ON (CO)HOMOLOGY OF 2-TYPES AND CROSSED MODULES

T. DATUASHVILI and T. PIRASHVILI

A. Razmadze Math. Inst., Aleksidze str. 1, Tbilisi, 380093. Republic of Georgia

1 Introduction

A connected CW -complex X is called 2-type if π_{i}X = 0 for i ≠ 1, 2. In [8] Ellis investigated the homology of 2-types by means of crossed modules, which are algebraic models for 2types (see [1] and [11] for this and more general results). On the other hand in [4] were defined cohomologies of internal categories in categories of groups with operations. Since internal categories in the category of groups are equivalent to crossed modules (see [11]) the approach of [4] gives also a cohomology theory for crossed modules. This theory was further developed in [5], where cohomologically trivial crossed modules are characterized.

The motivation for the present paper was an investigation of a possible relation between above mentioned two cohomologies and an application of arguments analogous of those given in [14] in order to obtain more information on (co)homology of 2-types.

The relation between cohomologies from [4] and [8] is given in the part i) of Theorem 5.3. For any crossed module (T, G, ∂) Ellis constructed a spectral sequence with abutment the (co)homology of (T, G, ∂) in the sense of [8] (see Corollary 3 of [8] or the first part of Theorem 5.3). It turns out that E_{2}^{∗1} is isomorphic to the cohomology theory considered in [4]. The result of [4] shows that E_{2}^{p1} = 0 provided p > 2. For crossed modules with trivial kernel it was proved in [13] in slightly different context that E_{pq}^{2} = 0 provided p > q. Moreover, it was proved in [14] a general result, which implies that for any crossed module E_{pq}^{2} = 0 and E_{2}^{pq} = 0 as soon as p > q. In the present paper we give a simpler proof of this particular fact. We also improve some results of Ellis [8]. Among other results obtained in this paper we mention a new proof of Theorem 8 of [8], which describes the homology of free crossed modules in dimensions ≤ 4. The original proof used the theory of quadratic complex of Baues [1]. Our proof is based on a theorem of Ratcliffe [15] and uses a standard argument with Serre spectral sequence. Our method works for projective crossed modules and gives an information on homology in dimensions ≤ 6.

The authors were partially supported by the grants INTAS-97-31961, INTAS-99-00817 and by the TMR network K-theory and algebraic groups, ERB FMRX CT-97-0107. The first author is grateful to DAAD for the support in preparing this paper (two months fellowship in Frankfurt University). The second author is gratefull to MPI at Bonn for hospitality.

2 Preliminaries on crossed modules

Let G be a group. A G-group is a group T together with a homomorphism ξ : G → Aut(T). For g ∈ G and t ∈ T we put ^{g}t := ξ(g)(t). A G-crossed module is a G-group T together with group homomorphism ∂ : T → G satisfying:

∂(^{g}t) = g∂(t)g^{−1} and ^{∂t}s = tst^{−1}, g ∈ G, t, s ∈ T.

Under the crossed module we shall mean a triple (T, G, ∂), where G is a group and ∂ : T → G is a G-crossed module. It follows from the definition that the image Im(∂) is a normal subgroup of G, and the kernel Ker(∂) is in the center of T. Moreover the action of G on T induces an action of Q on Ker(∂), where Q = Coker ∂.

A homomorphism of crossed modules is a pair of homomorphisms (ϕ, f) : (T, G, ∂) → (T′, ∂′, G′) such that

f ∘ ∂ = ∂′ ∘ ϕ, ϕ(^{g}t) =^{f(g)} ϕ(t), g ∈ G, t ∈ T.

We let XMod be the the category of crossed modules. For a group Q and for a Q-module A one denotes by X ext(Q, A) the category of short exact sequences

0 → A → T ⟶^{∂} G → Q → 0,

where (T, G, ∂) is a crossed module and the action of Q on A induced from the crossed module structure coincides with the prescribed one. The morphisms in X ext(Q, A) are commutative diagrams

0 → A → T ⟶^{∂} G → Q → 0
id ↓  ϕ ↓  f ↓  id ↓
0 → A → T′  ⟶^{∂′} G′ → Q → 0,

where (ϕ, f) is a morphism of crossed modules (T, G, ∂) → (T′, G′, ∂). We let Xext(Q, A) be the class of the connected components of the category X ext(Q, A).

For a simplicial group G_{∗} we let N_{∗}G_{∗} be the Moore normalization of G_{∗} (see [12]), which is a chain complex, whose n-dimensional component is given by

N_{n}G_{∗} = ⋂_{i>0} Ker(∂_{i}^{n} : G_{n} → G_{n−1}), n > 0

while the boundary map d : N_{n}G_{∗} → N_{n−1}G_{∗} is the restriction of ∂_{0} on N_{n}G_{∗}. By [11] we know that the Moore normalization yields an equivalence of categories N_{∗} : SG_{≤1} → XMod. Here SG_{≤1} is the category of simplicial groups of length one.

Let (T, G, ∂) be a crossed module. Since G acts on T, it acts also on T^{n} by^{g}(t_{1}, ⋯, t_{n}) = (^{g}t_{1}, ⋯,^{g} t_{n}). Hence we can take the semi-direct product G_{n} = T^{n} ⋊ G. Here G_{0} = G. Let N_{∗}^{−1}(T, G, ∂) be a simplicial group, which is given as follows

N_{n}^{−1}(T, G, ∂) = G_{n}, n ≥ 0,
∂_{i}(t_{1}, ⋯, t_{n}, g) = (t_{1}, ⋯, t̂_{i}, ⋯, t_{n}, g), 1 ≤ i ≤ n,
∂_{0}(t_{1}, ⋯, t_{n}, g) = (t_{2}t_{1}^{−1}, ⋯, t_{n}t_{1}^{−1}, ∂t_{1}g),
s_{i}(t_{1}, ⋯, t_{n}, g) = (t_{1}, ⋯, t_{i−1}, 0, t_{i}, ⋯, t_{n}, g).

One easily checks that N_{∗}(N_{∗}^{−1}(T, G, ∂)) ≅ (T, G, ∂). Thus N_{∗}^{−1} : XMod → SG_{≤1} is also an equivalence of categories and for any G_{∗} ∈ SG_{≤1} one has an isomorphism N_{∗}^{−1}(N_{∗}G_{∗}) ≅ G_{∗}

Now for a crossed module (T, G, ∂) we can define the classifying space B(T, G, ∂) to be the classifying space of the simplicial group N_{∗}^{−1}(T, G, ∂). Hence π_{i}B(T, G, ∂) = 0, i > 1 and

π_{1}B(T, G, ∂) ≅ Coker ∂, π_{2}B(T, G, ∂) ≅ Ker ∂.

By abuse of notation we shall write

π_{1}(T, G, ∂) = Coker ∂, π_{2}(T, G, ∂) = Ker ∂, π_{i}(T, G, ∂) = 0, i > 2.

Thus B(T, G, ∂) is a two-type, meaning a path-connected CW-complex X with π_{i}X = 0 for i ≥ 3. Therefore the homotopy type of B(T, G, ∂) is completely determined by π_{1}, π_{2} and the corresponding k-invariant of B(T, G, ∂) which is an element in H^{3}(π_{1}, π_{2}). We let k(T, G, ∂) be this element. So one gets a map

k : Xext(Q, A) → H^{3}(Q, A).

According to a result of Loday [10] this map is a bijection.

A morphism of crossed modules (ϕ, f) : (T, G, ∂) → (T′, ∂′, G′) is said to be a weak equivalence if it induces isomorphisms π_{1}(T, G, ∂) ≅ π_{1}(T′, ∂′, G′) and π_{2}(T, G, ∂) ≅ π_{2}(T′, ∂′, G′). It is well-known that there is an equivalence of categories

Ho (2 − types) → Ho (crossed modules).

between corresponding homotopy categories (see for example [11], [1]). Here Ho (2 − types) (resp. Ho (crossed modules)) is the localization of the category of 2-types (resp. crossed modules) with respect of weak equivalences.

3 On homology of crossed modules

Let (T, G, ∂) be a crossed module and let M be a π_{1}(T, G, ∂)-module. According to [8] we define the homology H_{∗}((T, G, ∂), M) and cohomology H^{∗}((T, G, ∂), M)) of (T, G, ∂) with coefficients in M to be the homology and cohomology of B(T, G, ∂) with coefficients in the local system corresponding to M. If (ϕ, f) : (T, G, ∂) → (T′, ∂′, G′) is a weak equivalence then it induces an isomorphism H_{∗}((T, G, ∂), M) → H_{∗}((T′, ∂′, G′), M). Of course the similar statement is true in cohomological framework. Let us note that if T = 0, then π_{1} = G and B(T, G, ∂) = K(G, 1). Hence in this case one obtains the classical homology and cohomology of groups. The second extreme case is when G = 1. In this case one has B(T, G, ∂) ≅ K(π_{2}, 2). The homology of Eilenberg - Mac Lane spaces K(A, 2) are well understood thanks to the work of Eilenberg and Mac Lane, Serre and Cartan. It is well known that

H_{1}K(A, 2) = 0 = H_{3}K(A, 2), H_{2}K(A, 2) ≅ A, H_{4}K(A, 2) ≅ Γ^{2}A.

Here Γ^{2}A is the Whitehead’s universal quadratic functor, which is the degree two part of the divided power algebra Γ^{∗}A generated by A [7]. Here we assume that elements of A have degree two. Moreover if A is a torsion free abelian group, then H_{∗}K(A, 2) ≅ Γ^{∗}A. Hence in this case H_{i}K(A, 2) = 0 for all odd i. For general abelian groups one can find in [7] that H_{6}K(A, 2) ≅ Γ^{3}A and H_{5}K(A, 2) ≅ R(A). Here R : Ab → Ab is the first simplicial derived functor of the functor Γ^{2}. It is known that R(A⊕B) ≅ R(A)⊕R(B)⊕T or(A, B). Moreover R preserves all filtered colimits, vanishes on torsion free groups and R(Z/n) = Z/(2, n). Here (2, n) is the ideal in Z generated by 2 and n.

If the action of π_{1} on π_{2} is trivial and k(T, G, ∂) = 0, then B(T, G, ∂) has the same homotopy type as the product K(π_{1}, 1) × K(π_{2}, 2) and hence the Künneth theorem describes the (co)homology of (T, G, ∂) in terms of (co)homology of π_{1} and K(π_{2}, 2). Such situation arises when (T, G, ∂) is abelian. We recall that (T, G, ∂) is abelian if both G and T are abelian groups and the action of G on T is trivial. In this case the k-invariant is trivial because it lies in the image of

Ext_{Z}^{2}(π_{1}, π_{2}) → Xext(π_{1}, π_{2}) ≅ H^{3}(π_{1}, π_{2})

and Ext_{Z}^{2} = 0. This simple observation shows that the homology of abelian crossed modules H_{∗}(T, G, ∂) completely can be described in terms of π_{1} and π_{2}. This improves Theorem 24 of [8].

In general the classifying space of a crossed module (T, G, ∂) fits in a fibration sequence

K(π_{2}, 2) → | B(T, G, ∂) | → K(π_{1}, 1).

Therefore there exists a spectral sequences

(3.1)  E_{pq}^{2} = H_{q}(π_{1}, H_{q}(K(π_{2}, 2), M)) ⟹ H_{p+q}((T, G, ∂), M)

E_{2}^{pq} = H^{q}(π_{1}, H^{q}(K(π_{2}, 2), M)) ⟹ H^{p+q}((T, G, ∂), M).

Since the space K(π_{2}, 2) is 1-connected one can use the universal coefficient theorem to get isomorphisms

E_{∗0}^{2} = H_{∗}(π_{1}, M), E_{∗1}^{2} = 0, E_{∗2}^{2} = H_{∗}(π_{1}, M ⊗ π_{2}),
E_{∗3}^{2} = H_{∗}(π_{1}, T or_{1}^{Z}(M, π_{2}), E_{∗4}^{2} = H_{∗}(π_{1}, M ⊗ Γ^{2}π_{2})

and exact sequences

⋯ → H_{∗}(π_{1}, M ⊗ R(π_{2})) → E_{∗5}^{2} → H_{∗}(π_{1}, T or_{1}^{Z}(M, Γ^{2}π_{2})) →
H_{∗−1}(π_{1}, M ⊗ R(π_{2})) → E_{∗−1,5}^{2} → H_{∗−1}(π_{1}, T or_{1}^{Z}(M, Γ^{2}π_{2})) → ⋯

and

⋯ → H_{∗}(π_{1}, M ⊗ Γ^{3}(π_{2})) → E_{∗6}^{2} → H_{∗}(π_{1}, T or_{1}^{Z}(M, R(π_{2}))) →
H_{∗−1}(π_{1}, M ⊗ Γ^{3}(π_{2})) → E_{∗−1,6}^{2} → H_{∗−1}(π_{1}, T or_{1}^{Z}(M, R(π_{2}))) → ⋯

It is well-known that the first nontrivial differential

d_{3} : E_{p0}^{2} = H_{p}(π_{1}, M) → H_{p−3}(π_{1}, M ⊗ π_{2}) = E_{p−3,2}^{2}

is equal to (−) ∩ k, where k ∈ H^{3}(π_{1}, π_{2}) is the k-invariant of (T, G, ∂) (see Theorem 10.4 of [9]). Of course the similar results hold for cohomology as well. For instance the differential d^{3} : H^{p−3}(π_{1}, Hom(π_{2}, M)) → H^{p}(π_{1}, M) fits in the commutative diagram

H^{p−3}(π_{1}, Hom(π_{2}, M))  ⟶^{∪ k}  H^{p}(π_{1}, π_{2} ⊗ Hom(π_{2}, M))
d^{3} ↘  ↙ ev_{∗}
H^{p}(π_{1}, M)

where ev : π_{2} ⊗ Hom(π_{2}, M) → M is given by x ⊗ f ↦ f(x).

4 On homology of projective crossed modules

Let G be a fixed group and δ : Z → G be a function from a set Z to G. Consider the category of crossed modules (T, G, ∂) together with maps α : Z → T satisfying ∂α = δ. One can prove that this category has an initial object, which is called free crossed module generated by δ (see [15] and references given there). A G-crossed module (T, G, ∂) is called projective if it is a projective object in the category of G-crossed modules (see [15]). Clearly any free crossed module is projective. A theorem of Ratcliffe [15] claims that (T, G, ∂) is projective iff T_{ab} is projective as π_{1} = Coker∂-module and ∂_{∗} : H_{2}T → H_{2}N is trivial. Here N = Im∂. The aim of this section is to prove the following theorem.

Theorem 4.1 Let G be a free group and let (T, G, ∂) be a projective G-crossed module. Let π_{1} = Coker∂ and π_{2} = Ker∂. Let M be a π_{1}-module. Then there are isomorphisms

H_{3}((T, G, ∂), M) = 0, H_{4}((T, G, ∂), M) ≅ H_{0}(π_{1}, Γ^{2}π_{2} ⊗ M)
H_{5}((T, G, ∂), M) ≅ H_{1}(π_{1}, Γ^{2}π_{2} ⊗ M)

and one has an exact sequence:

H_{3}(π_{1}, Γ^{2}π_{2} ⊗ M) → H_{0}(π_{1}, Γ^{3}π_{2} ⊗ M) →
→ H_{6}((T, G, ∂), M) → H_{2}(π_{1}, Γ^{2}π_{2} ⊗ M) → 0.

Remark. The results on H_{3} and H_{4} were obtained by Ellis (see Theorem 8 of [8]). His proof uses results of [1] on quadratic complexes. We leave the task to the interested reader to make the trivial reformulation for the corresponding result for cohomology groups.

Proof. Let N = Im∂. Then one has short exact sequences

0 → π_{2} → T → N → 0

and

0 → N → G → π_{1} → 0.

Since N is free and the first short exact sequence is central, it splits and hence induces short split exact sequences of abelian groups

0 → π_{2} → T_{ab} → N_{ab} → 0,
0 → π_{2} ⊗ M → T_{ab} ⊗ M → N_{ab} ⊗ N → 0.

However, these are in general nontrivial extensions in the category of π_{1}-modules. The class of the first short exact sequence in Ext_{Z[π_{1}]}^{1}(N_{ab}, π_{2}) is denoted by k_{1}. Since G is free, the second short exact sequence yields the following short exact sequence

0 → N_{ab} → G/[N, G] → π_{1} → 0.

Denote its corresponding class in H^{2}(π_{1}, N_{ab}) by k_{2}. Clearly k = k_{1} ∪ k_{2}. Since G is free it follows from the cup product reduced theorem [16] that

∩ k_{2} : H_{n}(π_{1}, M) → H_{n−2}(π_{1}, M ⊗ N_{ab})

is an isomorphism if n ≥ 3 (this fact is also a consequence of the Hochschild-Serre spectral sequence for the following short exact sequence of groups 1 → N → G → π_{1} → 1 and the fact that homologies of free groups vanishes in dimension ≥ 2). By Theorem of Ratcliffe T_{ab} is a projective π_{1}-module. Hence T_{ab} ⊗ M is also a projective π_{1}-module and

∩ k_{1} : H_{n}(π_{1}, M ⊗ N_{ab}) → H_{n−1}(π_{1}, M ⊗ π_{2})

is an isomorphism if n ≥ 2 and monomorphism for m = 1. Then follows that the differential

d_{3} = ∩ k : H_{n}(π_{1}, M) → H_{n−3}(π_{1}, M ⊗ π_{2})

of the Serre spectral sequence

E_{pq}^{2} = H_{p}(π_{1}, H_{q}(K(π_{2}, 2), M)) ⟹ H_{p+q}((T, G, ∂), M)

is an isomorphism for n ≥ 4 and is a monomorphism for n = 3. Since π_{2} is a subgroup of a projective Z[π_{1}]-module, it is free as an abelian group and by the arguments from Section 3 we have E_{pq}^{2} = 0 for odd q and E_{pq}^{2} = H_{p}(π_{,}M ⊗ Γ^{n}π_{2}) for q = 2n, n ≥ 0 and result follows.

5 Another spectral sequence

In this section we consider the spectral sequence constructed in Corollary 3 of [8] which converges to H_{∗}(T, G, ∂). It is a consequence of a general result of [14] that E_{pq}^{2} = 0 for p > q. We give here more direct proof of this fact, which allows also to get more informations on E^{2}.

Let G_{∗} be a simplicial group and let M be a π_{0}(G_{∗})-module. Since π_{1}BG_{∗} = π_{0}G_{∗} one can consider M as a local system on on the classifying space BG_{∗} of G_{∗}. By abuse of notation we will denote the homology and cohomology of BG_{∗} with coefficient in M by H_{∗}(G_{∗}, M) and H^{∗}(G_{∗}, M). Clearly for G_{∗} = N_{∗}^{−1}(T, G, ∂) one recovers the groups considered in section 3. Here N_{∗}^{−1} : Xmod → SG_{≤1} was constructed in Section 2.

We give also an algebraic definition of the same groups. For a group G and G-module A we let C_{∗}(G, A) (resp. C^{∗}(G, A)) be the standard chain (resp. cochain) complex for the group homology (resp. cohomology). Now if G_{∗} is a simplicial group and if M is a π_{0}(G_{∗})-module, then one can form the bicomplex C_{∗}(G_{∗}, M) (resp. C^{∗}(G_{∗}, M)) whose (n, m)-component is C_{n}(G_{m}, M) (resp. C^{n}(G_{m}, M)). The following fact is well-known.

Lemma 5.1 Let G_{∗} be a simplicial group and let M be a π_{0}(G_{∗})-module. Then one has the following isomorphisms and spectral sequences

H_{∗}(T ot(C_{∗}(G_{∗}, M))) ≅ H_{∗}(G_{∗}, M), H^{∗}(T ot(C^{∗}(G_{∗}, M))) ≅ H^{∗}(G_{∗}, M)
E_{pq}^{1} = H_{q}(G_{p}, M) ⟹ H_{p+q}(G_{∗}, M), E_{1}^{pq} = H^{q}(G_{p}, M) ⟹ H^{p+q}(G_{∗}, M).

Proof. If G_{∗} is a simplicial group, then B(G_{∗}) is a diagonal of the bisimplicial set associated to the simplicial set given by

[n] ↦ B(G_{n}).

Now the result is a consequence of the Eilenberg-Zilberg-Cartier theorem [6].

Let us observe that E_{∗q}^{1} is a simplicial abelian group. So we can take the Moore normalization of it to obtain N_{∗}E_{∗q}^{1}. Similarly, E_{1}^{∗q} is a cosimplicial abelian group and we still can take the normalization of it to obtain N^{∗}E_{1}^{∗q} (see [2]). Then the classical normalization theorem [12] together with Theorem 5.1 of [14] imply the following

Corollary 5.2 Let G_{∗} be a simplicial group and let M be a π_{0}(G_{∗})-module. Then there exists a spectral sequence

N_{p}E_{∗q}^{1} ⟹ H_{p+q}(G_{∗}, M),

which has the same E_{∗∗}^{r} terms as previous one as soon as r ≥ 2. Moreover if N_{n}G_{∗} = 0 for n > k then N_{p}E_{∗q}^{1} = 0 = E_{pq}^{r} provided p > kq and r ≥ 2. Similarly for cohomology.

Theorem 5.3 Let (T, G, ∂) be a crossed module and let M be a π_{1}(T, G, ∂)-module. i) Then there exist spectral sequences

E_{1}^{pq} = H^{q}(G_{p}, M) ⟹ H^{p+q}((T, G, ∂), M),
E_{pq}^{1} = H_{q}(G_{p}, M) ⟹ H_{p+q}((T, G, ∂), M).

Here G_{∗} = N_{∗}^{−1}(T, G, ∂) and hence G_{p} = T^{p} ⋊ G. The cohomology of (T, G, ∂) with coefficients in M in the sense of [4] is isomorphic to E_{2}^{p1} provided p > 0.

ii) One has N_{p}E_{∗q}^{1} = 0 = E_{pq}^{r} as soon as p > q and r ≥ 2. The similar vanishing result holds also for cohomology.

iii) One has isomorphisms

N_{p}E_{∗p}^{1} ≅ H_{0}(G, (T^{ab})^{⊗p} ⊗ M)
N^{p}E_{1}^{∗p} ≅ Hom_{G}((T^{ab})^{⊗p}, M).

Here p ≥ 1 and T^{ab} = T/[T, T].

Remarks. 1) The spectral sequence is the same as in Corollary 3 of [8].

2) The existence of the spectral sequences as well as part ii) are consequence of Corollary 5.2. Indeed it suffices to take G_{∗} = N^{−1}(T, G, ∂) in Corollary 5.2. However we give also a new proof for ii), which is far simpler then the original proof of the vanishing result in Corollary 5.2 given in [14].

The proof of part ii) and iii) of Theorem 5.3 is based on the notion of a degree of a functor [7]. Since we are in a slightly different situation then loc. cit. we give definitions of a cross-effect and of a degree of a functor in our framework. We let G_{G} be the category of G-groups. For a functor F : G_{G} → Ab with F (0) = 0 one can define the cross-effect functors cr_{n}F : G_{G}^{n} → Ab, n ≥ 0 as follows. We put cr_{1}F = F. Then the bifunctor cr_{2}F is defined by the short exact sequence

0 → cr_{2}F (T_{1}, T_{2}) → F (T_{1} × T_{2}) → F (T_{1}) × F (T_{2}) → 0.

In general

cr_{n}F (T_{1}, ⋯, T_{n}) =

Ker(F (T_{1} × ⋯ × T_{n}) → ∏_{i} F (T_{1} × ⋯ × T̂_{i} × ⋯ × T_{n})).

We claim that if

0 → F_{1} → F → F_{2} → 0

is a short exact sequence of functors, then

0 → cr_{n}F_{1} → cr_{n}F → cr_{n}F_{2} → 0

is also an exact sequence, n ≥ 1. The claim is obvious for n = 1 and for n = 2 it is a consequence of 3 × 3 lemma. The general case follows from this by induction, because

cr_{n}F (X_{1}, ⋯, X_{n}) ≅ cr_{2}(U(X_{1}, ⋯, X_{n−2}))(X_{n−1}, X_{n}).

Here

U(X_{1}, ⋯, X_{n−2}) : G_{G} → Ab

is given by

U(X_{1}, ⋯, X_{n−2})(Y) = cr_{n−1}F (X_{1}, ⋯, X_{n−2}, Y).

Following [7] we will say that a functor F is of degree ≤ n if cr_{n+1}F = 0. Thus if degF = n and F′ is a subquotient of F then degF′ ≤ n.

Proof of Theorem 5.3. i) Here we have only to establish relationship between the definition of cohomology in [4] and the one given in Section 3. Comparing the definitions one sees that the cochain complex K^{∗}((T, G, ∂), M) constructed in Section 3 of [4] is isomorphic to the cochain complex Der(G_{∗}, M) and the exact sequence of cosimplicial abelian groups

0 → H^{0}(G_{∗}, M) → M → Der(G_{∗}, M) → H^{1}(G_{∗}, M) → 0

shows that one has an isomorphism H^{p}K^{∗}((T, G, ∂), M) ≅ E_{2}^{p1} provided p > 0.

ii) and iii) We consider only homology. The dual argument works for cohomology as well. The case q = 0 is trivial, because in this case E_{∗0}^{1} = H_{0}(G_{∗}, M) is a constant cosimplicial abelian group with value H_{0}(π_{1}, M). So we assume that q > 0.

Let N be a G-module and let K be a G-group. We can consider N as a trivial Kmodule. It is a consequence of the Künneth theorem that the functor

H_{n}(−, N) : G_{G} → Ab, K ↦ H_{n}(K, N)

is of degree n and the last nontrivial cross-effect is

cr_{n}(H_{n}(−, N))(K_{1}, ⋯, K_{n}) = K_{1}^{ab} ⊗ ⋯ ⊗ K_{n}^{ab} ⊗ N.

For any p ≥ 0 and q ≥ 1 we let

F_{pq} : G_{G} → Ab

be a functor given by F_{pq}(K) = H_{p}(G, H_{q}(K, N)). Since cr_{n} is exact and preserves direct sums we see that deg(F_{pq}) ≤ q and

cr_{q}F_{pq}(K_{1}, ⋯, K_{q}) ≅ H_{p}(G, K_{1}^{ab} ⊗ ⋯ ⊗ K_{q}^{ab} ⊗ N).

We need also a functor F_{q} : G_{G} → Ab given by

F_{q}(K) := Ker(H_{q}(K ⋊ G, N) → H_{q}(G, N)).

It is clear that F_{q}(0) = 0 and by the Hochschild-Serre spectral sequence one has the spectral sequence of functors

E_{pq}^{2} = F_{pq} ⟹ F_{p+q}, q > 0.

Let us fix an integer n. Since the correspondence F ↦ cr_{n}F gives an exact functor we have aslo the spectral sequence E_{pq}^{2} = cr_{n}F_{pq} ⟹ cr_{n}F_{p+q}. Since degF_{pq} ≤ q we see that E_{pq}^{2} = 0 for q < n and E_{0n}^{2} = H_{0}(G, cr_{n}H_{n}). Therefore degF_{n} ≤ n and cr_{n}F_{n} = cr_{n}F_{0n} = H_{0}(G, cr_{n}H_{n}).

Now we take N = M. One observs that the constructions of G_{n} = T^{n} ⋊G and ∂_{i}, i > 0 use only the action of G on T and not the homomorphism ∂ : T → G, which appears only in the definition of ∂_{0}. Hence N_{p}E_{∗q}^{1} depends only on T considered as G-module and not on ∂. Comparing the definitions it is obvious that for any crossed module (T, G, ∂) one has the following isomorphism N_{p}E_{∗q}^{1}(T, G, ∂) ≅ cr_{p}F_{q}(T, ⋯, T). This completes the proof.

Let us recall that for any crossed module (T, G, ∂) there is a weak homotopic equivalence (T′, G′, ∂′) → (T, G, ∂) with free group G′ (see for example page 11 in [8]). Our next result is an immediate consequence of Theorem 5.3.

Corollary 5.4 For a crossed module (T, G, ∂) one has an exact sequence

H_{2}G → H_{2}(T, G, ∂) → H_{0}(G, T^{ab}) → G^{ab} → H_{1}(T, G, ∂) → 0.

Moreover if G is a free group then

H_{2}(T, G, ∂) ≅ Ker(T/[T, G] → G^{ab})

and there exists an exact sequence

H_{3}(T ⋊ G) → H_{4}(T, G, ∂) → T^{ab} ⊗_{G} T^{ab} ⟶^{α} H_{2}(T ⋊ G) → H_{3}(T, G, ∂) → 0.

The similar result is true also for cohomology.

Remarks. 1) The isomorphism H_{2}(T, G, ∂) ≅ Ker(T/[T, G] → G^{ab}) for free G was proved by Ellis (see Theorem 6 of [8]) using different methods.

2) The homomorphism α : T^{ab} ⊗_{G} T^{ab} → H_{2}(T ⋊ G) is the restriction of d_{0} : H_{2}(T^{2} ⋊ G) → H_{2}(T ⋊ G) on the subgroup T^{ab} ⊗_{G} T^{ab} ⊂ H_{2}(T^{2} ⋊ G) and hence in terms of Brown-Loday tensor product [3] it is given by x̄ ⊗ ȳ → (x^{−1}, ∂x) ⊗ (y, 1). Here x, y ∈ T and x̄ denotes the class of x in T^{ab}.

Corollary 5.5 Let (T, G, ∂) be a crossed module. Choise any crossed module (T′, G′, ∂′) which is weakly equivalent to (T, G, ∂), such that G′ is a free group. Then there is an isomorphism

H^{2}((T, G, ∂), A) ≅ H^{1}((T′, G′, ∂′), A),

where the right hand side is an internal category cohomology corresponding to the crossed module (T′, G′, ∂′) defined in [4].

The proof is the direct consequence of Theorem 3.1 of [4] and Theorem 6 of [8] (or the cohomological version of the previous Corollary).

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