(α, g)] ∈ H^{1}(G, C) and δ^{1}[(α, g)] = [γ]. Then one has a commutative diagram

M ⟶^{⟶}_{l_{1}}^{l_{0}} F ⟶^{τ} G
↓ γ  ↓ β  ↓ α
A ⟶^{φ} B ⟶^{ψ} C

where φγ(y) = βl_{1}(y)^{−1}βl_{0}(y), y ∈ M, and β is a crossed homomorphism, B being an F -group via τ. The existence of such β follows from the following assertion: if we have a surjective homomorphism ψ : B ⟶ C of F -groups and f : F ⟶ C is a crossed homomorphism where F is a free group, then there is a crossed homomorphism β : F ⟶ B such that ψβ = f. In effect, take the semi-direct product B ▷◁ F and consider a subgroup Y of B ▷◁ F consisting of all elements (b, x) such that ψ(b) = f(x). Then we have a commutative diagram

Y ⟶^{pr_{2}} F
↓ pr_{1}  ↓ f
B ⟶^{ψ} C

where pr_{i} is the projection, i = 1, 2, and pr_{2} is surjective. Thus, since F is a free group, there is a homomorphism f′ : F ⟶ Y such that ff′ = 1_{F}. Then pr_{1}f′ is the required crossed homomorphism.

It will be shown that (β, g) ∈ Der(F, (B, μ)). One has

gτ(x)g^{−1}τ(x)^{−1} = λατ(x) = λψβ(x) = μβ(x), x ∈ F.

This means (β, g) ∈ Der(F, (B, μ)). It is clear that φ^{2}δ^{1}([α, g]) = φ^{2}([γ]) = [(φγ, 1)] and (φγ, 1) ∼ (α_{0}, 1) (use (β, g) ∈ Der(F, (B, μ))), where α_{0}(x) = 1 for all x ∈ F. Therefore Im δ^{1} ⊂ ker φ^{2}.

Let [γ] ∈ H^{2}(G, A) such that φ^{2}([γ]) = [(φγ, 1)] = [(α_{0}, 1)]. Then there exists (β, h) ∈ Der(F, (B, μ)) such that

φγ(y) = βl_{1}(y)^{−1} · βl_{0}(y), y ∈ M,

whence ψβl_{0}(y) = ψβl_{1}(y), y ∈ M. It follows that there is a crossed homomorphism α : G ⟶ C such that ατ = ψβ. We have to show (α, h) ∈ Der_{G}(G, C). In effect, h τ(x) h^{−1} τ(x)^{−1} = μβ(x) = λψβ(x) = λατ(x), x ∈ F. This implies (α, h) ∈ Der_{G}(G, C). It is clear that δ^{1}([α, h]) = [γ]. Therefore, ker φ^{2} ⊂ Im δ^{1}.

It is obvious that Im φ^{2} ⊂ ker ψ^{2}.

Let [(α, 1)] ∈ H^{2}(G, B) such that ψ^{2}([α, 1]) = [(ψα, 1)] = [(α_{0}, 1)]. Then there exists (β, h) ∈ Der(F, (C, λ)) such that

ψα(y) = βl_{1}(y)^{−1}βl_{0}(y), y ∈ M.

It follows that there is a crossed homomorphism β′ : F ⟶ B such that ψβ′ = β. One gets the following commutative diagram:

\begin{aligned}M & ⟶^{⟶}_{l_{1}}^{l_{0}} & F & ⟶^{τ} & G \\ & ↘ α & ↓ β′ & ↘ β \\ A & ⟶^{φ} & B & ⟶^{ψ} & C \end{aligned} .

Thus, ψα(y)=ψβ′l_{1}(y)^{−1} ψβ′l_{0}(y), y ∈M, so that ψ(β′l_{1}(y)α(y) β′l_{0}(y)^{−1}) = 1, y ∈ M. One gets β′l_{1}(y)α(y)β′l_{0}(y)^{−1} ∈ φ(A), y ∈ M. Denote by γ : M ⟶ A the crossed homomorphism given by γ(y) = φ^{−1}(β′l_{1}(y)α(y)β′ l_{0}(y)^{−1}). Then ^{h^{−1}}γ(y) = φ^{−1}(^{h^{−1}}β′l_{1}(y)^{h^{−1}}α(y) ·^{h^{−1}} β′l_{0}(y)^{−1}), y ∈ M.

In the group Der(F, (B, μ)) consider the product

(β′l_{0}, h)^{−1}(α, 1)(β′l_{1}, h) = (η, 1)

where η(y) = ^{h^{−1}}β′l_{1}(y)^{h^{−1}}α(y)^{h^{−1}}β′l_{0}(y)^{−1}, y ∈ M.

This implies that the map given by

y ⟼^{h^{−1}} γ(y), y ∈ M,

is a crossed homomorphism.

Take [φ^{−1}η] ∈ H^{2}(G, A). Then we have φ^{2}([φ^{−1}η]) = [(η, 1)]. But (η, 1) ∼ (α, 1) by the above equality with (β′, h)^{−1} ∈ Der(F, (B, μ)). Therefore, ker ψ^{2} ⊂ Im φ^{2}.

Any crossed G-module (A, μ) induces the following short exact sequence of crossed G-modules:

1 ⟶ (ker μ, 1) ⟶^{φ} (A, μ) ⟶^{ψ} (Im μ, σ) ⟶ 1

where σ : Im μ ⟶ G denotes the inclusion and G acts on Im μ by conjugation.

Corollary 14. If (A, μ) is a crossed G-module there is an exact sequence

1 ⟶ H^{0}(G, ker μ) ⟶^{φ^{0}} H^{0}(G, A) ⟶^{ψ^{0}} H^{0}(G, Im μ) ⟶^{δ^{0}} H^{1}(G, ker μ) ⟶^{φ^{1}}

⟶^{φ^{1}} H^{1}(G, A) ⟶^{ψ^{1}} H^{1}(G, Im μ) ⟶^{δ^{1}} H^{2}(G, ker μ) ⟶^{ϑ′} H^{2}(G, A)

⟶ 1.

For the exact sequence (6) a connecting map

δ^{2} : H^{2}(G, C) ⟶ H^{3}(G, A)

will be defined, and for this we will use the equivalence of functors H^{n+1}(−, A) ≈ L_{n} Der(−, A), n ≥ 1 [9] when A is a Z[G]-module, where L_{n} Der(−, A) is the non-abelian nth derived functor of the contravariant functor Der(−, A) from the category \underline{D}_{G} of groups acting on A to the category of abelian groups.

Consider the following canonical free simplicial resolution of G in the category \underline{D}_{G}:

⋮_{⟶}^{⟶} F_{3} ⟶^{τ_{3}} M_{2} ⋮_{⟶_{l^{2}_{3}}}^{⟶^{l^{2}_{0}}} F_{2} ⟶^{τ_{2}} M_{1} ⟶_{⟶_{l^{1}_{2}}}^{⟶^{l^{1}_{0}}} F_{1} ⟶^{τ_{1}} M_{0} ⟶^{⟶}_{l_{1}^{0}}^{l_{0}^{0}} F_{0} ⟶^{τ_{0}} G  (7)

where F_{0} = F_{G}, F_{i} = F_{M_{i−1}}, i ≥ 1, τ_{i} is the canonical homomorphism, and (M_{i}, l_{0}^{i}, …, l_{i+1}^{i}) is the simplicial kernel of (l_{0}^{i−1}τ_{i}, …, l_{i}^{i−1}τ_{i}), i ≥ 0 (see [10]).

There is an action of Der(F_{0}, (C, λ)) on H^{3}(G, A) defined as follows. Let [f] ∈ H^{3}(G, A), where f : F_{2} ⟶ A is a crossed homomorphism such

that 3 ; ∏_{i=0} (fl_{i}^{1}τ_{3})^{ϵ} = 1, where ϵ = (−1)^{i}, and let (α, g) ∈ Der(F_{0}, (C, λ)).

Define

(α,g)[f] = [^{g}f].

We have first to show that ^{g}f is a crossed homomorphism.

In the group Der(F_{2}, (B, μ)) ( (B, μ) being a crossed F_{2}-G-bimodule induced by τ_{0}∂_{0}^{1}∂_{0}^{2} where ∂_{0}^{1} = l_{0}^{0}τ_{1}, ∂_{0}^{2} = l_{0}^{1}τ_{2}) take the product

(β∂_{0}^{1}∂_{0}^{2}, g)(φf, 1)(β∂_{0}^{1}∂_{0}^{2}, g)^{−1} = (f̃, 1),

where f̃is a crossed homomorphism and β : F_{0} ⟶ B is a crossed homomorphism too such that ψβ = α (such β exists, since F_{0} is a free group and ψ is surjective).

One has

f̃(x) = β∂_{0}^{1}∂_{0}^{2}(x)^{−1}^{g}φf(x)β∂_{0}^{1}∂_{0}^{2}(x) =^{g} φf(x), x ∈ F_{2}.

We show now the correctness. If f ∼ f′ then there is a crossed homomorphism η : F_{1} ⟶ A such that

f′(x) = f(x) 2 ; ∏_{i=0} (ηl_{i}^{1}τ_{2}(x))^{ϵ}, x ∈ F_{2},

where ϵ = (−1)^{i}. It can be shown in the same manner as for f that ^{g}η is a crossed homomorphism if (α, g) ∈ Der(F_{0}, (C, λ)).

Thus one gets

gf′ = ^{g}f ∏_{i=0}^{2} ^{g}(ηl_{i}^{1}τ_{2})^{ϵ},

where ϵ = (−1)^{i}. This implies

[^{g}f′] = [^{g}f].

Therefore the action of Der(F_{0}, (C, λ)) on H^{3}(G, A) is correctly defined.

Let (α, 1) ∈ D̃er(M_{0}, (C, λ)) and let β : F_{1} ⟶ B be a crossed homomorphism (F_{1} acts on B via τ_{0}l_{0}^{0}τ_{1} = τ_{0}l_{0}^{1}τ_{1}) such that ψβ = ατ_{1}. Then

we have β(F_{1}) ⊂ Z(B) and define a crossed homomorphism \overline{β} = 2 ; ∏_{i=0} (βl_{i}^{1})^{ϵ} :

M_{1} ⟶ B, ϵ = (−1)^{i}. Hence ψ\overline{β}(y) = 2 ; ∏_{i=0} ψβl_{i}^{1}(y)^{ϵ} = 2 ; ∏_{i=0} ατ_{1}l_{i}^{1}(y)^{ϵ}, y ∈ M_{1},

ϵ = (−1)^{i}.

It is easy to see that

τ_{1}l_{0}^{1}(y)τ_{1}l_{1}^{1}(y)^{−1}τ_{1}l_{2}^{1}(y) ∈ Δ, y ∈ M_{1}.

Since α(Δ) = 1, there is a crossed homomorphism γ : F_{2} ⟶ A such that φγ = \overline{β}τ_{2}.

Theorem 15. Let (6) be an exact sequence of crossed G-modules. If either the action of Der(F_{0}, (C, λ)) on H^{3}(G, A) is trivial (in particular, if G acts trivially on A) or Der(F_{0}, (C, λ)) = I Der(F_{0}, (C, λ)) (in particular, if either λ = 1 or G is abelian) then the connecting map δ^{2} : H^{2}(G, C) ⟶ H^{3}(G, A) is defined by

δ^{2}([α, 1]) = [γ], (α, 1) ∈ D̃er(M_{0}, (C, λ)),

and the sequence

H^{2}(G, B) ⟶^{ψ^{2}} H^{2}(G, C) ⟶^{δ^{2}} H^{3}(G, A)

is exact.

Proof. We have to show that δ^{2} is correctly defined.

Let Der(F_{0}, (C, λ)) act trivially on H^{3}(G, A). If (α′, 1) ∈ [(α, 1)] ∈ H^{2}(G, A), we have

α′(x) = ηl_{1}^{0}(x)^{−1}^{g}α(x)ηl_{0}^{0}(x), x ∈ F_{0},

with (η, g) ∈ Der(F_{0}, (C, λ)).

Take a crossed homomorphism η′ : F_{0} → B such that ψη′ = η. Recall that ψβ = ατ_{1}. Consider the product

(η′l_{0}^{0}τ_{1}, g)(β, 1)(η′l_{1}^{0}τ_{1}, g)^{−1} = (β′, 1)

in the group Der(F_{1}, (B, μ)). Then

β′(x) = η′l_{1}^{0}τ_{1}(x)^{−1}^{g}β(x)η′l_{0}^{0}τ_{1}(x), x ∈ F_{1}.

Thus ψβ′ = α′τ_{1} and β′(F_{1}) ⊂ Z(B). Note that this implies η′l_{1}^{0}(x)^{−1} η′L_{0}^{0}(x) ∈ Z(B) for all z ∈ M_{0}.

Further,

β′(l_{0}^{1}(y))β′(l_{1}^{1}(y)^{−1})β′(l_{2}^{1}(y)) = ^{g}β(l_{0}^{1}(y))η′l_{1}^{0}(l_{0}^{1}(y))^{−1}η′l_{0}^{0}(l_{0}^{1}(y))

gβ(l_{1}^{1}(y))^{−1}η′l_{1}^{0}(l_{1}^{1}(y)^{−1})^{−1}η′l_{0}^{0}(l_{1}^{1}(y)^{−1}^{g}β(l_{2}^{1}(y))η′l_{1}^{0}(l_{2}^{1}(y))^{−1}
η′l_{0}^{0}(l_{2}^{1}(y)) = ^{g}\overline{β}(y).

Therefore \overline{β′}τ_{2} = ^{g}(\overline{β}τ_{2}) and [γ′] = [^{g}γ] = [γ].

If Der(F_{0}, (C, λ)) = I Der(F_{0}(C, λ)) and α′, 1) ∼ (α, 1) then there is an element (η, g) ∈ Der(F_{0}, (C, λ)) such that

α′(x) = ηl_{1}^{0}(x)^{−1}α(x)ηl_{0}^{0}(x), x ∈ F_{0},

(see the proof of Proposition 10 (ii)) and it is clear that in both cases δ^{2} is correctly defined.

We will now prove the exactness. Let [(α, 1)] ∈ H^{2}(G, B). Then δ^{2}ψ^{2}[(α, 1)] = δ^{2}([(ψα, 1)]) = [γ], where φγ = \overline{β}τ_{2} and \overline{β} is taken such that β = ατ_{1}. Thus

\overline{β}(y) = ατ_{1}l_{0}^{1}(y)ατ_{1}l_{1}^{1}(y)^{−1}ατ_{1}l_{2}^{1}(y), y ∈ M_{1}.

Since τ_{1}L_{0}^{1}(y) τ_{1}L_{1}^{1}(y)^{−1} τ_{1}L_{2}^{1}(y) ∈ Δ, y ∈ M_{1} and α(Δ) = 1, this implies \overline{β} = 1. Thus we have Im ψ^{2} ⊂ ker δ^{2}.

Let [(α, 1)] ∈ H^{2}(G, C) such that δ^{2}([(α, 1)]) = 1. Then we have [γ] = 1 with φγ = \overline{β}τ_{2}, where \overline{β} : M_{+}1 → B is a crossed homomorphism such that \overline{β} = βl_{0}^{1}(βl_{1}^{1})^{−1}βl_{2}^{1} with ψβ = ατ_{1} and β : F_{1} → B a crossed homomorphism.

It follows that there exists a crossed homomorphism η : F_{1} → A such that γ = (ηl_{0}^{1}(ηl_{1}^{1})^{−1}ηl_{2}^{1})τ_{2}.

Thus we have \overline{β}τ_{2} = φ(ηl_{0}^{1}(ηl_{1}^{1})^{−1}ηl_{2}^{1})τ_{2}, whence \overline{β}(y) = φηl_{0}^{1}(y)φηl_{1}^{1}(y)^{−1} φηl_{2}^{1}, y ∈ M_{1}.

For y_{1}, y_{2} ∈ F_{1} such that τ_{1}(y_{1}) = τ_{1}(y_{2}) = x, since (y_{1}, y_{2}, s_{0}^{0}l_{1}^{0}(x)) ∈ M_{1}, one gets β(y_{1})β(y_{2})^{−1}βs_{0}^{0}l_{1}^{0}(x) = φη(y_{1})φη(y_{2})^{−1}φη(s_{0}^{0}l_{1}^{0}(x)), where s_{0}^{0} : F_{0} → F_{1} is the degeneracy map. In particular, if y_{1} = y_{2} we have

β(s_{0}^{0}l_{1}^{0}(x)) = φη(s_{0}^{0}l_{1}^{0}(x)), x ∈ M_{0}.

Therefore β(y_{1})φη(y_{1})^{−1} = β(y_{2})φη(y_{2})^{−1} if τ_{1}(y_{1}) = τ_{1}(y_{2}).

This implies a crossed homomorphism β′ : M_{0} → B given by

β′(x) = β(y)φη(y)^{−1}, x ∈ M_{0},

where τ_{1}(y) = x. Thus β′τ_{1} = β and β′(M_{0}) ⊂ Z(B). If x ∈ Δ ⊂ M_{0} then τ_{1}s_{0}^{0}l_{1}^{0}(x) = x. Therefore we have

β′(x) = β(s_{0}^{0}l_{1}^{0}(x)) = φη(s_{0}^{0}l_{1}^{0}(x))^{−1}, x ∈ Δ.

Whence β′(x) = 1 if x ∈ Δ.

On the other hand,

μβ′(x)μβ(y)μφη(y)^{−1} = μβ(y) = λψβ(y) = λατ_{1}(y) = 1.

We conclude that (β′, 1) ∈ Z̃^{1}(M_{0}, (B, μ)) and it is clear that ψ^{2}([(β′, 1)]) = [(α, 1)].

Acknowledgement

The research described in this publication was made possible in part by Grant MXH200 from the International Science Foundation and by INTAS Grant No 93-2618.

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(Received 2.06.1995)

Author’s address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia