Theory and Applications of Categories, Vol. 1, No. 3, 1995, pp. 54–71.

ON FINITE INDUCED CROSSED MODULES, AND THE HOMOTOPY 2-TYPE OF MAPPING CONES

Dedicated to the memory of J. H. C. Whitehead

RONALD BROWN AND CHRISTOPHER D. WENSLEY

Transmitted by Lawrence Breen

ABSTRACT. Results on the finiteness of induced crossed modules are proved both algebraically and topologically. Using the Van Kampen type theorem for the fundamental crossed module, applications are given to the 2-types of mapping cones of classifying spaces of groups. Calculations of the cohomology classes of some finite crossed modules are given, using crossed complex methods.

Introduction

Crossed modules were introduced by J.H.C. Whitehead in [31]. They form a part of what can be seen as his programme of testing the idea of extending to higher dimensions the methods of combinatorial group theory of the 1930’s, and of determining some of the extra structure that was necessary to model the geometry. Other papers of Whitehead of this era show this extension of combinatorial group theory tested in different directions.

In this case he was concerned with the algebraic properties satisfied by the boundary map

∂ : π_{2}(X, A) → π_{1}(A)

of the second relative homotopy group, together with the standard action on it of the fundamental group π_{1}(A). This is the fundamental crossed module Π_{2}(X, A) of the pair (X, A). In order to determine the second homotopy group of a CW -complex, he formulated and proved the following theorem for this structure: Theorem W Let X = A ∪ {e_{λ}^{2}} be obtained from the connected space A by attaching 2-cells. Then the second relative homotopy group π_{2}(X, A) may be described as the free crossed π_{1}(A)-module on the 2-cells. The proof in [32] uses transversality and knot theory ideas from the previous papers [30, 31]. See [5] for an exposition of this proof. Several other proofs are available. The survey by Brown and Huebschmann [12], and the book edited by Hog-Angeloni, Metzler and Sieradski [20], give wider applications.

The first author was supported by EPSRC grant GR/J94532 for a visit to Zaragoza in November, 1993, and is, with Dr T. Porter, supported for equipment and software by EPSRC Grant GR/J63552, “Non abelian homological algebra”.

Received by the editors 8 March 1995 and, in revised form, 12 April 1995.

Published on 10 May 1995

1991 Mathematics Subject Classification : 18G10, 20F38, 55P15, 55Q20.

Key words and phrases: crossed modules, homotopy 2-types, generalised van Kampen Theorem,crossed resolution, Postnikov invariant.

\begin{matrix}c \\ ◯ \end{matrix} Ronald Brown and Christopher D. Wensley 1995. Permission to copy for private use granted.

The paper of Mac Lane and Whitehead [26] uses Theorem W to show that the 2dimension-al homotopy theory of pointed, connected CW -complexes is completely modelled by the theory of crossed modules. This is an extra argument for regarding crossed modules as 2-dimensional versions of groups.

One of our aims is the explicit calculation of examples of the crossed module

(∂ : π_{2}(A ∪ ΓV, A) → π_{1}(A) )  (1)

of a mapping cone, when π_{1}(V), π_{1}(A) are finite. The key to this is the 2-dimensional Van Kampen Theorem (2-VKT) proved by Brown and Higgins in [9]. This implies a generalisation of Theorem W, namely that the crossed module (1) is induced from the identity crossed module (1 : π_{1}(V) → π_{1}(V)) by the morphism π_{1}(V) → π_{1}(A).

Presentations of induced crossed modules are given in [9], and from these we prove a principal theorem (Theorem 2.1), that crossed modules induced from finite crossed modules by morphisms of finite groups are finite. We also use topological methods to prove a similar result for finite p-groups (Corollary 4.3). These results give a new range of finite crossed modules.

Sequels to this paper discuss crossed modules induced by a normal inclusion [15], and calculations obtained using a group theory package [16].

The origin of the 2-VKT was the idea of extending to higher dimensions the notion of the fundamental groupoid, as suggested in 1967 in [4]. This led to the discovery of the relationship of 2-dimensional groupoids to crossed modules, in work with Spencer [14]. This relationship reinforces the idea of ‘higher dimensional group theory’, and was essential for the proof of the 2-VKT for the fundamental crossed module [9]. In view of the results of Mac Lane and Whitehead [26], and of methods of classifying spaces of crossed modules by Loday [25] and Brown and Higgins [11] (see section 3), the 2-VKT allows for the explicit computation of some homotopy 2-types, in the form of the crossed modules which model them.

In some cases, the Postnikov invariant of these 2-types can be calculated, as the following example shows. Corollary 5.5 Let C_{n} denote the cyclic group of order n, and let BC_{n} denote its classifying space. The second homotopy module of the mapping cone X = BC_{n^{2}} ∪ ΓBC_{n} is a particular cyclic C_{n}-module, A_{n}, say. The cohomology group H^{3}(C_{n}, A_{n}) is a cyclic group of order n, and the first Postnikov invariant of X is a generator of this group.

The method used for the calculation of the cohomology class here is also of interest. It introduces a new small free crossed resolution of the cyclic group of order n in order to construct an explicit 3-cocycle corresponding to the above crossed module. This indicates a wider possibility of using crossed resolutions for explicit calculations. It is also related to Whitehead’s use of what he called in [32] ‘homotopy systems’, and which are simply free crossed complexes.

The initial motivation of this paper was a conversation with Rafael Sivera in Zaragoza, in November, 1993, which suggested the lack of explicit calculations of induced crossed modules. This led to discussions at Bangor on the use of computational group theory packages which culminated in a GAP [29] program [16], and to the development of general theory.

1. Crossed modules and induced crossed modules

In this section, we recall the definition of induced crossed modules, and of results of [9] on presentations of induced crossed modules. We then give some basic examples of these.

Recall that a crossed module M = (μ : M → P) is a morphism of groups μ : M → P together with an action (m, p) ↦ m^{p} of P on M satisfying the two axioms

• CM1) μ(m^{p}) = p^{−1}μ(m)p

• CM2) m^{μn} = n^{−1}mn

for all m, n ∈ M and p ∈ P. We say that M is finite when M is finite.

The category X M of crossed modules has as objects all crossed modules. Morphisms in X M are pairs (g, f) forming commutative diagrams

[figure: M; μ; g; P; f; N; ν; Q]

in which the horizontal maps are crossed modules, and g, f preserve the action in the sense that for all m ∈ M, p ∈ P we have g(m^{p}) = (gm)^{fp}. If P is a group, then the category X M/P of crossed P -modules is the subcategory of X M whose objects are the crossed P -modules and in which a morphism (μ : M → P) → (ν : N → P) of crossed P -modules is a morphism g : M → N of groups such that g preserves the action ( g(m^{p}) = (gm)^{p}, for all m ∈ M, p ∈ P), and νg = μ.

Standard algebraic examples of crossed modules are:

(i) an inclusion of a normal subgroup, with action given by conjugation; (ii) an inner automorphism crossed module (χ : M → Aut M) in which χm is the automorphism n ↦ m^{−1}nm; (iii) a zero crossed module (0 : M → P) where M is a P -module; (iv) an epimorphism M → P with kernel contained in the centre of M.

Examples of finite crossed modules may be found among those above, the induced crossed modules of this paper and its sequels [15, 16], and coproducts [6] and tensor products [13, 19] of finite crossed P -modules.

Further important examples of crossed modules are the free crossed modules, referred to in the Introduction, which are rarely finite. They arise algebraically in considering identities among relations [12, 20], which are non-abelian forms of syzygies.

We next define pullback crossed modules. Let ι : P → Q be a morphism of groups and let (ν : N → Q) be a crossed Q-module. Let ν′ : ι^{∗}N → P be the pullback of N by ι, so that ι^{∗}N = {(p, n) ∈ P × N | ιp = νn}, and ν′ : (p, n) ↦ p. Let P act on ι^{∗}N by (p_{1}, n)^{p} = (p^{−1}p_{1}p, n^{ιp}). The verification of the axiom CM1) is immediate, while CM2) is proved as follows: Let (p, n), (p_{1}, n_{1}) ∈ ι^{∗}N. Then

(p, n)^{−1}(p_{1}, n_{1})(p, n) = (p^{−1}p_{1}p, n^{−1}n_{1}n)
= (p^{−1}p_{1}p, n_{1}^{νn})
= (p^{−1}p_{1}p, n_{1}^{ιp})
= (p_{1}, n_{1})^{ν′(p,n)}.

1.1. Proposition. The functor ι^{∗} : X M/Q → X M/P has a right adjoint ι_{∗}.

Proof. This follows from general considerations on Kan extensions.

The universal property of induced crossed modules is the following. Let (μ : M → P), (γ : C → Q) be crossed modules. In the diagram

[figure: M; f; μ; ῑ; P; ι; ι_{∗}M; δ; g; Q; C; γ]

the pair (ῑ, ι) is a morphism of crossed modules such that for any crossed Q-module (γ : C → Q) and morphism of crossed modules (f, ι), there is a unique morphism g : ι_{∗}M → C of crossed Q-modules such that gῑ = f.

It is a consequence of this universal property that if M = P = F (R), the free group on a set R, and if w : R → Q is the restriction of ι to the set R, then ι_{∗}F (R) is the free crossed module on w, in the sense of Whitehead [32] (see also [12, 20, 28]). Constructions of this free crossed module are given in these papers.

A presentation for induced crossed modules for a general morphism ι is given in Proposition 8 of [9]. We will need two more particular results. The first is Proposition 9 of that paper, and the second is a direct deduction from Proposition 10.

1.2. Proposition. If ι : P → Q is a surjection, and (μ : M → P) is a crossed Pmodule, then ι_{∗}M ≅ M/[M, K], where K = Ker ι, and [M, K] denotes the subgroup of M generated by all m^{−1}m^{k} for all m ∈ M, k ∈ K.

The following term and notation will be used frequently. Let P be a group and let T be a set. We define the copower P ∗ T to be the free product of groups P_{t}, t ∈ T, each with elements (p, t), p ∈ P, and isomorphic to P under the map (p, t) ↦ p. If Q is a group, then P ∗ Q will denote the copower of P with the underlying set of the group Q.

1.3. Proposition. If ι : P → Q is an injection, and (μ : M → P) is a crossed Pmodule, let T be a right transversal of ιP in Q. Let Q act on the copower M ∗ T by the rule (m, t)^{q} = (m^{p}, u), where p ∈ P, u ∈ T, and tq = (ιp)u. Let δ : M ∗ T → Q be defined by (m, t) ↦ t^{−1}(ιμm)t. Let S be a set of generators of M as a group, and let S^{P} = {x^{p} : x ∈ S, p ∈ P }. Then

ι_{∗}M = (M ∗ T)/R

where R is the normal closure in M ∗ T of the elements

⟨(r, t), (s, u)⟩ = (r, t)^{−1}(s, u)^{−1}(r, t)(s, u)^{δ(r,t)} (r, s ∈ S^{P} , t, u ∈ T).

Proof. Let N = M ∗ T. Proposition 10 of [9] yields that ι_{∗}M is the quotient of N by the subgroup ⟨N, N⟩ generated by ⟨n, n_{1}⟩ = n^{−1}n_{1}^{−1}nn_{1}^{δn}, n, n_{1} ∈ N, and which is called in [12] the Peiffer subgroup of N. Now N is generated by the set (S^{P} , T) = {(s^{p}, t) : s ∈ S, p ∈ P, t ∈ T }, and this set is Q-invariant since (s^{p}, t)^{q} = (s^{pp′}, u) where u ∈ T, p′ ∈ P satisfy tq = (ιp′)u. It follows from Proposition 3 of [12] that ⟨N, N⟩ is the normal closure of the set ⟨(S^{P} , T), (S^{P} , T)⟩ of basic Peiffer commutators.

1.4. Example. The dihedral crossed module. We show how this works out in the following case, which exhibits a number of typical features. We let Q be the dihedral group D_{n} with presentation ⟨x, y : x^{n} = y^{2} = xyxy = 1⟩, and let M = P be the cyclic subgroup C_{2} of order 2 generated by y. Let C_{n} = {0, 1, 2, …, n − 1} be the cyclic group of order n. A right transversal T of C_{2} in D_{n} is given by the elements x^{i}, i ∈ C_{n}. Hence ι_{∗}C_{2} has a presentation with generators a_{i} = (y, x^{i}), i ∈ C_{n}, and relations given by a_{i}^{2} = 1, i ∈ C_{n}, together with the Peiffer relations. Now δa_{i} = x^{−i}yx^{i} = yx^{2i}. Further the action is given by (a_{i})^{x} = a_{i+1}, (a_{i})^{y} = a_{n−i}. Hence (a_{i})^{δa_{j}} = a_{2j−i}, so that the Peiffer relations are a_{j}a_{i}a_{j} = a_{2j−i}. It is well known that we now have a presentation of the dihedral group D_{n}, from which we recover the standard presentation ⟨u, v : u^{n} = v^{2} = uvuv = 1⟩ by setting u = a_{0}a_{1}, v = a_{0}, so that u^{i} = a_{0}a_{i}. Then

δu = x^{2}, δv = y,

so that y acts on ι_{∗}C_{2} by conjugation by v. However x acts by

u^{x} = u, v^{x} = vu.

Note that this is consistent with the crossed module axiom CM2) since

v^{x^{2}} = (vu)^{x} = vuu = u^{−1}vu.

We call D_{n} = (δ : D_{n} → D_{n}) the dihedral crossed module. It follows from these formulae that δ is an isomorphism if n is odd, and has kernel and cokernel isomorphic to C_{2} if n is even. In particular, if n is even, then by results of section 3, π_{2}(BD_{n} ∪ ΓBC_{2}) can be regarded as having one non-trivial element represented by u^{n/2}.

1.5. Corollary. Assume ι : P → Q is injective. If M has a presentation as a group with g generators and r relations, the set of generators of M is P-invariant, and n = [Q : ιμ(M)], then ι_{∗}M has a presentation with gn generators and rn + g^{2}n(n − 1) relations.

Another corollary determines induced crossed modules under some abelian conditions. This result has useful applications. If M is an abelian group, or P -module, and T is a set, we define the copower of M with T, written M ⊕ T, to be the sum of copies of M, one for each element of T.

1.6. Corollary. Let (μ : M → P) be a crossed P -module and ι : P → Q a monomorphism of groups such that M is abelian and ιμ(M) is normal in Q. Then ι_{∗}M is abelian and as a Q-module is just the induced Q-module in the usual sense.

Proof. We use the result and notation of Proposition 1.3. Note that if u, t ∈ T and r ∈ S then uδ(r, t) = ut^{−1}(ιμr)t = (ιμm)ut^{−1}t = (ιμm)u for some m ∈ M, by the normality condition. The Peiffer commutator given in Proposition 1.3 can therefore be rewritten as

(r, t)^{−1}(s, u)^{−1}(r, t)(s, u)^{δ(r,t)} = (r^{−1}, t)(s, u)^{−1}(r, t)(s^{m}, u).

Since M is abelian, s^{m} = s. Thus the basic Peiffer commutators reduce to ordinary commutators. Hence ι_{∗}M is the copower M ⊕ T, and this, with the given action, is the usual presentation of the induced Q-module.

1.7. Example. Let M = P = Q be the infinite cyclic group C_{∞}, let μ be the identity, and let ι be multiplication by 2. Then ι_{∗}M ≅ C_{∞} × C_{∞}, and the action of a generator of Q on ι_{∗}M is to switch the two copies of C_{∞}. This result could also be deduced from well known results on free crossed modules. However, our results show that we get a similar conclusion simply by replacing each C_{∞} in the above by a finite cyclic group C_{2n}, and this fact is new.

2. On the finiteness of induced crossed modules

In this section we give an algebraic proof that a crossed module induced from a finite crossed module by a morphism with finite cokernel is also finite. In a later section we will prove a slightly less general result, but by topological methods which will also yield results on the preservation of the Serre class of a crossed module under the inducing process.

2.1. Theorem. Let (μ : M → P) be a crossed module and let ι : P → Q be a morphism of groups. Suppose that M and the index of ι(P) in Q are finite. Then the induced crossed module (δ : ι_{∗}M → Q) is finite.

Proof. Factor the morphism ι : P → Q as τσ where τ is injective and σ is surjective. Then ι_{∗}M is isomorphic to τ_{∗}σ_{∗}M. It is immediate from Proposition 1.2 that if M is finite then so also is σ_{∗}M. So it is enough to assume that ι is injective, and in fact we assume it is an inclusion.

Let T be a right transversal of ιP in Q. Let Y = M ∗ T be the copower of M and T, and let δ : Y → Q and the action of Q on Y be as in Proposition 1.3. The equations tq = (ιp)u which determine this action in fact provide a function

(ξ, η) : T × Q → P × T, (t, q) ↦ (p, u).

A basic Peiffer relation is then of the form

(m, t)(n, v) = (n, v)(m^{ξ(t,v^{−1}(ιμm)v)}, η(t, v^{−1}(ιμm)v)) = (n, v)(m^{p}, u)  (2)

where m, n ∈ M, t, u ∈ T and q = v^{−1}(ιμm)v.

We now assume that the finite set T has l elements and has been given the total order t_{1} < t_{2} < ⋯ < t_{l}. An element of Y may be represented as a word

(m_{1}, u_{1})(m_{2}, u_{2}) … (m_{e}, u_{e}).  (3)

Such a word is said to be reduced when u_{i} ≠ u_{i+1}, 1 ≤ i < e, and to be ordered if u_{1} < u_{2} < ⋯ < u_{e} in the given order on T. This yeilds a partial ordering of M ∗ T where (m_{i}, u_{i}) ≤ (m_{j}, u_{j}) whenever u_{i} ≤ u_{j}.

A twist uses the Peiffer relation (2) to replace a reduced word w = w_{1}(m, t)(n, v)w_{2}, with v < t, by w′ = w_{1}(n, v)(m^{p}, u)w_{2}. If the resulting word is not reduced, multiplication in M_{v} and M_{u} may be used to reduce it. In order to show that any word may be ordered by a finite sequence of twists and reductions, we define an integer weight function on the set W_{n} of non-empty words of length at most n by

Ω_{n} : W_{n} → ℤ^{+}, (m_{1}, t_{j_{1}})(m_{2}, t_{j_{2}}) … (m_{e}, t_{j_{e}}) ↦ l^{e} ∑_{i=1}^{e} l^{n−i}j_{i}.

It is easy to see that Ω_{n}(w′) < Ω_{n}(w) when w → w′ is a reduction. Similarly, for a twist

w = w_{1}(m_{i}, t_{j_{i}})(m_{i+1}, t_{j_{i+1}})w_{2} → w′ = w_{1}(m_{i+1}, t_{j_{i+1}})(n, t_{k})w_{2}

the weight reduction is

Ω_{n}(w) − Ω_{n}(w′) = l^{n+e−i−1}( l(j_{i} − j_{i+1}) + j_{i+1} − j_{k} ) ≥ l^{n+e−i−1},

so the process terminates in a finite number of moves.

This ordering process is a special case of a purely combinatorial sorting algorithm discussed in [17].

We now specify an algorithm for converting a reduced word to an ordered word. Various algorithms are possible, some more efficient than others, but we are not interested in efficiency here. We call a reduced word k-ordered if the subword consisting of the first k elements is ordered and the remaining elements are greater than these. Every reduced word is at least 0-ordered. Given a k-ordered, reduced word, find the rightmost minimal element to the right of the k-th position. Move this element one place to the left with a twist, and reduce if necessary. The resulting word may only be j-ordered, with j < k, but its weight will be less than that of the original word. Repeat until an ordered word is obtained.

Let Z = M_{t_{1}} × M_{t_{2}} × … × M_{t_{l}} be the product of the sets M_{t_{i}} = M × {t_{i}}. Then the algorithm yeilds a function ϕ : Y → Z such that the quotient morphism Y → ι_{∗}M factors through ϕ. Since Z is finite, it follows that ι_{∗}M is finite.

2.2. Remark. In this last proof, it is in general not possible to give a group structure on the set Z such that the quotient morphism Y → ι_{∗}M factors through a morphism to Z. For example, in the dihedral crossed module of example 1.4, with n = 3, the set Z will have 8 elements, and so has no group structure admitting a morphism onto D_{3}. This explains why the above method does not give an algebraic proof of Corollary 4.3, which gives conditions for ι_{∗}M to be a finite p-group. However, in [15], we will give an algebraic proof for the case P is normal in Q.

3. Topological applications

As explained in the Introduction, the fundamental crossed module functor Π_{2} assigns a crossed module (∂ : π_{2}(X, A) → π_{1}(A)) to any base pointed pair of spaces (X, A). We will use the following consequence of Theorem C of [9], which is a 2-dimensional Van Kampen type theorem for this functor.

3.1. Theorem. ([9], Theorem D) Let (B, V) be a cofibred pair of spaces, let f : V → A be a map, and let X = A ∪_{f} B. Suppose that A, B, V are path-connected, and the pair (B, V) is 1-connected. Then the pair (X, A) is 1-connected and the diagram

[figure: π_{2}(B, V); δ; ϵ; π_{1}(V); λ; π_{2}(X, A); δ′; π_{1}(A)]

presents π_{2}(X, A) as the crossed π_{1}(A)-module λ_{∗}(π_{2}(B, V)) induced from the crossed π_{1}(V)-module π_{2}(B, V) by the group morphism λ : π_{1}(V) → π_{1}(A) induced by f.

As pointed out earlier, in the case P is a free group on a set R, and μ is the identity, then the induced crossed module ι_{∗}P is the free crossed Q-module on the function ι|R : R → Q. Thus Theorem 3.1 implies Whitehead’s Theorem W of the Introduction. A considerable amount of work has been done on this case, because of the connections with identities among relations, and methods such as transversality theory and “pictures” have proved successful ([12, 28]), particularly in the homotopy theory of 2-dimensional complexes [20]. However, the only route so far available to the wider geometric applications of induced crossed modules is Theorem 3.1. We also note that this Theorem includes the relative Hurewicz Theorem in this dimension, on putting A = ΓV, and f : V → ΓV the inclusion.

We will apply this Theorem 3.1 to the classifying space of a crossed module, as defined by Loday in [25] or Brown and Higgins in [11]. This classifying space is a functor B assigning to a crossed module M = (μ : M → P) a pointed CW -space BM with the following properties:

3.2. The homotopy groups of the classifying space of the crossed module M = (μ : M → P) are given by

π_{i}(BM) ≅ \{ \begin{matrix}Coker μ & for i = 1 \\ Ker μ & for i = 2 \\ 0 & for i > 2. \end{matrix}

3.3. The classifying space B(ι : 1 → P) is the usual classifying space BP of the group P, and BP is a subcomplex of BM. Further, there is a natural isomorphism of crossed modules

Π_{2}(BM, BP) ≅ M.

3.4. If X is a reduced CW -complex with 1-skeleton X^{1}, then there is a map

X → B(Π_{2}(X, X^{1}))

inducing an isomorphism of π_{1} and π_{2}.

It is in these senses that it is reasonable to say, as in the Introduction, that crossed modules model all pointed homotopy 2-types.

We now give two direct applications of Theorem 3.1.

3.5. Corollary. Let M = (μ : M → P) be a crossed module, and let ι : P → Q be a morphism of groups. Let β : BP → BM be the inclusion. Consider the pushout

[figure: BP; β; Bι; BM; BQ; β′; X.]  (4)

Then the fundamental crossed module of the pair (X, BQ) is isomorphic to the induced crossed module (δ : ι_{∗}M → Q), and this crossed module determines the 2-type of X.

Proof. The first statement is immediate from Theorem 3.1. The final statement follows from results of [11], since the morphism Q → π_{1}(X) is surjective.

3.6. Remark. An interesting special case of the last corollary is when M is an inclusion of a normal subgroup, since then BM is of the homotopy type of B(P/M). So we have determined the 2-type of a homotopy pushout

[figure: BP; Bp; Bι; BR; BQ; p′; X]

in which p : P → R is surjective.

We write ΓV for the cone on a space V.

3.7. Corollary. Let ι : P → Q be a morphism of groups. Then the fundamental crossed module Π_{2}(BQ ∪_{Bι} ΓBP, BQ) is isomorphic to the induced crossed module (δ : ι_{∗}P → Q).

4. Finiteness theorems by topological methods

The aim of this section is to show that the property of being a finite p-group is preserved by the process of induced crossed modules. We use topological methods.

An outline of the method is as follows. Suppose that Q is a finite p-group. To prove that ι_{∗}M is a finite p-group, it is enough to prove that Ker (ι_{∗}M → Q) is a finite p-group. But this kernel is the second homotopy group of the space X of the pushout (4), and so is also isomorphic to the second homology group of the universal cover X̃ of X. In order to apply the homology Mayer-Vietoris sequence to this universal cover, we need to show that it may be represented as a pushout, and we need information on the homology of the spaces determining this pushout. So we start with the necessary information on covering spaces.

We work in the convenient category T OP of weakly Hausdorff k-spaces [24]. Let α : X̃ → X be a map of spaces. In the examples we will use, α will be a covering map. Then α induces a functor

α^{∗} : T OP/X → T OP/X̃.

It is known that α has a right adjoint and so preserves colimits [2, 1, 24].

For regular spaces, the pullback of a covering space in the above category is again a covering space. These results enable us to identify a covering space of an adjunction space as an adjunction space obtained from the induced covering spaces.

If further, α is a covering map, and X is a CW -complex, then X̃ may be given the structure of a CW -complex [27].

We also need a special case of the basic facts on the path components and fundamental group of induced covering maps ([7, 8, 27]). Given the following pullback

[figure: Â; α′; X̃; α; A; f; X]

and points a ∈ A, x̃ ∈ X̃ such that fa = αx̃, in which α is a universal covering map and X, A, X̃ are path connected, then there is a sequence

1 → π_{1}(Â, (a, x̃)) → π_{1}(A, a) →^{f_{∗}} π_{1}(X, fa) → π_{0}(Â) → 1.  (5)

This sequence is exact in the sense of sequences arising from fibrations of groupoids [7], which involves an operation of the fundamental group π_{1}(X, fa) on the set π_{0}(Â) of path components of Â. It follows that the fundamental group of Â is isomorphic to Ker f_{∗}, and that π_{0}(Â) is bijective with the set of cosets (π_{1}(X, fa))/(f_{∗}π_{1}(A, a)). It is also clear that the covering Â → A is regular and that all the components of Â are homeomorphic.

Let M be the crossed module (μ : M → P) and let ι : P → Q be a morphism of groups. Let

X = BQ ∪_{Bι} BM

as in diagram (4). Let α : X̃ → X be the universal covering map, and let B̂Q, BM^{}, B̂P be the pullbacks of X̃ under the maps BQ → X, BM → X, BP → X. Then we may write

X̃ ≅ B̂Q ∪_{Bι} BM^{},  (6)

by the results of section 3.

From the exact sequence (5) we obtain the following exact sequences, in which π_{1}X ≅ Q ∗_{P} (P/μM):

1 → π_{1}(B̂Q) → Q → Q ∗_{P} (P/μM) → 1,
1 → π_{1}(BM^{}) → P/μM → Q ∗_{P} (P/μM) → π_{0}BM^{}→ 1,
1 → π_{1}(B̂P) → P  → Q ∗_{P} (P/μM) → π_{0}B̂P → 1.

4.1. Proposition. Under the above situation, let the groups π_{1}(B̂P), π_{1}(B̂M), π_{1}(B̂Q) be denoted by P′, M′, Q′ respectively, and let BM′ denote a component of B̂M. Then there is an exact sequence

H_{2}(P′) ⊕ π_{0}B̂P → (H_{2}(BM′) ⊕ π_{0}B̂M) ⊕ H_{2}(Q′) → π_{2}(X) →
→ H_{1}(P′) ⊕ π_{0}B̂P → (H_{1}(M′) ⊕ π_{0}B̂M) ⊕ H_{1}(Q′) → 0.

Proof. This is immediate from the Mayer-Vietoris sequence for the pushout (6) and the fact that H_{2}(X̃) ≅ π_{2}(X).

4.2. Corollary. If ι : P → Q is the inclusion of a normal subgroup, and X = BQ ∪_{Bι} ΓBP, then π_{2}(X) is isomorphic to H_{1}(P)⊗I(Q/P), where I(G) denotes the augmentation ideal of a group G.

This result agrees with Corollary 2.5 of [15], in which the induced crossed module itself is computed, in the case P is normal in Q, via the use of coproducts of crossed P -modules.

4.3. Corollary. Let M = (μ : M → P) be a crossed module and let ι : P → Q be a morphism of groups. If M, P and Q are finite p-groups, then so also is ι_{∗}M.

Proof. It is standard that the (reduced) homology groups of a finite p-group are finite p-groups. The same applies to the reduced homology of the classifying space of a crossed module of finite p-groups. The latter may be proved using the spectral sequence of a covering, and Serre C theory, as in Chapters IX and X of [21]. In the present case, we need information only on H_{2}(BM), and some of its connected covering spaces, and this may be deduced from the exact sequence due to Hopf

H_{3}K → H_{3}G → (π_{2}K) ⊗_{ℤG} ℤ → H_{2}K → H_{2}G → 0

for any connected space K with fundamental group G (see for example Exercise 6 on p.175 of [3]). Proposition 4.1 shows that Ker (ι_{∗}(M) → Q) ≅ π_{2}(X) is a finite p-group. Since Q is a finite p-group, it follows that ι_{∗}M is a finite p-group.

Note that these methods extend also to results on the Serre class of an induced crossed module, which we leave the reader to formulate.

5. Cohomology classes

Recall [22, 3] that if G is a group and A is a G-module, then elements of H^{3}(G, A) may be represented by equivalence classes of crossed sequences

0 → A → M ⟶^{μ} P → G → 1,  (7)

namely exact sequences as above such that (μ : M → P) is a crossed module. The equivalence relation between such crossed sequences is generated by the basic equivalences, namely the existence of a commutative diagram of morphisms of groups as follows

[figure: 0; A; 1; M; μ; f; P; g; G; 1; 1; 0; A; M′; μ′; P′; G; 1]

such that (f, g) is a morphism of crossed modules. Such a diagram is called a morphism of crossed sequences.

The zero cohomology class is represented by the crossed sequence

0 → A ⟶^{1} A ⟶^{0} G ⟶^{1} G → 1,

which we sometimes abbreviate to

A ⟶^{0} G.

In a similar spirit, we say that a crossed module (μ : M → P) represents a cohomology class, namely an element of H^{3}(Coker μ, Ker μ).

5.1. Example. Let C_{n^{2}} denote the cyclic group of order n^{2}, written multiplicatively, with generator u. Let γ_{n} : C_{n^{2}} → C_{n^{2}} be given by u ↦ u^{n}. This defines a crossed module, with trivial operations. This crossed module represents the trivial cohomology class in H^{3}(C_{n}, C_{n}), in view of the morphism of crossed sequences

[figure: 0; C_{n}; 1; 1; C_{n}; 0; λ; C_{n}; 1; λ; C_{n}; 1; 0; 0; C_{n}; C_{n^{2}}; γ_{n}; C_{n^{2}}; C_{n}; 0]

where, if t is the generator of the top C_{n}, then λ(t) = u^{n}.

5.2. Example. We show that the dihedral crossed module D_{n} of Example 1.4 represents the trivial cohomology class. This is clear for n odd, since then δ is an isomorphism. For n even, we simply construct a morphism of crossed sequences as in the following diagram

[figure: 0; C_{2}; 1; ≅; C_{2}; f_{2}; 0; C_{2}; f_{1}; 1; C_{2}; ≅; 0; 0; C_{2}; D_{n}; δ; D_{n}; C_{2}; 0]

where if t denotes the non trivial element of C_{2} then f_{1}(t) = x, f_{2}(t) = u^{n/2}. Just for interest, we leave it to the reader to prove that there is no morphism in the other direction between these crossed sequences.

A crossed module M = (μ : M → P) determines a cohomology class

k_{M} ∈ H^{3}(Coker μ, Ker μ).

If X is a connected, pointed CW -complex with 1-skeleton X^{1}, then the class

k_{X}^{3} ∈ H^{3}(π_{1}X, π_{2}X)

of the crossed module Π_{2}(X, X^{1}) is called the first Postnikov invariant of X. This class is also represented by Π_{2}(X, A) for any connected subcomplex A of X such that (X, A) is 1-connected and π_{2}(A) = 0. It may be quite difficult to determine this Postnikov invariant from a presentation of this last crossed module, and even the meaning of the word “determine” in this case is not so clear. There are practical advantages in working directly with the crossed module, since it is an algebraic object, and so it, or families of such objects, may be manipulated in many convenient and useful ways. Thus the advantages of crossed modules over the corresponding 3-cocycles are analogous to some of the advantages of homology groups over Betti numbers and torsion coefficients.

However, in work with crossed modules, and in applications to homotopy theory, information on the corresponding cohomology classes, such as their non-triviality, or their order, is also of interest. The aim of this section is to give background to such a determination, and to give two example of finite crossed modules representing non-trivial elements of the corresponding cohomology groups.

The following general problem remains. If G, A are finite, where A is a G-module, how can one characterise the subset of H^{3}(G, A) of elements represented by finite crossed modules? This subset is a subgroup, since the addition may be defined by a sum of crossed sequences, of the Baer type. (An exposition of this is given by Danas in [18].) It might always be the whole group.

The natural context in which to show how a crossed sequence gives rise to a 3-cocycle is not the traditional chain complexes with operators but that of crossed complexes [22]. We explain how this works here. For more information on the relations between crossed complexes and the traditional chain complexes with operators, see [10].

Recall that a free crossed resolution of the group G is a free aspherical crossed complex F_{∗} together with an epimorphism ϕ : F_{1} → G with kernel δ_{2}(F_{2}).

5.3. Example. The cyclic group C_{n} of order n is written multiplicatively, with generator t. We give for it a free crossed resolution F_{∗} as follows. Set F_{1} = C_{∞}, with generator w, and for r ≥ 2, set F_{r} = (C_{∞})^{n}. Here for r ≥ 2, F_{r} is regarded as the free C_{n}-module on one generator w_{0}, and we set w_{i} = (w_{0})^{t^{i}}. The morphism ϕ : C_{∞} → C_{n} sends w to t, and the operation of F_{1} on F_{r} for r ≥ 2 is via ϕ. The boundaries are given by

1. δ_{2}(w_{i}) = w^{n},

2. for r odd, δ_{r}(w_{i}) = w_{i}w_{i+1}^{−1},

3. for r even and greater than 2, δ_{r}(w_{i}) = w_{0}w_{1} … w_{n−1}.

Previous calculations show that δ_{2} is the free crossed C_{∞}-module on the element w^{n} ∈ C_{∞}. Thus F_{∗} is a free crossed complex. It is easily checked to be aspherical, and so is, with ϕ, a crossed resolution of C_{n}.

Let A be a G-module. Let C(G, A, 3) denote the crossed complex C which is G in dimension 1, A in dimension 3, with the given action of G on A, and which is 0 elsewhere, as in the following diagram

\xymatrix{⋯ \ar[r] & 0 \ar[r] & A \ar[r] & 0 \ar[r] & G.}

Let (F_{∗}, ϕ) be a free crossed resolution of G. It follows from the discussions in [10, 11] that a 3-cocycle of G with coefficients in A can be represented as a morphism of crossed complexes f : F_{∗} → C(G, A, 3) over ϕ. This cocycle is a coboundary if there is an operator morphism l : F_{2} → A over ϕ : F_{1} → G such that lδ_{3} = f_{3}.

[figure: F_{4}; δ_{4}; F_{3}; f_{3}; δ_{3}; F_{2}; l; δ_{2}; F_{1}; ϕ; 0; A; 0; G]

To construct a 3-cocycle on F_{∗} from the crossed sequence (7), first construct a morphism of crossed complexes as in the diagram

[figure: F_{4}; F_{3}; f_{3}; F_{2}; f_{2}; F_{1}; ϕ; f_{1}; G; 1; 0; A; M; μ; P; ψ; G]  (8)

using the freeness of F_{∗} and the exactness of the bottom row. Then compose this with the morphism of crossed sequences

[figure: 0; A; 1; M; μ; P; ψ; ψ; G; 1; 0; A; 0; G; 1; G]

Hence it is reasonable to say that the morphism f_{3} of diagram (8) is a 3-cocycle corresponding to the crossed sequence.

We now use these methods in an example.

5.4. Theorem. Let n ≥ 2, and let ι : C_{n} → C_{n^{2}} denote the injection sending a generator t of C_{n} to u^{n}, where u denotes a generator of C_{n^{2}.} Let A_{n} denote the C_{n}-module which is the kernel of the induced crossed module N = (∂ : ι_{∗}C_{n} → C_{n^{2}}). Then H^{3}(C_{n}, A_{n}) is cyclic of order n and has as generator the class of this induced crossed module.

Proof. By Corollary 1.6 the abelian group ι_{∗}C_{n} is the product V = (C_{n})^{n}. As a C_{n}- module it is cyclic, with generator v, say. Write v_{i} = v^{t^{i}}, i = 0, 1, …, n − 1. Then each v_{i} is a generator of a C_{n} factor of V. The kernel A_{n} of ∂ is a cyclic C_{n}-module on the generator a = v_{0}v_{1}^{−1}. Write a_{i} = a^{t^{i}} = v_{i}v_{i+1}^{−1}. As an abelian group, A_{n} has generators a_{0}, a_{1}, …, a_{n−1} with relations a_{i}^{n} = 1, a_{0}a_{1} … a_{n−1} = 1.

We define a morphism f_{∗} from F_{∗} to the crossed sequence containing N as in diagram (9), where

1. f_{1} maps w to u,

2. f_{2} maps the module generator w_{0} of F_{2} to v = v_{0}.

3. f_{3} maps the module generator w_{0} of F_{3} to a_{0}.

[figure: (C_{∞})^{n}; 0; δ_{4}; (C_{∞})^{n}; f_{3}; δ_{3}; (C_{∞})^{n}; l; f_{2}; δ_{2}; C_{∞}; f_{1}; C_{n}; 1; 0; A_{n}; (C_{n})^{n}; ∂; C_{n^{2}}; C_{n}]  (9)

The operator morphisms f_{r} over f_{1} are defined completely by these conditions.

The group of operator morphisms g : (C_{∞})^{n} → A_{n} over f_{1} may be identified with A_{n} under g ↦ g(w_{0}). Under this identification, the boundaries δ_{4}, δ_{3} are transformed respectively to 0 and to a_{i} ↦ a_{i}(a_{i}^{t})^{−1}. So the 3-dimensional cohomology group is the group A_{n} with a_{i} identified with a_{i+1}, i = 0, …, n−1. This cohomology group is therefore isomorphic to C_{n}, and a generator is the class of the above cocycle f_{3}.

5.5. Corollary. The mapping cone X = BC_{n^{2}} ∪_{Bι} ΓBC_{n} satisfies π_{1}X = C_{n}, and π_{2}X is the C_{n}-module A_{n} of Theorem 5.4. The first Postnikov invariant of X is a generator of the cohomology group H^{3}(π_{1}X, π_{2}X), which is a cyclic group of order n.

The following is another example of a determination of a non-trivial cohomology class by a crossed module. The method of proof is similar to that of Theorem 5.4, and is left to the reader.

5.6. Example. Let n be even. Let C′_{n} denote the C_{n}-module which is C_{n} as an abelian group but in which the generator t of the group C_{n} acts on the generator t′ of C′_{n} by sending it to its inverse. Then H^{3}(C_{n}, C′_{n}) ≅ C_{2} and a generator of this group is represented by the crossed module (ν_{n} : C_{n} × C_{n} → C_{n^{2}}), with generators t_{0}, t_{1}, u say, and where ν_{n}t_{0} = ν_{n}t_{1} = u^{n}. Here u ∈ C_{n^{2}} operates by switching t_{0}, t_{1}. It is not clear if this crossed module can be an induced crossed module for n > 2. However, n = 2 gives the case n = 2 of Theorem 5.4.

5.7. Remark. The crossed module (ν_{2} : C_{2}×C_{2} → C_{4}) also appears as an example in [23, pp.332-333]. The proof given there that its corresponding cohomology class is non-trivial is obtained by relating this class to the obstruction to a certain kind of extension.

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School of Mathematics University of Wales, Bangor Gwynedd LL57 1UT, U.K.

Email: r.brown@bangor.ac.uk and mas023@clvax.bangor.ac.uk

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