Homology, Homotopy and Applications, vol.2, No.7, 2000, pp.105–114 ISSN 1512–0139

MORE ABOUT HOMOLOGICAL PROPERTIES OF PRECROSSED MODULES

NICK INASSARIDZE and EMZAR KHMALADZE

(communicated by Graham Ellis)

Abstract Homology groups modulo q of a precrossed P -module in any dimensions are defined in terms of nonabelian derived functors, where q is a nonnegative integer. The Hopf formula is proved for the second homology group modulo q of a precrossed P -module which shows that for q = 0 our definition is a natural extension of Conduché and Ellis’ definition [CE]. Some other properties of homologies of precrossed P -modules are investigated.

Introduction

The homology of precrossed modules was introduced by Conduché and Ellis in [CE]. The aim of this paper is to pursue their line of investigation homological properties of precrossed modules.

Let P be a group. A precrossed P -module (M, μ) is a group homomorphism μ : M → P together with an action of P on M denoted by ^{p}m for p ∈ P and m ∈ M, which satisfies the following condition:

μ(^{p}m) = pμ(m)p^{−1}.

If in addition the following Peiffer identity holds

μ(m)m′ = mm′m^{−1},

(M, μ) is a crossed P -module (see e.x. [BH]).

A morphism φ : (M, μ) → (N, ν) of (pre)crossed P -modules is a commutative triangle

\begin{matrix}M ⟶^{φ} N \\ _{μ} ↘ ↙ _{ν} \\ P \end{matrix} ,

where φ is a P -equivariant group homomorphism i.e. φ(^{p}m) = ^{p}φ(m) for all m ∈ M, p ∈ P. Let us denote the category of precrossed (crossed) P -modules by PCM(P) (CM(P)).

Further we shall occasionally supress explicit mention of the homomorphism μ in a precrossed P -module (M, μ) and write simply M.

Precrossed modules form a model of homotopy type in dimensions 1 and 2 for connected CW-complexes. Precisly Kan’s 𝔾 functor establishes an equivalence relation between the category of connected CW-complexes and the category of free simplicial groups [K] and the first two terms of the Moore chain complex associated to the simplicial group gives a precrossed module.

During this work both authors were supported by INTAS Georgia grant No 213 and NATO linkage grant No 975316. The first author was also supported by INTAS 93-436 and INTAS Fellowship grant for Young Scientists No 151. Received 17 March 2000, revised 3 August 2000; published on 12 September 2000. 2000 Mathematics Subject Classification: 20J05, 18G10, 18G50 Key words and phrases: precrossed module, Peiffer abelianization, homology group, nonabelian derived functor \begin{matrix}c \\ ◯ \end{matrix} 2000, Nick Inassaridze and Emzar Khmaladze. Permission to copy for private use granted.

Let (M, μ) be a precrossed P -module. The following type elements in M

⟨m, m′⟩ = mm′m^{−1}^{μ(m)}m′^{−1}, m, m′ ∈ M

are called Peiffer commutators, and now we give some identities for them from [BC]

⟨m, m′m′′⟩ = ⟨m, m′⟩^{μ(m)}m′⟨m, m′′⟩^{μ(m)}m′^{−1},  (1)
⟨mm′, m′′⟩ = m⟨m′, m′′⟩m^{−1}⟨m,^{μ(m′)}m′′⟩,  (2)
p⟨m, m′⟩ = ⟨^{p}m,^{p}m′⟩,  (3)
⟨k, m⟩ = kmk^{−1}m^{−1},  (4)
⟨k, m⟩⟨m, k⟩ = k^{μ(m)}k^{−1}  (5)

for all m, m′, m′′ ∈ M, p ∈ P and k ∈ Kerμ.

The Peiffer commutator subgroup ⟨M, M⟩, which is a subgroup of the group M generated by the Peiffer commutators, plays the same role for precrossed modules as the commutator subgroup plays for groups. Analogously as a lower central series in a group, a lower Peiffer central series in a precrossed P -module is defined by Baues and Conduché [BC]

M^{(1)} = M ⊃ M^{(2)} ⊃ ….

This series has properties like classical central series giving a hope to generalize some methods of Curtis [CU1, CU2] and Quillen [Q] for nonsimply connected spaces.

The crossed P -module μ′ : M/⟨M, M⟩ → P associated to the precrossed P -module μ : M → P, where M/⟨M, M⟩ is a factor group of M by the Peiffer commutator subgroup, the homomorphism μ′ and the action of P on M/⟨M, M⟩ are induced by μ and the action of P on M respectively, plays the role of abelianization of a group which we further call Peiffer abelianization. As an analog of the classical first group homology, Conduché and Ellis [CE] defined the first homology of a precrossed P -module (M, μ) by Peiffer abelianization i.e.

H_{1}(M)_{P} = M/⟨M, M⟩ .

We point out that despite its name, the Peiffer abelianization can be nonabelian. Let (L, λ), (M, μ) and (N, ν) be precrossed P -modules. A short exact sequence of groups

1 ⟶ L ⟶^{φ} M ⟶^{ψ} N ⟶ 1

is called short exact sequence of precrossed P -modules if φ and ψ are morphisms of the category PCM(P).

Let X be a set and δ : X → P a map to the group P. Then the free precrossed P -module ∂ : F → P with base (X, δ) is defined as follows: F is the free group generated by the set X × P, ∂ is defined on generators by ∂(x, p) = pδ(x)p^{−1} and the action of P on F is given by p(x, p′) = (x, pp′).

Conduché and Ellis in [CE] also defined the second homology group of a precrossed P - module (M, μ) by the Hopf formula

H_{2}(M)_{P} = R ∩ ⟨F, F ⟩/⟨⟨F, R⟩⟩ ,

where 1 → R → F → M → 1 is a short exact sequence of precrossed P -modules, (F, ∂) is a free precrossed P -module with some base (X, δ) which is called free presentation of the precrossed P -module (M, μ). They studied some properties so defined low dimensional homology groups of precrossed P -modules and hoped that higher homology could be defined analogously using Hopf formulas for higher homology groups (see [BE]). Using this way to define all homology groups of a precrossed P -module (M, μ), H_{n}(M)_{P}, one should have some difficulties, for n ⩾ 3, to prove that the definition does not depend on the free presentation of the precrossed P -module (M, μ).

In the present paper we have another conception to define all homology groups of a precrossed P -module, particulary the use of nonabelian derived functors.

All treatments with homology of precrossed P -modules we consider in the q modular aspect, where q is a nonnegative integer, and for q = 0 it gives homology groups of precrossed modules introduced in [CE]. Thus, for nonnegative integer q, we define homology groups modulo q of precrossed P -module (M, μ) in any dimension n ⩾ 1, denoted by H_{n}(M, q)_{P}, and study their properties generalizing the classical homology of groups with coefficients in ℤ_{q} = ℤ/qℤ.

1. Construction

Let us denote by Set(P) the category of sets over the group P, whose objects are all sets with a map to P and morphisms are all maps of sets such that the corresponding triangles are commutative.

Consider the functor F : Set(P) → PCM(P) defined as follows: for an object X ⟶^{α} P

of the category Set(P), let F(X ⟶^{α} P) be a free precrossed P -module with base (X, α);

for a morphism X ⟶^{κ} X′, let F(κ) be the canonical homomorphism induced by κ.

It is known that the forgetful functor from the category PCM(P) to the category Set(P) is a right adjoint of the functor F. This adjunction induces the cotriple (F, τ, δ) in the category PCM(P). Let P be the projective class in the category PCM(P) induced by the cotriple (F, τ, δ) (see [TV], [IH]).

First we describe the projective class P and the corresponding P-epimorphisms.

Proposition 1.1. A morphism M ⟶^{φ} N of the category PCM(P) is a P-epimorphism if and only if φ is surjective (as map of sets).

Proposition 1.2. In the category PCM(P) the following conditions are equivalent:

(i) A precrossed P -module (Q, ν) belongs to the projective class P;

(ii) (Q, ν) is a free precrossed P -module with base (X, α) for some object X ⟶^{α} P of the category Set(P)

The proof of these propositions is left to the reader.

A precrossed P -module (N, ν) is a precrossed P -submodule of a precrossed P -module (M, μ) if N is a subgroup of M, the action of P on N is induced by the action of P on M and ν is the restriction of μ on N. If, in addition, N is a normal subgroup of the group M then we write N <_{P} M.

Let (M, μ) be a precrossed P -module, N, N′ be two subgroups of M and q be a nonnegative integer. We denote by ⟨N, N′⟩_{(q)} the subgroup of M generated by the elements ⟨n, n′⟩ and k^{q} for all n ∈ N, n′ ∈ N′, k ∈ N ∩ N′ ∩ Kerμ. Let ⟨⟨N, N′⟩⟩_{(q)} = ⟨N, N′⟩_{(q)}⟨N′, N⟩_{(q)}. One has the following

Lemma 1.3. (i) If N and N′ are precrossed P -submodules of M then ⟨N, N′⟩_{(q)} and

⟨⟨N, N′⟩⟩_{(q)} are precrossed P -submodules of M. (ii) If N <_{P} M then ⟨M, N⟩_{(q)} <_{P} M, ⟨N, M⟩_{(q)} <_{P} M, ⟨⟨M, N⟩⟩_{(q)} <_{P} M.

Proof. (i) Follows from the relation (3) and the equality ^{p}(k^{q}) = (^{p}k)^{q}, p ∈ P, k ∈ N ∩ N′ ∩ Kerμ. (ii) Follows from the relations (1), (2) and the equality mk^{q}m^{−1} = (mkm^{−1})^{q}, m ∈ M, k ∈ N ∩ Kerμ. □

Using Lemma 1.3 one can define a covariant functor T_{(q)} from the category PCM(P) to the category 𝔊𝔯 of groups by the following way: for any precrossed P -module (M, μ),

let T_{(q)}(M) = M/⟨⟨M, M⟩⟩_{(q)} = M/⟨M, M⟩_{(q)}; for a morphism (M, μ) ⟶^{φ} (M′, μ′), let

T_{(q)}(φ) be a group homomorphism induced by φ. Note that for q = 0 the functor T_{(q)} is the Peiffer abelianization functor.

In the category PCM(P) there exist finite limits (easy to show). Let us consider the nonabelian left derived functors L_{n}^{P}T_{(q)}, n ⩾ 0, of the functor T_{(q)} : PCM(P) → 𝔊𝔯 relative to the projective class P induced by the cotriple (F, τ, δ) in the category PCM(P) [IH].

Definition 1.4. Let P be a group, (M, μ) be a precrossed P -module and q be a nonnegative integer. Define the n-th homology group modulo q of the precrossed P -module (M, μ) by

H_{n}(M, q)_{P} = L_{n−1}^{P}T_{(q)}(M), n ⩾ 1 .

Proposition 1.5. Let μ : M → P be a precrossed P -module such that μ(m) = 1 for all m ∈ M. Then one has

H_{n}(M, q)_{P} = H_{n}(M, ℤ_{q}), n ⩾ 1.

Proof. Consider a P-projective pseudosimplicial resolution (see [IH]) of (M, μ) in the category PCM(P)

⋯ ⟶ ; ⋮ ; ⟶ F_{2} ⟶^{κ_{2}} Y_{1} \begin{matrix}⟶ \\ ⟶ \\ ⟶ \end{matrix} F_{1} ⟶^{κ_{1}} Y_{0} \overrightarrow{⟶} F_{0} → M ,  (6)

where F_{n} ∈ P and Y_{n} is a simplicial kernel in the category PCM(P). By Propositions 1 and 2 all F_{n} are free groups and all κ_{n} are surjective group homomorphisms, implying that (6) is a projective resolution of the group M in the category 𝔊𝔯. Since μ is a trivial group homomorphism, T_{(q)}(F_{n}) = F_{n}^{ab}/qF_{n}^{ab}. Using [BB] one gets the assertion. □

2. Main Results

In this section we give our main results. We investigate the functor T_{(q)} and prove a Hopf type formula for the second homology modulo q of precrossed P -modules, generalizing the classical one (see [BR], [E]).

Let C be a category with finite limits, Q be a projective class in the category C and T be a covariant functor from the category C to the category 𝔊𝔯 of groups.

Definition 2.1 ([P]). The functor T is called a cosheaf over (C, Q) if for any Q-epimorphism X → A the sequence of groups

T (X ×_{A} X) \overrightarrow{⟶} T (X) ⟶ T (A) ⟶ 1 ,

is simplicially exact, where X ×_{A} X is the pullback of the diagram

\begin{aligned}& & X \\ & & ↓ \\ X & ⟶ & A \end{aligned} .

Lemma 2.2. Let P be a group and q be a nonegative integer. Then the functor T_{(q)} : PCM(P) → 𝔊𝔯 is a cosheaf over (PCM(P), P), where P is the projective class induced by the cotriple (F, τ, δ) (see above).

Proof. It is easy to verify that for a short exact sequence of precrossed P -modules

1 ⟶ L ⟶ M ⟶ N ⟶ 1

there is an exact sequence of groups

T_{(q)}(L) ⟶ T_{(q)}(M) ⟶ T_{(q)}(N) ⟶ 1 .  (7)

Consider a P-epimorphism Q ⟶^{α} M in the category PCM(P). We have to show that the diagram of groups

T_{(q)}(Q ×_{M} Q) ⟶^{d_{0}} ; ⟶_{d_{1}} T_{(q)}(Q) ⟶^{T_{(q)}(α)} T_{(q)}(M) ⟶ 1

is exact. In effect, there is the following commutative diagram of groups

\begin{matrix}T_{(q)}(R) & ⟶ & T_{(q)}(Q) & ⟶^{T_{(q)}(α)} & T_{(q)}(M) & ⟶ & 1 \\ ↓ ^{λ} & & ∥ & & ∥ \\ Ker d_{0} & ⟶^{d_{1}} & T_{(q)}(Q) & ⟶^{T_{(q)}(α)} & T_{(q)}(M) & ⟶ & 1 \end{matrix} ,

where R is the kernel of α : Q → M, λ is a homomorphism induced by the inclusion R ↪ Q ×_{M} Q, r ↦ (r, 1), and the top row is exact by (7). Hence the bottom row of this diagram is also exact. □

Proposition 2.3. Let P be a group, (M, μ) be a precrossed P -module and q be a nonnegative integer. Then there is a natural isomorphism

H_{1}(M, q)_{P} ≈ M/⟨M, M⟩_{(q)}

Proof. Follows by Lemma 2.2 and [P or IH, Proposition 2.26].□

Theorem 2.4 (Hopf Formula). Let P be a group, (M, μ) be a precrossed P -module and q be a nonnegative integer. Then there is an isomorphism

H_{2}(M, q)_{P} ≈ R ∩ ⟨F, F ⟩_{(q)}/⟨⟨F, R⟩⟩_{(q)} ,

where 1 ⟶ R ⟶ F ⟶^{φ} M ⟶ 1 is any free presentation of the precrossed P -module (M, μ) i.e. using Propositions 1 and 2, F is an object of the projective class P and φ is a P-epimorphism.

Proof. Consider the Ĉech resolution of (M, μ) ∈ PCM(P) for φ : F → M [P or IH, Definition 2.31, Examples]

⋯ \begin{matrix}⟶ \\ ⟶ \\ ⟶ \\ ⟶ \end{matrix} F ×_{M} F ×_{M} F ⟶^{d_{0}} ; ⟶^{d_{1}} ; ⟶^{d_{2}} F ×_{M} F ⟶^{d_{0}} ; ⟶_{d_{1}} F ⟶^{φ} M.

By Lemma 2.2 T_{(q)} is a cosheaf over (PCM(P), P) and using [P or IH, Theorem 2.39(ii)] there is an isomorphism

L_{1}^{P}T_{(q)}(M) ≈ π_{1}C_{∗},

where C_{∗} is the following simplicial group

C_{∗} ≡ ⋯ \begin{matrix}⟶ \\ ⟶ \\ ⟶ \\ ⟶ \end{matrix} T_{(q)}(F ×_{M} F ×_{M} F) ⟶^{T_{q}(d_{0})} ; ⟶^{T_{q}(d_{1})} ; ⟶^{T_{q}(d_{2})} T_{(q)}(F ×_{M} F) ⟶^{T_{q}(d_{0})} ; ⟶_{T_{q}(d_{1})} T_{(q)}(F) .

The Moore complex NC_{∗} of the simplicial group C_{∗} has length 1 i.e. (NC_{∗})_{n} = 0, n ⩾ 2. This follows from the fact that the Moore complex of the Ĉech resolution has length 1.

Hence π_{1}C_{∗} = Ker T_{q}(d_{0}) ∩ Ker T_{q}(d_{1}).

Furthermore, one has the following isomorphism of precrossed P -modules F ×_{M} F ⟶^{≈} R ⋊ F, defined by (r, f) ↦ (rf, f), where the precrossed P -module structure on the group R ⋊ F is given by the following way: a homomorphism R ⋊ F → P is defined by (r, f) ↦ μφ(f) and an action of P on R ⋊ F by ^{p}(r, f) = (^{p}r,^{p} f) for all p ∈ P, r ∈ R, f ∈ F.

One gets R ⋊ F ⟶^{d_{0}} ; ⟶_{d_{1}} F, d_{0}(r, f) = f, d_{1}(r, f) = rf.

It only remains and easy to prove that the homomorphism

α : (R/⟨⟨F, R⟩⟩_{(q)}) × F/⟨F, F ⟩_{(q)} ⟶ T_{(q)}(R ⋊ F) ,

defined by α([r], [f]) = [(r, f)], is an isomorphism. □

Remark 2.5. For μ = 0 Theorem 2.4 generalizes the classical Hopf Formula from [BR] and for q = 0 Proposition 2.3 and Theorem 2.4 show that one can get the first and the second homology of precrossed P -modules of Conduché and Ellis [CE] as nonabelian derived functors of the Peiffer abelianization functor.

Conjecture. Let P be a group, (M, μ) be a precrossed P -module and q be a nonnegative integer. Choose a precrossed P -module (F, ∂) and R_{1}, …, R_{n} <_{P} F such that: F/ ∏_{1⩽i⩽n} R_{i} ≈ M, H_{2}(F, q)_{P} = 1 and H_{r}(F/ ∏_{i∈A} R_{i}, q)_{P} = 1 for every proper subset A ≠ ∅ of ⟨n⟩ = {1, …, n}, r = |A| + 1 and |A| + 2 (for example, the precrossed P -modules F/ ∏_{i∈A} R_{i} are free for A ≠ ⟨n⟩). Then there is an isomorphism

H_{n+1}(M, q)_{P} ≈ {∩_{i=1}^{n}R_{i} ∩ ⟨F, F ⟩_{(q)}}/{ ∏_{A⊆⟨n⟩}⟨⟨∩_{i∈A}R_{i}, ∩_{i∉A}R_{i}⟩⟩_{(q)}}

3. Some other results

In this section we investigate low dimensional, first and second, homologies modulo q of precrossed P -modules, always have in mind Proposition 2.3 and Theorem 2.4 and give some results generalizing in q modular aspect the results of Conduché and Ellis [CE].

Proposition 3.1. Let P be a group, q be a nonnegative integer and

1 → L → M → N → 1

a short exact sequence of precrossed P -modules. Then there is an exact sequence of groups

\begin{matrix}H_{2}(M, q)_{P} → H_{2}(N, q)_{P} → L/⟨⟨M, L⟩⟩_{(q)} \\ → H_{1}(M, q)_{P} → H_{1}(N, q)_{P} → 1 \end{matrix} .  (8)

Proof. Suppose 1 → R → F → M → 1 is a free presentation of the precrossed P -module M, and hence 1 → R′ → F → N → 1 is a free presentation of the precrossed P -module N. Therefore R ⊂ R′ implying R ∩ ⟨F, F ⟩_{(q)} ⊂ R′ ∩ ⟨F, F ⟩_{(q)}, ⟨⟨F, R⟩⟩_{(q)} ⊂ ⟨⟨F, R′⟩⟩_{(q)} and there is the canonical group homomorphism H_{2}(M, q)_{P} → H_{2}(N, q)_{P}.

The following commutative diagram of groups with exact rows

1 ⟶ R′ ⟶ F ⟶ N ⟶ 1
↓  ↓  ∥
1 ⟶ L ⟶ M ⟶ N ⟶ 1

induces a homomorphism H_{2}(N, q)_{P} → L/⟨⟨M, L⟩⟩_{(q)}.

Other homomorphisms are defined naturally and it is easy to check that the sequence (8) is exact. □

Remark 3.2. One can extend the sequence (8) to any dimensions using the long exact sequence of the nonabelian derived functors and recovering for μ = 0 the eight term exact homology sequence of groups with coefficients in ℤ_{q} [ER].

The following result generalises the classical group result and uses the standard proof, originally due to [S].

For any precrossed P -module (M, μ) and any nonnegative integer q there is the following family of precrossed P -submodules

M_{(q)}^{(1)} = M, M_{(q)}^{(2)} = ⟨⟨M, M⟩⟩_{(q)}, …, M_{(q)}^{(n+1)} = ⟨⟨M, M_{(q)}^{(n)}⟩⟩_{(q)}.

Theorem 3.3. Let P be a group, q be a nonnegative integer and φ : M → N be a morphism of precrossed P -modules such that the following properties hold:

(i) the natural homomorphism H_{1}(M, q)_{P} → H_{1}(N, q)_{P}, induced by φ, is an isomorphism; (ii) the natural homomorphism H_{2}(M, q)_{P} → H_{2}(N, q)_{P}, induced by φ, is a surjection. Then φ induces a natural isomorphism of precrossed P -modules

M/M_{(q)}^{(n)} ⟶^{≈} N/N_{(q)}^{(n)} for n ⩾ 2 .

Proof. By induction. For n = 2 the theorem is true. Suppose it is true for n. By Proposition 3.1 and the following commutative diagram of groups with exact rows

\begin{aligned}1 & ⟶ & M_{(q)}^{(n)} & ⟶ & M & ⟶ & M/M_{(q)}^{(n)} & ⟶ & 1 \\ & & ↓ & & ^{φ}↓ & & ↓ \\ 1 & ⟶ & N_{(q)}^{(n)} & ⟶ & N & ⟶ & N/N_{(q)}^{(n)} & ⟶ & 1 \end{aligned} ,

one has the following commutative diagram of groups with exact rows

H_{2}(M, q)_{P} ⟶ H_{2}(M/M_{(q)}^{(n)}, q)_{P} ⟶ M_{(q)}^{(n)}/M_{(q)}^{(n+1)} ⟶
↓  ↓  ↓
H_{2}(N, q)_{P} ⟶ H_{2}(N/N_{(q)}^{(n)}, q)_{P} ⟶ N_{(q)}^{(n)}/N_{(q)}^{(n+1)} ⟶

\begin{aligned}⟶ & H_{1}(M, q)_{P} & ⟶ & H_{1}(M/M_{(q)}^{(n)}, q)_{P} & ⟶ & 1 \\ & ↓ & & ↓ \\ ⟶ & H_{1}(N, q)_{P} & ⟶ & H_{1}(N/N_{(q)}^{(n)}, q)_{P} & ⟶ & 1 \end{aligned} .

Using the “five lemma” M_{(q)}^{(n)}/M_{(q)}^{(n+1)} is isomorphic to N_{(q)}^{(n)}/N_{(q)}^{(n+1)}. Then the following commutative diagram of groups

1 ⟶ M_{(q)}^{(n)}/M_{(q)}^{(n+1)} ⟶ M/M_{(q)}^{(n+1)} ⟶ M/M_{(q)}^{(n)} ⟶ 1
↓  ↓  ↓
1 ⟶ N_{(q)}^{(n)}/N_{(q)}^{(n+1)} ⟶ N/N_{(q)}^{(n+1)} ⟶ N/N_{(q)}^{(n)} ⟶ 1

gives the result for n + 1. □

For any precrossed P -module μ : M → P, let M ∧_{P}^{q} M be the group generated by the symbols m ∧ m′ and {k}, m, m′ ∈ M, k ∈ Kerμ subject to the following relations:

m ∧ m′m′′ = (m ∧ m′)(m ∧ m′′)(⟨m, m′′⟩^{−1} ∧^{μm}m′),  (9)
mm′ ∧ m′′ = (m ∧ m′m′′m′^{−1})(^{μm}m′ ∧^{μm}m′′),  (10)
⟨m, m′⟩ ∧ ⟨n, n′⟩ = (m ∧ m′)(n ∧ n′)(m ∧ m′)^{−1}(n ∧ n′)^{−1},  (11)
(⟨m, m′⟩ ∧ m′′)(m′′ ∧ ⟨m, m′⟩) = (m ∧ m′)(^{μm′′}m ∧^{μm′′}m′)^{−1},  (12)
k ∧ k = 1,  (13)
{k}(m ∧ m′){k}^{−1} = (k^{q}m ∧ m′)(k^{q} ∧^{μm}m′)^{−1},  (14)
{kk′} = {k} ∏_{i=1}^{q−1}(k^{−1} ∧ k^{1−q+i}(k′)^{i}k^{q−1−i}){k′},  (15)
{k}{k′}{k}^{−1}{k′}^{−1} = k^{q} ∧ k′^{q},  (16)
{⟨m, m′⟩} = (m ∧ m′)^{q}  (17)

for all m, m′, m′′, n, n′ ∈ M and k, k′ ∈ Kerμ.

Note that (9)-(13) are the defining relations for the group M ∧_{P} M defined in [CE]. Furthermore, when P = 1 or μ = 0 the group M ∧_{P}^{q} M coincides with the nonabelian exterior product modulo q, M ∧^{q} M, introduced by Conduché and Rodriguez-Fernandez [CR] (see also [B], [ER], [E], [IN]).

There is an action of the group P on the group M ∧_{P}^{q} M given by ^{p}(m ∧ m′) = ^{p}m ∧^{p}m′ and ^{p}{k} = {^{p}k} for all m, m′ ∈ M, k ∈ Kerμ. Moreover, there exists a P -equivariant group homomorphism ∂_{2}^{q} : M ∧_{P}^{q} M → M defined by ∂_{2}^{q}(m ∧ m′) = ⟨m, m′⟩ and ∂_{2}^{q}({k}) = k^{q}. It is clear that ∂_{2}^{q}(M ∧_{M}^{q} M) = M_{(q)}^{(2)}.

Note that the complex of groups M ∧_{P}^{q} M ⟶^{∂_{2}^{q}} M ⟶^{μ} P is a 2-crossed module in

the sence of Conduché [C].

Proposition 3.4. Let (M, μ) be a precrossed P -module, q > 0 and

1 ⟶ R ⟶ F ⟶^{φ} M ⟶ 1

a short exact sequence of precrossed P -modules, where (F, ν) is a free precrossed P -module. If the homomorphism ∂_{2}^{q} : F ∧_{P}^{q} F → F is injective then the group M ∧_{P}^{q} M is isomorphic to the group F_{(q)}^{(2)}/⟨⟨F, R⟩⟩_{(q)}.

Proof. Let L_{F} (resp. L_{M}) be the free group generated by the set (F × F) ∪ Kerν (resp. (M × M) ∪ Kerμ). There is a commutative diagram of groups

L_{F} ⟶  L_{M}
π_{F} ↓  ↓^{π_{M}}
F ∧_{P}^{q} F ⟶ M ∧_{P}^{q} M ,

where the horizontal homomorphisms are surjective and π_{F} and π_{M} are canonical homomorphisms defined by π_{F} (f, f′) = f ∧ f′, π_{F} (g) = {g} and π_{F} (m, m′) = m ∧ m′, π_{F} (k) = {k} for all f, f′ ∈ F, g ∈ Kerν, m, m′ ∈ M and k ∈ Kerμ. It is easy to get that Ker(F ∧_{P}^{q} F → M ∧_{P}^{q} M) is the homomorphic image of Ker(L_{F} → L_{M}) by π_{F}. It is also easy to check that Ker(L_{F} → L_{M}) is the normal subgroup of L_{F} generated by the elements (f_{1}, f_{2})(f′_{1}, f′_{2})^{−1} and f_{3}f′_{3}^{−1} such that φf_{i} = φf′_{i}, f_{i}, f′_{i} ∈ F (i = 1, 2) and φf_{3} = φf′_{3}, f_{3}, f′_{3} ∈ Kerν. Thus its image in F ∧_{P}^{q} F is the normal subgroup generated by the elements (f_{1} ∧ f_{2})(f′_{1} ∧ f′_{2})^{−1} and {f_{3}}{f′_{3}}^{−1}, which by the formulas (9), (10) and (16) coincides with the normal subgroup of F ∧_{P}^{q} F generated by the elements f ∧ r, r ∧ f and {r}, f ∈ F, r ∈ R. Then the image of this subgroup by the isomorphism F ∧_{P}^{q} F ≈ ∂_{2}^{q}(F ∧_{P}^{q} F) = F_{(q)}^{(2)} is ⟨⟨F, R⟩⟩_{(q)} and thus

F_{(q)}^{(2)}/⟨⟨F, R⟩⟩_{(q)} ≈ M ∧_{P}^{q} M. □

Lemma 3.5. Let h : A → B and g : B → C be group homomorphisms. If h is surjective then the following sequence of groups

1 ⟶ Ker(h) ⟶ Ker(gh) ⟶ Ker(g) ⟶ 1.

is exact.

Theorem 3.6. Let μ : M → P be a precrossed P -module, q > 0 and

1 ⟶ R ⟶ F ⟶^{φ} M ⟶ 1

a short exact sequence of precrossed P -modules, where F is a free precrossed P -module. If the homomorphism ∂_{2}^{q} : F ∧_{P}^{q} F → F_{(q)}^{(2)} is an isomorphism then there is an isomorphism of groups

H_{2}(M, q) ≈ Ker(M ∧_{P}^{q} M → M).

Proof. By Lemma 3.5 one has the following exact sequence of groups

1 ⟶ Ker(φ ∧_{P}^{q} φ) ⟶ Ker(∂_{2}^{q}(φ ∧_{P}^{q} φ)) ⟶ Ker∂_{2}^{q} ⟶ 1.

From the commutative diagram of groups

F ∧_{P}^{q} F ⟶^{φ∧_{P}^{q} φ} M ∧_{P}^{q} M
≈↓  ↓^{∂_{2}^{q}}
F_{(q)}^{(2)} ⟶^{φ_{(q)}^{(2)}} M_{(q)}^{(2)}

one gets Ker(∂_{2}^{q}(φ ∧_{P}^{q} φ)) ≈ Kerφ_{(q)}^{(2)} = R ∩ F_{(q)}^{(2)}. Then Ker(∂_{2}^{q} : M ∧_{P}^{q} M → M) ≈

R ∩ F_{(q)}^{(2)}/⟨⟨F, R⟩⟩_{(q)} = H_{2}(M, q)_{P}. □

Finally we give an example showing that there exists such a group P and a free precrossed P -module F that the homomorphism ∂_{2}^{q} : F ∧_{P}^{q} F → F is injective.

Lemma 3.7. Let (M, μ) be precrossed P -module and q be a nonnegative integer. Then there is an exact sequence of groups

M ∧_{P} M ⟶^{φ} M ∧_{P}^{q} M ⟶ ker μ/⟨M, M⟩ ⟶ 1 .

Proof. The homomorphism φ is given by φ(m ∧ m′) = m ∧ m′. The required exacteness is easy to check. □

Proposition-Example 3.8. Let P be a free group, μ : F → P be a free precrossed P -module and q > 0. Then the homomorphism ∂_{2}^{q} : F ∧_{P}^{q} F → F_{(q)}^{(2)} is an isomorphism.

Proof. Using Lemma 3.7 one has the following commutative diagram of groups with exact rows

\begin{aligned}1 & ⟶ & F ∧_{P} F & ⟶^{φ} & F ∧_{P}^{q} F & ⟶ & ker μ/F^{(2)} & ⟶ & 1 \\ & & ↓ ^{∂_{2}} & & ↓ ^{∂_{2}^{q}} & & ↓ ^{α} \\ 1 & ⟶ & F^{(2)} & ⟶ & F_{(q)}^{(2)} & ⟶ & F_{(q)}^{(2)}/F^{(2)} & ⟶ & 1 \end{aligned} ,

where F^{(2)} = ⟨F, F ⟩, ∂_{2} is an isomorphism [BC] proved applying a theorem of J.H.C. Whitehead [W] on 2-dimensional CW-complexes and the theorem of Kan [K] (see above) and hence φ is injective. One can directly check that α is an isomorphism and so is ∂_{2}^{q}. □

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Nick Inassaridze inas@rmi.acnet.ge

A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia

Emzar Khmaladze khmal@rmi.acnet.ge

A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia