THE MILGRAM-STEENROD CONSTRUCTION OF CLASSIFYING

SPACES FOR TOPOLOGICAL GROUPS^{†}

Renzo A. Piccinini and Mauro Spreafico^{∗}

Introduction

A classifying space for a topological group G is the base space of a principal G bundle (E_{G}, p_{G}, B_{G}) with (weakly) contractible total space E_{G}. Classifying spaces have been constructed using several different methods; amongst the most popular constructions we recall the Milnor construction, the Dold-Lashof-Fuchs construction (see [7] for an up-dated version) and the Milgram-Steenrod contruction. The latter was introduced in [8] and then, taken up and reformulated by N. Steenrod (see [12]). The objective of this article is to review the contents of Steenrod’s paper in the light of more streamlined techniques and, as a consequence, obtain new results not contained in [12] about the classifying space functor B which transforms a topological group G into a space B_{G}. In fact, we shall prove that B is exact and preserves products, normality, closed inclusions, proclusions and closed cofibrations; moreover, we shall see that if Z is a central subgroup of a topological group G such that the pair (G, Z) is a Z-equivariant closed cofibration, the map B_{q} : B_{G} → B_{G/Z} induced by the quotient map q : G → G/Z gives rise to a locally trivial principal B_{Z}-bundle.

The paper is largely self-contained; however, we do not hesitate in giving several references (particularly to [12]) in order to make it more easily readable.

In his version of Milgram’s construction, Steenrod selected as his basic category the category CG of compactly generated topological spaces defined in [11]. In this paper we choose to work in the more convenient (and larger) category wHk(T op) of weak Hausdorff k-spaces. For the reader’s convenience we recall here the definition of the category wHk(T op). We begin with a couple of definitions. Let T op be the category of topological spaces and maps (continuous functions) and let X ∈ T op be given; a subset A ⊂ X is compactly closed if, for every compact Hausdorff space K and every map f : K X, f^{−1}(A) ⊂ K is closed in K; the space X is said to be a k-space if all of its compactly closed subsets are closed. Next, we define a functor k : T op T op by associating to each X ∈ T op the space k(X) with the same underlying set as X but with the topology given by taking as closed sets the compactly closed sets with respect to the topology of X. As for the morphisms, notice that if Y is a k-space, a function f : Y X is continuous if and only if f : Y k(X) is continuous; thus, for any morphism f ∈ T op(Y, X), we simply take k(f) = f : k(Y) k(X). The functor k is called k-ification and its image k(T op) is a full subcategory of T op. The category k(T op) is both complete and cocomplete, with the product of k-spaces given by the rule

X × Y = k(X ×_{c} Y)

where X ×_{c} Y is the usual cartesian product in T op. A k-space X is said to be weak Hausdorff if, and only if, the diagonal Δ_{X} : X X × X is closed in X × X. The weak Hausdorff k-spaces generate a full subcategory wHk(T op) of k(T op); the objects of wHk(T op) are called weak Hausdorff k-spaces. The category wHk(T op) is closed under the formation of subspaces, finite products, quotients by closed subspaces, adjunction spaces and function spaces (see [4, Appendix]).

†1991 Mathematics Subject Classification: Primary: 55R35; Secondary: 54B17

∗Work partially supported by the European Union HCMR Grant CT940560 .

Besides the categories T op, k(T op) and wHk(T op), we shall also deal with other categories like E of sets and functions, T opGr of topological groups and continuous homomorphisms, and the corresponding based categories (these will be distinguished from the non-based ones by a star subscript). In particular, the base point of an object G ∈ T opGr_{∗} shall always be taken as its identity element. We shall indicate that X is an object of a category C simply by writing X ∈ C; moreover, C(X, Y) denotes the set of all morphisms of C from X to Y.

We wish to thank Wilson Sutherland for many useful comments.

1 Preliminary notions

We begin by recalling that a space X ∈ T op has the initial topology induced by a map i : X Y if, for every Z ∈ T op, f ∈ T op(Z, X) ⟺ if ∈ T op(Z, Y). Dually, a space X has the final topology induced by a map p : Y X if, for every space Z ∈ T op, g ∈ T op(X, Z) ⟺ gp ∈ T op(Y, Z). If X has the initial topology induced by i : X Y in T op and i is injective, we say that i is an inclusion; then U ⊂ X is open if, and only if, i(U) ⊂ i(X) is open.^{1} If X has the final topology induced by p : Y X and p is surjective, we say that p is a proclusion; then U ⊂ X is open if, and only if, p^{−1}(U) ⊂ Y is open. It is very easy to show that the composition of two inclusions (resp. proclusions) is an inclusion (resp. proclusion). Furthermore, a finite product of inclusions in T op is an inclusion (see [1, 1.4.1, Corollary to Proposition 3]); the corresponding property for proclusions fails to be true in T op but not in wHk(T op):

Lemma 1.1 Suppose that f : X  Y and f′ : X′  Y′ are proclusions

in wHk(T op). Then f ×f′ : X ×X′ Y ×Y′ is a proclusion in wHk(T op).

Proof – Since the functor X × − preserves colimits (see [4, Appendix]) the space X × Y′ has the final topology with respect to 1_{X} × f′; a similar argument applied to − × Y′ shows that X′ × Y′ has the final topology with respect to f ×1_{Y′} and therefore, X′ ×Y′ has the final topology with respect to f ×f′. □

Lemma 1.2 The k-ification functor k : T op T op preserves inclusions.

Proof – Let i ∈ T op(X, Y) be an inclusion. Clearly, ki is injective. Now we must show that, given any Z ∈ k(T op), a function g : Z k(X) is continuous ⟺ kig is continuous. In fact, kig is continuous ⟺ ig : Z Y is continuous ⟺ g : Z X is continuous ⟺ g : Z k(X) is continuous. □

We now recall that if R is an equivalence relation on X ∈ k(T op), because X/R has the final topology induced by the quotient map p : X X/R, then X/R ∈ k(T op) (see [4, page 242]). Let R and R′ be two given equivalence relations on X and X′ respectively. A map f ∈ k(T op)(X, X′) is said to be relation preserving if x_{1}Rx_{2} implies f(x_{1})R′f(x_{2}) for each x_{1}, x_{2} in X, and relation bipreserving if x_{1}Rx_{2} ⟺ f(x_{1})R′f(x_{2}) for each x_{1}, x_{2} in X. A relation preserving map f ∈ k(T op)(X, X′) defines a unique map f̂ ∈ k(T op)(X/R, X′/R′) such that p′f = f̂p. Notice that if f is relation bipreserving then f̂ is injective; moreover, if f is an injective relation preserving map and f̂ is injective, then f is actually relation bipreserving. Finally, if f is a proclusion, so is f̂. The last statement holds true in wHk(T op) provided X/R and X′/R′ are weak-Hausdorff (for example, if R and R′ are the relations induced by closed subsets A ⊂ X and A′ ⊂ X′).

1The following observation will be useful later on: an injection of T op is closed ⟺ it is a closed inclusion.

We say that M ∈ k(T op) is a topological monoid with multiplication τ ∈ k(T op)(M × M, M) if τ is associative with identity element u_{M} ∈ M. A topological monoid M acts on X ∈ k(T op) if there exists an action^{2} φ ∈ k(T op)(X × M, X) such that φ(φ × 1_{M}) = φ(1_{X} × τ) and, for all x ∈ X, φ(x, u_{M}) = x. With the previous conditions, X is said to be an M-space. An equivalence relation R on X is said to be consistent with the action φ if, for every m ∈ M and x, x′ ∈ X, xRx′ implies that (xm)R(x′m).

Now suppose that the topological monoid M is actually a topological group G. The action of G on a G-space X ∈ k(T op) determines an equivalence relation R on X : (xRx′ ⟺ (∃g ∈ G) x′ = xg) which is consistent with the action in the sense that, for every g ∈ G, xRx′ implies (xg)R(x′g). The k-space X/R, also denoted by X/G, is said to be an orbit space.

Let ϑ : G G′ be a continuous homomorphism between the topological groups G and G′. Take a G-space X and a G′-space X′; a map f : X X′ is ϑ-equivariant if, for every (x, g) ∈ X × G, f(xg) = f(x)ϑ(g). Consider the equivalence relation S in X × G given by

(x_{1}, g_{1})S(x_{2}, g_{2}) ⟺ x_{1}g_{1} = x_{2}g_{2} ⟺ (∃g ∈ G) x_{2} = x_{1}g and g_{2} = g^{−1}g_{1}

(similarly, for X′ × G′).

The proof of the next result is straightforward.

Lemma 1.3 Under the conditions set before, the following statements hold true:

i) if f is ϑ-equivariant, both f and f × ϑ are relation preserving; ii) if f is ϑ-equivariant and injective, then f × ϑ is relation bipreserving.

Suppose that we are given a closed subspace A ⊂ X ∈ wHk(T op) and an action ϕ : A × G A of a weak Hausdorff k-group G (that is, an object of T opGr_{∗} ∩ wHk(T op_{∗})) . Then A × G , X × G ∈ wHk(T op) and the pushout space \overline{X} of the diagram

[figure: X × G; i × 1_{G}; A × G; \overline{ϕ}; A]

is also a weak Hausdorff k-space (see [4, Corollary A.1.4]). Going quickly over the construction of the pushout, we recall that one produces a commutative diagram

2To simplify the notation, we shall normally write xm for φ(x, m).

A × G \overline{ϕ} \overline{}A

[figure: i × 1_{G}; \overline{ı × 1_{G}}]

X × G \overline{ϕ̄} \overline{}\overline{X}

in which the maps ϕ̄ and \overline{ı × 1_{G}} are the compositions of the appropriate inclusion maps with a proclusion p : A ⊔ (X × G) \overline{X}. Notice that because ϕ : A × G A is an epimorphism, the elements of \overline{X} can be represented by classes [x, g] given by the equivalence relation on X × G obtained by extending ϕ:

(x, g)R(x′, g′) ⇔ \{ \begin{matrix}(x, g) = (x′, g′) & if (x, g), (x′, g′) ∈ (X \ A) × G \\ xg = x′g′ & if (x, g), (x′, g′) ∈ A × G ; \end{matrix}

then \overline{X} is the quotient space (X × G)/R and ϕ̄ is the quotient map.

Proposition 1.4 The space \overline{X} is a G-space; the action Φ : \overline{X} × G \overline{X} extends ϕ.

Proof – The functor − × G preserves colimits (see [4, Appendix A.1]) and therefore, \overline{X} × G is the pushout of the diagram

[figure: (X × G) × G; (i × 1_{G}) × 1_{G}; (A × G) × G; \overline{ϕ × 1_{G}}; A × G]

Let τ_{G} : G × G G be the multiplication of G. The maps

ϕ̄(1_{X} × τ_{G}) : (X × G) × G \overline{X}
(\overline{i × 1_{G}})ϕ : A × G \overline{X}

are such that

(\overline{i × 1_{G}})ϕ(ϕ × 1_{G}) = ϕ̄(1_{X} × τ_{G})(i × 1_{G} × 1_{G})

and therefore, by the universal property of pushouts, there exists a unique map

Φ : \overline{X} × G \overline{X}

rendering commutative the resulting diagram. It is a straightforward matter to prove that Φ is an action defined by Φ([x, g], g′) = [x, gg′], for every [x, g] ∈ \overline{X} and every g′ ∈ G. The action Φ gives rise to an equivalence relation \overline{R} on \overline{X}. □

The adjunction space \overline{X} is the enlargement of the G-action of A (to X). Remark - To carry on this construction within the category CG of compactly generated topological spaces we must require that both pairs (X, A) and (G, {u_{G}}) are closed cofibrations; this is another instance where there is a distinct advantage to work in wHk(T op).

Proposition 1.5 The space X is homeomorphic to a closed subset of \overline{X}.

Proof – The restriction to X × {u_{G}} of the map

ϕ̄: X × G ^{i_{2}} A ⊔ (X × G) ^{p} \overline{X}

is induced by the homeomorphism ϕ(−, u_{G}) : A A; hence it is a homeomorphism onto the closed set ϕ̄(X × \{u_{G}\}) of \overline{X}. □

Our objective now is to clarify the question whever the inclusion (\overline{X}, X) is a closed cofibration; to do this we shall give an auxiliary result which relates relative homeomorphisms and adjunction spaces. We first recall that a map of pairs (ϕ′, ϕ) : (X, A) (Y, B) in wHk(T op) is a relative homeomorphism if A is closed in X, the map ϕ′ is proclusive and induces a homeomorphism (X \ A) (Y \ B).

Lemma 1.6 A map of pairs (ϕ′, ϕ) : (X, A) (Y, B) is a relative homeomorphism ⟺ Y ≅ B ⊔_{ϕ} X.

Proof – Let (ϕ′, ϕ) : (X, A) (Y, B) be a relative homeomorphism and let Z be the adjunction space of X to B via ϕ. Consider the following commutative diagram where the square is a pushout:

A  \overline{ϕ} \overline{}B

[figure]

X  \overline{ϕ̄} \overline{}Z

[figure: ϕ′] [figure: f]

Y

By the universal property of pushouts and the definition of relative homeomorphism, the sets Z ≅ Y; thus, the map f is bijective. Since ϕ̄ = f^{−1}ϕ′ and ϕ′ is a proclusion, f^{−1} is continuous and hence, Z ≅ Y as topological spaces.

The converse is given by [4, Proposition A.4.8 (iii)] . □

Proposition 1.7 Suppose that (X, A) and (G, {u_{G}}) are closed cofibrations in wHk(T op); then (\overline{X}, X) is a closed cofibration in wHk(T op).

Proof – The following pairs are closed cofibrations: (X ×G, A×G), (X ×G, X × \{u_{G}\} ∪ A × G) and (\overline{X}, A) (see [4, Proposition A.4.2 (iii), (iv) and Proposition A.4.8 (ii)] , respectively). Now apply Lemma 1.6 to the map of pairs

(X × G, X × \{u_{G}\} ∪ A × G) (\overline{X}, X)

to end the proof. □

We conclude our observations about the enlargement of actions noticing that the universal property of pushouts proves easily that we can interpret the enlargement of an action as a functor:

Proposition 1.8 For every ϑ ∈ (T opGr ∩ wHk(T op))(G, H) and every map of pairs (f, u) : (X, A) (Y, B) in wHk(T op) such that A is a G-space, B is an H-space and u is ϑ-equivariant, then f × ϑ : X × G Y × H is ϑ-equivariant with respect to the trivial actions 1_{X} × τ_{G}, 1_{Y} × τ_{H} and induces a ϑ-equivariant quotient mapping \overline{f} : \overline{X} \overline{Y}.

We note explicitly that

(∀[x, g] ∈ \overline{X}) , \overline{f}([x, g]) = [f(x), ϑ(g)] .

Now we start to talk about contractions. Regard the unit interval I = [0, 1] as a topological monoid under ordinary multiplication and define an I-action as a map h′ : X ×I X such that h′(−, 1) = 1_{X} and h′(x, tt′) = h′(h′(x, t), t′), for every x ∈ X and t, t′ ∈ I. Let X ∈ wHk(T op) be based, with base point x_{0}; a contraction of X to x_{0} is an I-action h′ : X × I X that factors through

the smash product h′ : X ×I ^{p} X ∧I ^{h} X (we take 0 ∈ I as base point). In particular, the ordinary multiplication on I is a contraction of I to 0. Denote the base point of X ∧ I by \overline{x_{0}} ; then the trivial action

τ : (X × I) × I X × I , ((x, t), s) (x, ts)

induces a contraction of X ∧ I to \overline{x_{0}}, called canonical contraction. In fact, τ factors through X ∨ I and gives rise to a map

c′ : (X ∧ I) × I X ∧ I , (x ∧ t, s) x ∧ ts

which is the desired contraction.

Suppose that we are given x_{0} ∈ A ⊂ X (with A closed, X ∈ wHk(T op)) and a contraction h′ = hp : A × I A of A to x_{0}. The adjunction space X̃ := A ⊔_{h} (X ∧ I) ∈ wHk(T op) is the enlargement to X of the contraction^{3} h′ : A × I A. Let x̃_{0} be the base point of X̃.

Arguments similar to those used for the proofs of propositions 1.4, 1.5, 1.7 and 1.8 show the following results:

Proposition 1.9 The space X̃ has a contraction h̃′ : X̃ × I  X̃ that ex-

tends the contraction h′ : A × I A.

Proposition 1.10 The space X is homeomorphic to a closed subset of X̃.

Proposition 1.11 If the pairs (X, A), (X, {x_{0}}) and (A, {x_{0}}) of wHk(T op) are closed cofibrations, then (X̃, A), (X̃, x_{0}) and (X̃, X) are closed cofibrations in wHk(T op).

Proposition 1.12 Let (f, u) : (X, A) (Y, B) be a based map of pairs of weak Hausdorff based k-spaces and let h′ : A × I A, k′ : B × I B be contractions such that k(u ∧ 1_{I}) = uh. Then there exists a unique based map f̃: X̃ Ỹ which commutes with the extensions of the contractions h′ and k′ .

The last proposition shows that we can view the process of enlargement of contractions functorially.

3If we stay in the category CG, we must require that (X, A), (X, {x_{0}}) and (A, {x_{0}}) are n.d.r.’s in order to guarantee that _{X} ∈ CG (see [12, Lemma 3.3]).

2 Preservation of inclusions and proclusions

In this section we prove that the functorial processes of enlargement of actions and contractions preserve closed inclusions and proclusions.

Proposition 2.1 Let X, Y ∈ wHk(T op) with closed subsets A ⊂ X and B ⊂ Y; the corresponding inclusions are indicated by

i : A  X , j : B  Y .

Let G, H be weak Hausdorff k-groups acting on A and B respectively, with actions

ϕ : X × G  X , ψ : Y × H  Y .

Take ϑ ∈ T opGr ∩ wHk(T op_{∗})(G, H) and a map of pairs (f, u) : (X, A) (Y, B) with u : A B ϑ-equivariant; finally, suppose that ϑ and f are closed inclusions and f(X) ∩ B = u(A). Then the induced map \overline{f} : \overline{X} \overline{Y} is an inclusion, and \overline{f}(\overline{X}) ∩ \overline{r}(Y) = \overline{f}\overline{s}(X), where \overline{s} : X \overline{X} and \overline{r} : Y \overline{Y} are the inclusions obtained in the construction of \overline{X} and \overline{Y} (see Proposition 1.5).

Proof – Since u is ϑ-equivariant and injective, u×ϑ is relation bipreserving (see Lemma 1.3); then f × ϑ is relation bipreserving (with respect to the extended equivalence relations) and consequently, \overline{f} is injective.

It remains to prove that \overline{f} is closed. We begin by observing that in view of [1, 1.10.1, Propositions 2 and 4], both f × ϑ and u × ϑ are closed maps. Now take a closed subset C ⊂ \overline{X}; set K = ϕ̄^{−1}(C) and take (f × ϑ)(K) ⊂ Y × H. Because of [3, VI.6.2]

\overline{f}(C) = ψ̄(f × ϑ)(K) closed ⟺ ψ((f × ϑ)(K) ∩ (B × H)) closed .

Hence, we must prove that ψ((f × ϑ)(K) ∩ (B × H)) is closed in B. Notice that because ϕ̄(K) = C is closed, [3, VI.6.2] can be used again to prove that ϕ(K∩(A×G)) is closed. The injectivity of u, f, ϑ and the assumption f(X)∩B = u(A) imply that

(f × ϑ)(K) ∩ (B × H) = (u × ϑ)(K ∩ (A × G))

and thus,

ψ((f × ϑ)(K) ∩ (B × H)) = uϕ(K ∩ (A × G))

is closed, because u is a closed map.

The diagram below portrays the relationship between the various spaces of the proposition:

A × G  \overline{ϕ}  \overline{}A

[figure: u × ϑ; u]

B × H \overline{ψ} \overline{}B

[figure: i × 1_{G}; j × 1_{H}]

Y × H \overline{ψ̄} \overline{}\overline{Y}

[figure: f × ϑ; \overline{f}]

X × G  \overline{ϕ̄}  \overline{}\overline{X}

□

Corollary 2.2 In the situation of the proposition, if u is relation bipreserving with respect to the equivalence relations defined in A and B by the actions of G and H respectively, then \overline{f} and \overline{f} × ϑ are relation bipreserving.

Proof – The first statement is proved by direct computation; as for the second, notice that \overline{f} is ϑ-equivariant and injective and then use Lemma 1.3. □

Before we prove a result similar to Proposition 2.1 for the enlargement of a contraction, we give an auxiliary lemma.

Lemma 2.3 If f ∈ wHk(T op_{∗})((X, x_{0}), (Y, y_{0})) is a closed inclusion, then f ∧ 1 : X ∧ I Y ∧ I is a closed inclusion.

Proof – Construct the commutative diagram

X ∨ I  \overline{c_{∗}}  \overline{}(x_{0}, 0)

[figure: f ∨ 1_{I}]

Y ∨ I \overline{c′_{∗}} \overline{}(y_{0}, 0)

[figure: i; j; j̄; ī]

Y × I \overline{\overline{c′_{∗}}} \overline{}Y ∧ I

[figure: f × 1_{I}; f ∧ 1_{I}]

X × I  \overline{\overline{c_{∗}}}  \overline{}X ∧ I

and observe that X ∧ I and Y ∧ I are quotient spaces of X × I and Y × I by the equivalence relations R and S respectively, defined by

(x, t)R(x′, t′) ⇔ x = x′ = x_{0} or t = t′ = 0,
(y, t)S(y′, t′) ⇔ y = y′ = y_{0} or t = t′ = 0 .

Since f is injective, f × 1_{I} is relation bipreserving and thus, f ∧ 1_{I} is injective.

Let C ⊂ X × I be closed; because f × 1_{I} is a closed map, (f × 1_{I})(p^{−1}(C)) is closed in X × I, implying that (f ∧ 1_{I})(C) is closed in Y ∧ I. □

Proposition 2.4 Let (f, u) : (X, A, x_{0}) (Y, B, y_{0}) be a based map of pairs in wHk(T op_{∗}) such that A is a closed subset of X with a contraction h′ to x_{0}, B is a closed subset of Y with a contraction k′ to y_{0}, and u is an I-mapping^{4}. Suppose that f is a closed inclusion and that f(X) ∩ (B) = u(A). Then the quotient map f̃: X̃ Ỹ is a closed inclusion, and f̃(X̃) ∩ r̃(Y) = f̃s̃(X), where s̃ and r̃ are the inclusions (X̃, X) and (Ỹ, Y) respectively.

4i.e., for every (x, t) ∈ A × I, u(xt) = u(x)t

Proof – Take the commutative diagram

A ∧ I  \overline{h}  \overline{}A

[figure: u ∧ 1_{I}; u]

B ∧ I \overline{k} \overline{}B

[figure: i ∧ 1_{I}; j ∧ 1_{I}; \overline{j ∧ 1_{I}}; \overline{i ∧ 1_{I}}]

Y ∧ I \overline{k̄} \overline{}Ỹ

[figure: f ∧ 1_{I}; f̃]

X ∧ I  \overline{h̄}  \overline{}X̃

recall that f ∧ 1_{I} is a closed inclusion and proceed as in Proposition 2.1. □

Now we consider the preservation of proclusions.

Proposition 2.5 Let ϑ : G H be in T opGr ∩ wHk(T op); let (f, u) : (X, A) (Y, B) be a mapping of pairs in wHk(T op) such that A is a closed subset of X and a G-space with action ϕ, B is a closed subset of Y and an H-space with action ψ, and u is ϑ-equivariant. Suppose that ϑ and f are proclusions. Then the quotient map \overline{f} : \overline{X} \overline{Y} is a proclusion.

Proof – Just recall that a relation preserving proclusion f ∈ k(T op)(X, X′) defines a unique proclusion f̂ ∈ k(T op)(X/R, X′/R′). □

Proposition 2.6 Let (f, u) : (X, A, x_{0}) (Y, B, y_{0}) be a based map of pairs in wHk(T op_{∗}) such that A is a closed subset of X with a contraction h′ to x_{0}, B is a closed subset of Y with a contraction k′ to y_{0}, and u is an I-mapping. Suppose that f is a proclusion. Then the quotient map f̃ : X̃ Ỹ is a proclusion.

3 The construction of the classifying space

Let G be a weak Hausdorff k-group. We are going to construct two expanding sequences in wHk(T op_{∗}) with homeomorphic union spaces.^{5}

We begin by taking

E_{−1} := ∅ , D_{0} := {u_{G}}

5For the definition and properties of expanding sequences and their union spaces, please consult [4, A.5]. Note that in the present paper although in either expanding sequence each space is a closed subspace of the next, these inclusions are not necessarily cofibrations (they will be such whenever the inclusion of {u_{G}} in G is a closed cofibration); nevertheless, the results of [4] are still valid.

and the maps

ϕ_{−1} : ∅ × G  ∅ , h′_{0} : {u_{G}} × I  {u_{G}} .

Notice that we can view ϕ_{−1} as a group action and h′_{0} as a contraction; then construct

E_{0} := E_{−1} ⊔_{ϕ_{−1}} (D_{0} × G) and D_{1} := D_{0} ⊔_{h_{0}} (E_{0} ∧ I) .

The weak Hausdorff k-spaces E_{0}, D_{0} and D_{1} are based, with base point u_{G}. According to Proposition 1.4 there exists a G-action ϕ_{0} : E_{0} × G E_{0}. Moreover, in view of Proposition 1.10, E_{0} is homeomorphic to a closed subset of D_{1}; from this we construct E_{1} as the enlargement of the action ϕ_{0} namely, E_{1} := E_{0} ⊔_{ϕ_{0}} (D_{1} ×G). On the other hand, there is a contraction h′_{1} : D_{1} ×I D_{1} (see Proposition 1.9) and we know that D_{1} is homeomorphic to a closed subset of E_{1} (see Proposition 1.5); then we construct the enlargement of the contraction D_{2} := D_{1} ⊔_{h_{1}} (E_{1} ∧ I) . Up to now we have a finite string of inclusions

E_{−1} ⊂ D_{0} ⊂ E_{0} ⊂ D_{1} ⊂ E_{1} ⊂ D_{2} ;

we construct an infinite string

(ED) E_{−1} ⊂ D_{0} ⊂ E_{0} ⊂ D_{1} ⊂ E_{1} ⊂ … ⊂ D_{n} ⊂ E_{n} ⊂ D_{n+1} ⊂ …

by induction :

E_{n−1} := E_{n−2} ⊔_{ϕ_{n−2}} (D_{n−1} × G) ,
D_{n} := D_{n−1} ⊔_{h_{n−1}} (E_{n−1} ∧ I) .

Because of Propositions 1.4 and 1.9 we can also construct the following actions and contractions:

ϕ_{n−2} : E_{n−2} × G E_{n−2} ,
h_{n−1} := D_{n−1} ∧ I D_{n−1}

The next diagram is useful to understand the construction of E_{n−1} and D_{n}:

[figure: D_{n−1} × I; \overline{p_{n−1}}; D_{n−1} ∧ I; \overline{h_{n−1}}; D_{n−1}]

[figure]

[figure: (D_{n−1} × G) × I; E_{n−1} × I; E_{n−1} ∧ I; D_{n}] [figure]

(E_{n−2} × G) × I \overline{ϕ_{n−2} × 1_{I}} \overline{}E_{n−2} × I

The reader should observe that if the pair (G, {u_{G}}) is a closed cofibration in wHk(T op_{∗}), then all the inclusions in the infinite sequence of spaces (ED) above are closed cofibrations: in fact, (E_{0}, D_{0}) is a closed cofibration under the assumption on (G, {u_{G}}); for the other inclusions we proceed by induction and use Propositions 1.7 and 1.11.

We obtained two expanding sequences of spaces in wHk(T op_{∗}),

E_{−1} ⊂ E_{0} ⊂ E_{1} ⊂ … ⊂ E_{n} ⊂ … and
D_{0} ⊂ D_{1} ⊂ D_{2} ⊂ … ⊂ D_{n} ⊂ …,

whose union spaces are homeomorphic:

E_{G} := ⋃_{n=−1}^{∞} E_{n} = ⋃_{n=0}^{∞} D_{n}

(by definition, the topology of E_{G} is determined by either of the families {E_{n}|n ≥ −1} or {D_{n}|n ≥ 0}). Note that E_{G} ∈ wHk(T op) by [4, Proposition A.5.2]. Furthermore, for every n ≥ 0, the inclusion i_{n} : D_{n−1} D_{n} is nulhomotopic (take the homotopy i_{n}h_{n−1}p_{n−1} : D_{n−1} × I D_{n}); hence E_{G} ≅ ⋃_{n=0}^{∞} D_{n} is weakly contractible.

The union space of the expanding sequence {E_{n} × G|n ≥ −1} is homeomorphic to E_{G} × G (see [4, Proposition A.5.1]); this fact determines the existence of an action

ϕ := ⋃_{n=0}^{∞} ϕ_{n} : E_{G} × G E_{G} .

By the same token ⋃_{n=−1}^{∞}(E_{n} × I) = E_{G} × I. Moreover, because I is compact Hausdorff, there is a bijection between the sets of based maps

wHk(T op_{∗})(X ∧ I, Y) ⇀ ↽ wHk(T op_{∗})(X, Map_{∗}(I, Y))

that is to say, −∧I is a left adjoint functor and thus, preserves colimits; therefore ⋃_{n=−1}^{∞}(E_{n}∧I) ≅ E_{G}∧I. Under these circumstances, we can define a contraction

h := ⋃_{n=0}^{∞} h_{n} : E_{G} ∧ I E_{G}

and thus E_{G} is actually a contractible space.

Now we construct the classifying space B_{G}. For every n ≥ −1, let B_{n} be the orbit space determined by the action ϕ_{n} : E_{n} × G E_{n} and let p_{n} : E_{n} B_{n} be the identification map. We define the classifying space B_{G} of G to be the union space of the expanding sequence {B_{n}|n ≥ −1}. Notice that B_{G} is homeomorphic to the orbit space determined by the action ϕ of G on E_{G}; by an abuse of notation we denote the identification map p_{G} : E_{G} B_{G} by p_{G} = ⋃_{n=−1}^{∞} p_{n}.

The process just described, which gives rise to the based weak Hausdorff k-spaces E_{G} and B_{G} associated to a weak Hausdorff k-group G, is known as the Milgram-Steenrod construction. The next results show that the Milgram-Steenrod construction is functorial.

Lemma 3.1 Let ϑ : G H be a homomorphism between two weak Hausdorff k-groups. Then ϑ determines an equivariant map E_{ϑ} : E_{G} E_{H}.

Proof – The proof is done by induction and using Proposition 1.8. By the Milgram-Steenrod construction E_{G} is the union space of the expanding sequence {E_{n}|n ≥ −1} with the action ϕ := ⋃_{n=0}^{∞} ϕ_{n}. Likewise, E_{H} is the union space of the expanding sequence {Ê_{n}|n ≥ −1} with action ϕ̂ := ⋃_{n=0}^{∞} ϕ̂_{n}. The first steps of the induction are trivial. Assume that we have defined the equivariant

map E_{ϑ,n−1} : E_{n−1}  Ê_{n−1}; the next step in the construction is depicted

by the following commutative diagram with pushout squares:

E_{n−1} × G  \overline{ϕ_{n−1}}  \overline{}E_{n−1}

[figure: E_{ϑ,n−1} × ϑ; E_{ϑ,n−1}]

Ê_{n−1} × H \overline{ϕ̂_{n−1}} \overline{}Ê_{n−1}

[figure]

D̂_{n} × H  \overline{}Ê_{n}

[figure: D_{ϑ,n−1} × ϑ; E_{ϑ,n}]

D_{n} × G  \overline{}E_{n}
□

Corollary 3.2 Let ϑ : G H be a homomorphism between two weak Hausdorff k-groups. Then ϑ determines a map B_{ϑ} : B_{G} B_{H}.

Proof – The existence of B_{ϑ} is clear. We only wish to notice, for the sake of completeness, that B_{ϑ} = ⋃_{n=0}^{∞} B_{ϑ,n}, where the maps B_{ϑ,n} : B_{n} B̂_{n} are induced by the corresponding maps E_{ϑ,n}. □

Let us put together the lemma and its corollary. Denote by EG the category of weak Hausdorff k-spaces together with the action of a weak Hausdorff k-group and equivariant k-maps.

Theorem 3.3 The Milgram-Steenrod construction defines two functors

E : T opGr_{∗} ∩ wHk(T op_{∗}) EG , G E_{G}
B : T opGr_{∗} ∩ wHk(T op_{∗}) wHk(T op_{∗}) , G B_{G} .

With the aid of the mapping track we can replace p_{G} by a fibration whose fibre is weakly homotopic to ΩB_{G}; however, under a mild assumption on the pair (G, {u_{G}}) we can obtain much more (see [12, Theorem 8.3] and [6, Theorem 4.2]):

Theorem 3.4 Let G be a weak Hausdorff k-group with identity element u_{G} such that (G, {u_{G}}) is a closed cofibration in wHk(T op_{∗}). Then (E_{G}, p_{G}, B_{G}) is a numerable principal G-bundle with contractible total space E_{G}.^{6}

6Thus (E_{G}, p_{G}, B_{G}) is a universal bundle for numerable principal G-bundles.

4 Inclusions, proclusions and cofibrations

Theorem 4.1 The functors E and B preserve closed inclusions.

Proof – Let ϑ : G H ∈ T opGr ∩ wHk(T op) be an inclusion. According to the notation of Section 3 we write

E_{−1} := ∅ , D_{0} := {u_{G}} ,

ϕ_{−1} : ∅ × G  ∅ , h′_{0} : {u_{G}} × I  {u_{G}} ,

Ê_{−1} := ∅ , D̂_{0} := {u_{H}} ,

ϕ̂_{−1} : ∅ × H  ∅ , ĥ′_{0} : {u_{H}} × I  {u_{H}} ,

E_{G} := ⋃_{n=−1}^{∞} E_{n} = ⋃_{n=0}^{∞} D_{n} ,

E_{H} := ⋃_{n=−1}^{∞} Ê_{n} = ⋃_{n=0}^{∞} D̂_{n} ,

E_{ϑ,n} : E_{n}  Ê_{n} , D_{ϑ,n} : D_{n}  D̂_{n} ,

and

E_{ϑ} = ⋃_{n=−1}^{∞} E_{ϑ,n} = ⋃_{n=0}^{∞} D_{ϑ,n} : E_{G} E_{H} .

The hypotheses of Proposition 2.1 are satisfied for the map of pairs

(D_{ϑ,0}, E_{ϑ,−1}) : (D_{0}, E_{−1}) (D̂_{0}, Ê_{−1})

and those of Proposition 2.4 hold true for the map of pairs

(E_{ϑ,0}, D_{ϑ,0}) : (E_{0}, D_{0}) (Ê_{0}, D̂_{0}) ;

therefore, D_{ϑ,1} and E_{ϑ,0} are closed inclusions; an induction argument plus Propositions 2.1 and 2.4 prove that, for every n ≥ 0, both D_{ϑ,n} and E_{ϑ,n} are closed inclusions (this assertion is also true for E_{ϑ,−1}). But the topology of E_{G} coincides with the final topology of the family {ι_{n} : E_{n} E_{G}|n ≥ −1} (or of the family {ι′_{n} : D_{n} E_{G}|n ≥ 0}) as one can see from [4, Lemma A.2.4] (and similarly for E_{H}). It follows that the family of closed inclusions {E_{ϑ,n}|n ≥ −1} (or {D_{ϑ,n}|n ≥ 0}) induces a closed inclusion E_{ϑ} : E_{G} E_{H}. This shows that E preserves closed inclusions.

Now we study the map B_{ϑ} : B_{G} B_{H} induced by E_{ϑ}. We notice first that the map E_{ϑ,0} : E_{0} Ê_{0} is relation bipreserving in view of Corollary 2.2; by induction, all maps E_{ϑ,n} are relation bipreserving, for every n ≥ −1 and thus, E_{ϑ} is relation bipreserving. This shows that the induced map B_{ϑ} is injective.

It remains to prove that B_{ϑ} is closed. This is done by induction on the components of B_{ϑ}. Clearly, B_{ϑ,0} is closed; assume that B_{ϑ,n−1} is closed and take the commutative diagram at the end of the proof.

Let C ⊂ B_{n} be closed; then K := p̄_{G,n−1}^{−1}(C) is closed in D_{n} and, because D_{ϑ,n} is a closed map, D_{ϑ,n}(K) is closed in D̂_{n}. Using [3, VI.6.2] we conclude that

B_{ϑ,n}(C) ⊂ B̂_{n} closed ⟺ p_{H,n−1}(D_{ϑ,n}(K) ∩ Ê_{n−1}) ⊂ B̂_{n−1} closed .

Since D_{ϑ,n} is injective and D_{ϑ,n}(D_{n}) ∩ Ê_{n−1} = E_{ϑ,n−1}(E_{n−1}) (compare with Proposition 2.1), it follows that D_{ϑ,n}(K) ∩ Ê_{n−1} = E_{ϑ,n−1}(K ∩ E_{n−1}). Therefore, p_{H,n−1}(D_{ϑ,n}(K)∩Ê_{n−1}) = p_{H,n−1}E_{ϑ,n−1}(K ∩E_{n−1}) = B_{ϑ,n−1}p_{G,n−1}(K ∩ E_{n−1}), which is closed in B̂_{n−1} since B_{ϑ,n−1} is a closed map, and p_{G,n−1}(K ∩ E_{n−1}) is closed in B_{n−1}, again by [3, VI.6.2], because p̄_{G,n−1}(K) = C is closed in B_{n}.

E_{n−1}  \overline{E_{ϑ,n−1}}  \overline{}Ê_{n−1}

[figure: i_{n−1}; i′_{n−1}]

D_{n}  \overline{D_{ϑ,n}} \overline{}D̂_{n}

[figure: p_{G,n−1}; p̄_{G,n−1}; p̄_{H,n−1}; p_{H,n−1}]

B_{n}  \overline{B_{ϑ,n}} \overline{}B̂_{n}

[figure: ī_{n−1}; _{ī′_{n−1}}]

B_{n−1}  \overline{B_{ϑ,n−1}}  \overline{}B̂_{n−1}
□

Theorem 4.2 The functors E and B preserve proclusions.

Proof – Let ϑ : G H be a proclusion of two weak Hausdorff k-groups. According to the notation of Lemma 3.1

E_{G} = ⋃_{n=−1}^{∞} E_{n} = ⋃_{n=0}^{∞} D_{n} ,

E_{H} = ⋃_{n=−1}^{∞} Ê_{n} = ⋃_{n=0}^{∞} D̂_{n}

and

E_{ϑ} = ⋃_{n=−1}^{∞} E_{ϑ,n} : E_{G} E_{H} .

Induction and Proposition 2.6 show that the maps D_{ϑ,n} : D_{n}  D̂_{n} (see

notation of Lemma 3.1) are proclusions; this, coupled with Proposition 2.5 and induction, shows that the maps E_{ϑ,n} : E_{n} Ê_{n} are proclusions. The map E_{θ} is clearly surjective as a union function of surjective functions; it remains to prove that its image has the final topology. To this end, take a subset V of E_{H} and suppose that E_{h}^{−1}(V) is open in E_{G}; we wish to prove that V is open in E_{H}. For every integer n ≥ −1, E_{ϑ}^{−1}(V) ∩ E_{n} is open in E_{n}; but

E_{ϑ}^{−1}(V) ∩ E_{n} = ⋃_{i=−1}^{∞} E_{ϑ,i}^{−1}(V ∩ Ê_{n}) ∩ E_{n} = E_{ϑ,n}^{−1}(V ∩ Ê_{n})

and therefore, because E_{ϑ,n} is a proclusion, V ∩ Ê_{n} is open in Ê_{n}. This means that V is open in E_{H}.

To complete the proof, first notice that B_{ϑ}p_{G} = p_{H}E_{ϑ} and that p_{G}, p_{H} and E_{ϑ} are proclusions and then apply [1, 1.2.4, Proposition 7]. □

Theorem 4.3 The functors E and B preserve closed cofibrations between weak Hausdorff k-groups with non-degenerate unit.

Proof – Let G, H ∈ (T opGr∩wHk(T op) be such that (G, {u_{G}}) and (H, {u_{H}}) are closed cofibrations (i.e., u_{G} and u_{H} are non-degenerate). Suppose that the pair (G, H) is a closed cofibration; we wish to prove that the pairs (E_{G}, E_{H}) and (B_{G}, B_{H}) are closed cofibrations.

We first recall that under the non-degeneracy hypothesis of u_{G} all pairs in the infinite sequence of spaces (ED) defined during the construction of E_{G} are closed cofibrations (similarly for E_{H}). Then, we prove that, for every n ≥ −1, (Ê_{n}, E_{n}) is a closed cofibration and use the appropriate result analogous to [4, Proposition A.5.5]. The proof follows the same lines as Theorem 4.1.

Now for (B_{G}, B_{H}): we already know that all maps B_{ϑ,n} are closed injections; in order to prove that the pairs (B̂_{n}, B_{n}) are all closed cofibrations we proceed again by induction; this is a routine exercise (refer back to the diagram of Theorem 4.1). □

We conclude this section by observing that if G is a compact Hausdorff space, then E_{G} and B_{G} are paracompact and normal; again, this is done by induction and at the end, we use [4, Proposition A.5.1].

5 The group structure of E_{G}

We begin by observing that the results stated in sections 5, 6 and 7 of [12] are still valid if one works withing the category of weak Hausdorff k-spaces. We have already noted that E_{G} is a contractible space; moreover, as proved by N. Steenrod, E_{G} is a topological group [12, Theorem 7.6 (e)]. Our immediate objective is to review this group structure; to do this, it is necessary to conduct a refined analysis on the nature of E_{G}. First recall how to express the elements of E_{G} (see details in [12]). Let Δ_{n} be the n-simplex of IR^{n} defined by the inequalities 0 ≤ t_{1} ≤ … ≤ t_{n} ≤ 1 and let δ_{n} be its interior; imbed Δ_{n} in Δ_{n+1} by adding the (n + 1)^{th} coordinate t_{n+1} = 1. A point of G^{n} × Δ_{n} is represented by its coordinates in the shuffled form [g_{1}, t_{1}, …, g_{n}, t_{n}]. Imbed G^{n} × Δ_{n} in G^{n+1} × Δ_{n+1} by adding the coordinates g_{n+1} = u_{G} and t_{n+1} = 1. Theorem 5.1 of [12] shows that, for every n ≥ 0, there exists a proclusion k_{n} : G^{n} × Δ_{n} D_{n}, whose restriction to G^{n−1} × Δ_{n−1} is k_{n−1} and thus, the union map

k := ⋃_{n=0}^{∞} k_{n} : ⋃_{n=0}^{∞}(G^{n} × Δ_{n}) E_{G}

is a well defined proclusion. Thus, for every x ∈ E_{G} := ⋃_{n=0}^{∞} D_{n}, there exist a non negative integer n and an element u ∈ G^{n} × Δ_{n} such that x ∈ D_{n} and k_{n}(u) = x. Two elements u, v ∈ G^{n} × Δ_{n} are said to be equivalent if k_{n}(u) = k_{n}(v); the resulting quotient space can be viewed as the space

N_{n} = ⋃_{j=0}^{n}(G \ {u_{G}})^{j} × (δ_{j} ∪ δ_{j−1})

and the restriction of k_{n} to N_{n} is a bijection to D_{n} (see [12, Corollary 5.4]).

At this point consider the free abstract group Ẽ_{G} generated by all pairs (g, t) ∈ G × I: this is the set ⋃_{n=0}^{∞}(G × I)^{n} with multiplication μ′ : Ẽ_{G} × Ẽ_{G} Ẽ_{G} defined by juxtaposition of monomials (the unit element is the empty monomial ⊥ corresponding to n = 0). Let E′_{G} be the quotient group obtained from Ẽ_{G} using the following Fundamental Relations: for every g, g′ ∈ G and every t, t′ ∈ I,

1. (g, 0) = (u_{G}, t) = uẼ_{G} = ⊥ ,

2. (g, t)(g′, t) = (gg′, t) ,

3. if 0 < t′ < t ≤ 1, then (g, t)(g′, t′) = (g · g′ · g^{−1}, t′)(g, t) .

Notice that if G is abelian, then E′_{G} is also abelian.

A monomial (g_{1}, t_{1}) … (g_{k}, t_{k}) is said to be in normal form if it is the empty monomial or if 0 < t_{1} < … < t_{k} ≤ 1 and each g_{i} ∈ G \ {u_{G}}. Moreover, each monomial is equivalent to one and only one monomial in normal form; the equivalence is obtained by using the previous Fundamental Relations. Thus E′_{G} is isomorphic to the abstract subgroup of Ẽ_{G} determined by all the monomials in normal form (for example, u_{E′_{G}} corresponds to the empty monomial ⊥). Next, define f : E′_{G} E_{G} as the function which assigns to each element (g_{1}, t_{1}) … (g_{m}, t_{m}) in normal form the element k_{m}([g_{1}, t_{1}, …, g_{m}, t_{m}]) of E_{G} (if m = 0, f(u_{E′_{G}}) = f(⊥) = u_{G} ∈ E_{G}). According to [12, Theorem 7.6 (a)], f is a bijection. One should observe that, even if the two sets Ẽ_{G} and ⋃_{n=0}^{∞} G^{n} × Δ_{n} are not isomorphic, the sets E′_{G} and N_{∞} are indeed isomorphic, via the function

ϵ : E′_{G}  N_{∞} , (g_{1}, t_{1}) … (g_{n}, t_{n})  [g_{1}, t_{1}, …, g_{n}, t_{n}]

(in normal form); then f = kϵ. Finally, define a multiplication μ : E_{G} × E_{G} E_{G} as the composition

E_{G} × E_{G} ^{f^{−1}×f^{−1}} E′_{G} × E′_{G} ^{μ′} E′_{G} ^{f} E_{G}

(the continuity of μ is shown in [12, Theorem 7.6 (d)]).

Let ϑ : G H be a homomorphism between two weak Hausdorff kgroups. An analysis of the construction of E_{ϑ} shows that

E_{ϑ}((g_{1}, t_{1}) … (g_{n}, t_{n})) = (ϑ(g_{1}), t_{1}) … (ϑ(g_{n}), t_{n})

In particular, we obtain the following result (cf. Theorem 3.3):

Proposition 5.1 E is a covariant functor of the category of weak Hausdorff k-groups to itself.

Let E_{G}^{′0} be the set of all elements (g, 1) ∈ E′_{G}; with the identification (u_{G}, 1) ≡ ⊥, the elements of E_{G}^{′0} are all in normal form and f : E_{G}^{′0} E_{0} ⊂ E_{G} (see the definition of E_{0} in Section 3). On the other hand, we can identify E_{G}^{′0} with the topological group G by the function i_{G} : G E′_{G} which takes an arbitrary g ∈ G into (g, 1); hence, G can be viewed as a closed topological subgroup of E_{G}. Furthermore, the multiplication μ restricted to E_{G}×G ⊂ E_{G}×E_{G} coincides with the action ϕ : E_{G} × G E_{G} defined somewhat abstractly in Section 3; in other words, the following diagram commutes:

E_{G} × G \overline{ϕ} \overline{}E_{G}

f^{−1} × i_{G} f

E′_{G} × E′_{G} \overline{μ′} \overline{}E′_{G}

Therefore, ϕ acts on the elements of E_{G} × G as follows:

ϕ(((g_{1}, t_{1}) … (g_{k}, t_{k})), g) = \{ \begin{matrix}f((g_{1}, t_{1}) … (g_{k}g, t_{k})) & if t_{k} = 1 \\ f((g_{1}, t_{1}) … (g_{k}, t_{k})(g, 1)) & if t_{k} < 1 \end{matrix}

The previous considerations also show that the space B_{G} = E_{G}/G of G-orbits in E_{G} can be identified to the space of right cosets of the topological group E_{G} with respect to its subgroup G. In particular, if G is normal in E_{G}, the classifying space B_{G} is a topological group.

We simplify the notation by omitting f and identifying implicitly the elements of E_{G} with the corresponding monomials in E′_{G} in normal form (this can be done as f is bijective and thus, the equivalence classes of E_{G} are equivalence classes of E′_{G}).

The orbit of an element x = (g_{1}, t_{1}) … (g_{n}, t_{n}) ∈ E_{G} (in normal form) is the set

xG = {(g_{1}, t_{1}) … (g_{n}, t_{n})(g, 1) | g ∈ G} .

Thus, according to the rules established, if t_{n} = 1 the elements of xG have the form (g_{1}, t_{1}) ⋯ (g_{n}g, 1) and thus, we take (g_{1}, t_{1}) ⋯ (g_{n−1}, t_{n−1}) to represent it. This shows that as sets

B_{G} ≅ {(g_{1}, t_{1}) … (g_{n}, t_{n}) ∈ E_{G} | t_{n} < 1, n = 0, ⋯, ∞}

(the elements of E_{G} are always taken in normal form and if n = 0, we have the neutral element ⊥).

If ϑ : G H is a morphism between two weak Hausdorff k-groups,

B_{ϑ} : xG ≡ [x] E_{ϑ}(x)H ≡ [E_{ϑ}(x)] .

Proposition 5.2 The continuous group multiplication in E_{G} induces a continuous group multiplication in B_{G} if, and only if, G is abelian.

Proof – It is enough to prove that i_{G}(G) is normal in E_{G} ⟺ G is abelian.

Recall that i_{G}(G) is the set of monoids of the form (g, 1) with g ∈ G. Thus, we should check that ∀x ∈ E_{G} and (g, 1) ∈ i_{G}(G)

x(g, 1)x^{−1} ∈ i_{G}(G),

iff G is abelian.

If G is abelian, the Fundamental Relations 2. and 3. show that, for every monomial (h, s) ∈ E_{G} and every (g, 1) ∈ i_{G}(G),

(g, 1)(h, s) = (ghg^{−1}, s)(g, 1) = (h, s)(g, 1)

if s < 1, and

(g, 1)(h, s) = (gh, 1) = (hg, 1) = (h, 1)(g, 1)

if s = 1. Hence, for every x ∈ E_{G},

x(g, 1)x^{−1} = (g, 1)xx^{−1} = (g, 1) ∈ i_{G}(G) ,

and thus i_{G}(G) is normal in E_{G}.

Conversely, if i_{G}(G) is normal in E_{G}, for every g, h ∈ G and every s < 1, (h, s)(g, 1)(h^{−1}, s) ∈ i_{G}(G) and hence,

(h, s)(g, 1)(h^{−1}, s) = (h, s)(gh^{−1}g^{−1}, s)(g, 1) = (hgh^{−1}g^{−1}, s)(g, 1) ∈ i_{G}(G) ;

it follows that hgh^{−1}g^{−1} = u_{G} and thus, G is abelian since g and h were taken arbitrarly. □

Let G and H be two weak Hausdorff k-groups; the function

ξ_{G,H} : E_{G×H} E_{G} × E_{H}

ξ_{G,H}{((g_{1}, h_{1}), t_{1}) … ((g_{n}, h_{n}), t_{n})} = ((g_{1}, t_{1}) … (g_{n}, t_{n}), (h_{1}, t_{1}) … (h_{n}, t_{n}))

is a natural homeomorphism of weak Hausdorff k-groups (see of [12, Theorems 6.2 and 7.6 (g)]); in other words, the functor E preserves finite products.

The natural homeomorphism ξ_{G,H} passes on to the orbit spaces, that is to say, there exists a map

ρ_{G,H} : B_{G×H} B_{G} × B_{H}
ρ_{G,H}{[((g_{1}, h_{1}), t_{1}) … ((g_{n}, h_{n}), t_{n})]} =
([(g_{1}, t_{1}) … (g_{n}, t_{n})], [(h_{1}, t_{1}) … (h_{n}, t_{n})])

such that the following diagram commutes

E_{G×H} \overline{ξ_{G,H}}\overline{}E_{G} × E_{H}

[figure: p_{G×H}; p_{G} × p_{H}]

B_{G×H} \overline{ρ_{G,H}} \overline{}B_{G} × B_{H}

The map ρ_{G,H} is actually a natural homeomorphism; in other words, the functor B preserves finite products.

6 Exact sequences, subgroups and quotient spaces of groups

We begin this section by proving the following result:

Theorem 6.1 The functors E and B are exact.

Proof – Let

A ^{ϑ} G ^{φ} C

be a sequence in T opGr∩wHk(T op) such that im ϑ = ker φ; then, the following diagram commutes:

[figure: A; \overline{ϑ}; G; \overline{φ}; C]

[figure: i_{A}; i_{G}; i_{C}]

[figure: E_{A}; \overline{E_{ϑ}}; E_{G}; \overline{E_{φ}}; E_{C}]

[figure: p_{A}; p_{G}; p_{C}]

[figure: B_{A}; \overline{B_{ϑ}}; B_{G}; \overline{B_{φ}}; B_{C}]

Take arbitrarily z = (a_{1}, s_{1}) … (a_{k}, s_{k}) ∈ E_{A}. Then

E_{φ}E_{ϑ}(z) = E_{φϑ}(z) =
= (φϑ(a_{1}), s_{1}) … (φϑ(a_{k}), s_{k}) = (u_{C}, s_{1}) … (u_{C}, s_{k}) = ⊥_{E_{C}} .

Hence, im E_{ϑ} ⊆ ker E_{φ}. Now, take x = (g_{1}, t_{1}) … (g_{n}, t_{n}) ∈ ker E_{φ} in normal form; then

E_{φ}(x) = (φ(g_{1}), t_{1}) … (φ(g_{n}), t_{n}) = ⊥_{E_{C}}

and since 0 < t_{i} < t_{j} for all 0 ≤ i < j ≤ n, φ(g_{j}) = u_{G}; thus, for every j ∈ [0, n], there exists an element a_{j} in A such that g_{j} = ϑ(a_{j}). Therefore, x = E_{ϑ}(z) with z = (a_{1}, t_{1}) … (a_{n}, t_{n}), and ker E_{φ} ⊆ im E_{ϑ}.

Now let us look at the sequence of classifying (based) spaces. Let [z] ∈ B_{A} be given; then, for every z ∈ [z] in E_{A},

B_{φ}B_{ϑ}([z]) = [E_{φ}E_{ϑ}(z)] = [⊥_{E_{C}}] ,

and thus, im B_{ϑ} ⊆ ker B_{φ}.

Suppose that [x] ∈ ker B_{φ} is such that B_{φ}([x]) = [⊥_{E_{C}}]. But B_{φ}([x]) = [E_{φ}(x)] for every representative x ∈ [x] and thus, E_{φ}(x) = ⊥_{E_{C}} (c, 1), for some c ∈ C. Suppose that x = (g_{1}, t_{1}) … (g_{n}, t_{n}) in [x] with t_{n} < 1^{7}; then c = u_{C}, and E_{φ}(x) = ⊥_{E_{C}}. The exactness of the middle row of the previous diagram now implies that there exists z in E_{A}, such that x = E_{ϑ}(z), and hence [x] = [E_{ϑ}(z)] = B_{ϑ}([z]), that is to say, ker B_{φ} ⊆ im B_{ϑ}. □

The functor E behaves well with respect to subgroups, normality and related properties. We start this series of results by observing that if G is a weak Hausdorff k-group and K is a topological subgroup of G, then E_{K} is a topological subgroup of E_{G}: in fact, the closed inclusion ι : K G gives rise to a closed inclusion E_{ι} : E_{K} E_{G} (see Theorem 4.1) and E_{ι}(E_{K}) is an abstract subgroup of E_{G}.

7This is possible because if x′ = (g′_{1}, t′_{1}) … (g′_{m}, t′_{m}) and t′_{m} = 1, it is sufficient to take x = x′(g_{m}^{′−1}, t′_{m}).

Corollary 6.2 The functor E preserves normality of subgroups.

Proof – Let N be a normal subgroup of G; Theorem 6.1 applied to the short exact sequence

[figure: 0; N; \overline{i}; G; \overline{p}; G/N; 0]

gives rise to a short exact sequence of groups

[figure: 0; E_{N}; \overline{E_{i}}; E_{G}; \overline{E_{p}}; E_{G/N}; 0 .]

Then, for every x ∈ E_{G} and every y ∈ E_{N}, xyx^{−1} ∈ ker E_{p} = im E_{i} and thus xyx^{−1} ∈ E_{N}. □

We recall from [12] that if A is an abelian subgroup of G, then E_{A} is an abelian subgroup of E_{G}; in particular, if ZG is the center of G, then E_{ZG} is an abelian subgroup of E_{G}. Actually, we can go one step further:

Lemma 6.3 E_{ZG} coincides with the center Z(E_{G}) of E_{G}.

Proof – Take two monomials (g, s) ∈ E_{G} and (z, t) ∈ E_{ZG}. If s > t

(g, s)(z, t) = (gzg^{−1}, t)(g, s) = (z, t)(g, s) ;

if s < t

(z, t)(g, s) = (zgz^{−1}, s)(z, t) = (g, s)(z, t) ;

if t = s

(g, s)(z, t) = (gz, s) = (zg, s) = (z, t)(g, s) .

For arbitrary elements in normal form, the proof follows after successive applications of the Fundamental Relations. □

More generally,

Corollary 6.4 Let Z be a central topological subgroup of G ∈ T opGr∩wHk(T op). Then E_{Z} is a central subgroup of E_{G}; in short, E preserves centrality.

Now we wish to prove that E preserves quotient groups obtained from normal subgroups: more precisely:

Theorem 6.5 Let N be a normal subgroup of the weak Hausdorff k-group G. Then E_{G/N} ≅ E_{G}/E_{N} in T opGr ∩ wHk(T op).

Proof – Take the short exact sequence of weak Hausdorff k-groups

[figure: 0; N; \overline{ι}; G; \overline{π}; G/N; 0]

and notice the following: (i) E_{ι} : E_{N} E_{G} is a closed inclusion in T opGr ∩ wHk(T op) (Theorem 4.1); (ii) E_{π} : E_{G} E_{G/N} is a proclusion in T opGr ∩ wHk(T op) (Theorem 4.2); (iii) the sequence

[figure: 0; E_{N}; \overline{E_{ι}}; E_{G}; \overline{E_{π}}; E_{G/N}; 0]

is exact (Theorem 6.1). The last observation shows that as abstract groups, E_{G/N} ≅ E_{G}/E_{N}; the question is now to prove that actually we have a homeomorphism in T opGr ∩ wHk(T op). To this end, we refer to [10, Theorem 12, III.19] and notice that we have to adjust Pontrjagin’s proof in the sense that in his treatment E_{π} is an open map, while in our case, E_{π} is a proclusion. Let f : E_{G/N} E_{G}/E_{N} be the function that maps an element E_{π}(x) ∈ E_{G/N} into the coset xE_{N} and consider the diagram

E_{G}

[figure: E_{π}; q]

E_{G/N}  \overline{f} \overline{}E_{G}/E_{N}

in which E_{G}/E_{N} has the final topology determined by q. According to [10, Theorem 12, III.19] f is a bijection and its inverse f^{−1} is continuous; the continuity of f follows from the fact that fp = q is continuous. □

There is a similar result for the classifying spaces in the particular case in which N is a central subgroup of the weak Hausdorff k-group G. However, in order to prove this fact we need some additional considerations. We begin with the following lemma.

Lemma 6.6 Let G and H be two weak Hausdorff k-groups which act on X ∈ wHk(T op) via the actions ϕ and ψ respectively. Suppose that the two actions commute, i.e.

(∀x ∈ X, g ∈ G, h ∈ H) ψ(ϕ(x, g), h) = ϕ(ψ(x, h), g) .

Then the spaces (X/G)/H and (X/H)/G are well defined and homeomorphic.

Proof – Because of [2, 3.2.5, Proposition 11] G acts on X/H by

ϕ′ : X/H × G X/H , (xH, g) (xg)H

and H acts on X/G by

ψ′ : X/G × H X/G , (xG, h) (xh)G ;

thus the quotient spaces (X/H)/G and (X/G)/H are well defined.

Consider the compositions p′q and q′p of the natural projections

X ^{q} X/H ^{^{p′}} (X/H)/G

and

X ^{p} X/G ^{^{q′}} (X/G)/H .

The maps p′q and q′p define the equivalence relations R and S in X

xRx′ ⇔ ∃g ∈ G, h ∈ H | x′ = x.h.g ,
xSx′ ⇔ ∃g ∈ G, h ∈ H | x′ = x.g.h .

The commutativity of the actions ϕ and ψ implies that

(∀x, x′ ∈ X) xRx′ ⇔ xSx′ .

Hence, the sets X/S = (X/H)/G and X/R = (X/G)/H coincide (up to bijection). Furthermore, p′q = q′p and so, the final topology determined by p′q on (X/H)/G coincides with that determined by q′p on the (same) set (X/G)/H. Then, the two spaces are homeomorphic. □

Let G and H be topological groups and let r : G × H G be a right action of H on G; in general, r is not a group homomorphism. The next result characterizes this situation.

Lemma 6.7 A right action r : G×H G is a group homomorphism ⟺ r commutes with the right and left translations of G. , i.e. iff for each g, g′ in G and h in H

gr(g′, h) = r(gg′, h) = r(g, h)g′ .

Proof – ⇒ For every g, g′ ∈ G and every h, h′ ∈ H

r((g, h)(g′, h′)) = r(gg′, hh′) = r(g, h)r(g′, h′) .

In particular, if we take h′ = u_{H} we obtain that r(gg′, h) = r(g, h)g′, while taking h = u_{H} we get r(gg′, h′) = gr(g′, h′).

⇐ With arbitrary g, g′, h, h′ as before

r((g, h)(g′, h′)) = r(gg′, hh′) = r(r(gg′, h), h′) =
= r(r(g, h)g′, h′) = r(g, h)r(g′, h′) .
□

As an example, we take G ∈ T opGr ∩ wHk(T op), H a central subgroup of G and r the group multiplication. Notice that in this example r is free.

For the next lemma we shall assume that H ∈ T opGr_{∗} ∩ wHk(T op_{∗}) is abelian; to emphasize this point we shall use the name A rather than H for the group acting on the right.

Let r : G × A G be an action that commutes with the translations of G. Then the group E_{A} is abelian and B_{A} := E_{A}/A is an abelian group with the natural quotient structure (see Section 5). Because E and B preserve finite products (see the end of Section 5) we can build up the commutative diagram in wHk(T op_{∗})

[figure: G × A; \overline{1}; G × A; \overline{r}; G]

[figure: i_{G} × i_{A}; i_{G} × i_{A}; i_{G}]

[figure: E_{G} × E_{A}; \overline{ξ_{G,A}^{−1}}; E_{G×A}; \overline{E_{r}}; E_{G}]

[figure: p_{G} × p_{A}; p_{G×A}; p_{G}]

[figure: B_{G} × B_{A}; \overline{ρ_{G,A}^{−1}}; B_{G×A}; \overline{B_{r}}; B_{G}]

where ξ_{G,A}^{−1} and ρ_{G,A}^{−1} are natural homeomorphisms in T opGr ∩ wHk(T op) and wHk(T op), respectively.

Lemma 6.8 If r : G × A G is a right action which commutes with the translations of G, then the maps

E(r) := E_{r}ξ_{G,A}^{−1} and B(r) := B_{r}ρ_{G,A}^{−1}

are right actions.

The lemma says that, modulo certain natural homeomorphisms, the functors E and B preserve the right actions of abelian groups which commute with translations.

Proof – Let x = (g_{1}, s_{1}) … (g_{n}, s_{n}) ∈ E_{G} be in normal form. Then

E(r)(x, ⊥_{E_{A}}) = E_{r}((g_{1}, u_{A}), s_{1}) … ((g_{n}, u_{A}), s_{n})) =
= (r(g_{1}, u_{A}), s_{1}) … (r(g_{n}, u_{A}), s_{n}) = x ;

furthermore, for any representative z ∈ [⊥_{E_{A}}],

B(r)([x], [⊥_{E_{A}}]) = B_{r}([ξ_{G,A}^{−1}(x, z)]) = B_{r}([x]) = [E_{r}(x)] = [x] .

The proof of the associativity is based on the fact that in the present case r and the multiplication m of E_{A} are homomorphisms. Therefore, we can construct the commutative diagram

[figure: E_{G} × E_{A} × E_{A}; \overline{≅}; E_{G×A} × E_{A}; \overline{E_{r} × 1}; E_{G} × E_{A}]

[figure: ≅; ≅; ≅; ≅]

[figure: E_{G} × E_{A×A}; \overline{≅}; E_{G×A×A}; \overline{E_{r×1}}; E_{G×A}]

[figure: 1 × E_{m}; E_{1} × m; E_{r(1×m)}; E_{r(r×1)}; E_{r}]

[figure: E_{G} × E_{A}; \overline{≅}; E_{G×A}; \overline{E_{r}}; E_{G}]

where the isomorphisms are given by the maps ξ^{−1} defined at the end of Section 5. The associativity of B(r) then comes by projection of the diagram onto the corresponding diagram of classifying spaces. □

Notice that, since E(r) is a group homomorphism, it commutes with the multiplication in E_{G} (see Lemma 6.7).

Theorem 6.9 For any G ∈ T opGr ∩ wHk(T op) and any central subgroup Z ⊂ G, B_{G/Z} ≅ B_{G}/B_{Z}.

Proof – The sequences

[figure: 0; E_{Z}; \overline{E_{ι}}; E_{G}; \overline{E_{π}}; E_{G/Z}; 0]

and

[figure: 0; B_{Z}; \overline{B_{ι}}; B_{G}; \overline{B_{π}}; B_{G/Z}; 0]

are exact in the respective categories by Theorem 6.1 and E_{G/Z} ≅ E_{G}/E_{Z} because of Theorem 6.5. ¿From [2, 3.2.8, Proposition 22] we conclude that

B_{G} = E_{G}/G ≅ (E_{G}/Z)/(G/Z) .

This, together with the fact that B_{Z} acts on B_{G} (see Lemma 6.8) and Lemma 6.6 imply the following sequence of homeomorphisms:

B_{G}/B_{Z} ≅ ((E_{G}/Z)/(G/Z))/(E_{Z}/Z) ≅
≅ ((E_{G}/Z)/(E_{Z}/Z))/(G/Z) ≅ (E_{G}/E_{Z})/(G/Z) ≅
≅ E_{G/Z}/(G/Z) ≅ B_{G/Z} .
□

7 Classifying spaces and principal bundles

The last section of this paper is concerned with quotient maps of weak Hausdorff k-groups q : G G/N which give rise to (locally trivial) principal B_{N}-bundles B_{q} : B_{G} B_{G/N}. To achieve this goal we must prove some auxiliary lemmas and introduce the concept of G-equivariant closed cofibrations.

Lemma 7.1 Let ϑ ∈ (T opGr ∩ wHk(T op))(G, H). Let X be a G-space, Y be an H-space Y, and f : X Y be a ϑ-equivariant relation bipreserving map. Furthermore, assume that f is an inclusion and that f(X) is closed in Y. Then the quotient map f̂: X/G Y/H is an inclusion.

Proof – Since f is relation bipreserving, f̂ is injective. In order to prove that f̂ is an inclusion we show that f̂: X/G f̂(X/G) is open.

For every h ∈ H, the map ψ_{h} : Y Y, ψ_{h}(y) := y.h is a homeomorphism; thus, because f(X) ⊂ Y is closed, every f(X).h is closed in Y. Let f(X).H = ⋃_{h∈H} f(X).h with the topology determined by the closed subspaces f(X).h , h ∈ H; one now shows that this topology is equivalent to the relative topology induced by Y.

Next, let p : X X/G be the identification map and let U ⊂ X/G be an open set; then V := p^{−1}(U) is open in X and, f(V) is open in f(X) since f is an inclusion. Because f is relation bipreserving and ϑ-equivariant f(X) ∩ f(V).H = f(V); moreover, for every h ∈ H, f(V).h = f(X).h ∩ f(V).H is open in f(X).h and hence, f(V).H is open in f(X).H. Using the relative topology of the latter space, this means that there exists an open set B in Y such that f(V).H = f(X).H ∩ B.

At this point we take the identification map q : Y Y/H and observe that

q(B) ∩ f̂(X/G) = q(B ∩ f(X).H) = q(f(V).H) = q(f(V)) = f̂(U) ;

since q(B) is open, it follows that f̂(U) is open and therefore, f̂ is a homeomorphism onto its image f̂(X/G), that is to say, f̂ is an inclusion. □

Lemma 7.2 Let G and G′ be weak Hausdorff k-groups whose intersection G ∩ G′ = Z is a central subgroup of both G and G′. Let X be a space on which both G and G′ act; suppose these two actions commute. Then

(X/G′)/(G/Z) ≅ (X/G)/(G′/Z)

Proof – We wish to construct a commutative diagram

[figure: Z; \overline{i}; G; \overline{p}; G/Z]

[figure: i′; i_{G}; î_{G} = i_{G/Z}]

[figure: G′; \overline{i′_{G}}; X; \overline{p′_{G}}; X/G′]

[figure: p′; p_{G}; p_{G/Z} = p̂_{G}]

[figure: G′/Z; î′_{G} = i_{G′/Z}; X/G; p_{G′/Z} = p_{G′}; Y]

in which i_{G} and i′_{G} are the restrictions of the actions of G and G′ to {x_{0}} × G and {x_{0}} × G′ respectively (for some x_{0} fixed in X); p, p′, p_{G} and p′_{G} are the natural projections; î_{G} and î_{G′} are the quotient maps induced by i_{G} and i_{G′} respectively.

Since the actions of G and G′ on X commute, G′ acts on X/G and G on X/G′; since Z is central in both groups, these actions pass to the quotient giving rise to actions of G/Z on X/G′ and of G′/Z on X/G. At this point define the quotient spaces with relative projections

p_{G/Z} : X/G′  \overline{}(X/G′)/(G/Z) ,
p_{G′/Z} : X/G \overline{}(X/G)/(G′/Z) .

Because of Lemma 7.1 and [2, 3.2.8, Proposition 22] we conclude that

Y := (X/G′)/(G/Z) ≅ ((X/Z)/(G′/Z))/(G/Z) ≅
≅ ((X/Z)/(G/Z))/(G′/Z) ≅ (X/G)/(G′/Z) .

It is easy to verify that p̂_{G} = p_{G/Z} and p̂_{G′} = p_{G′/Z}. □

For a given topological group G and a G-space E, define E^{∗} to be the subspace of all (x, xg) ∈ E × E where x ∈ E and g ∈ G. Define the function τ : E^{∗} G such that

(∀(x, x′) ∈ E^{∗}) xτ(x, x′) = x′ .

The function τ just defined is called a translation function; the action of G on E is principal if τ is a continuous function (see [5, Chapter 4]).

Proposition 7.3 Let K be a weak Hausdorff k-group together with two subgroups G and H whose intersection G ∩ H = Z is a central subgroup. Now, let E be a principal H-space in wHk(T op) such that E^{∗} is closed in E × E. Suppose that G also acts freely on E and that this action commutes with the action of H. Then E/G is a principal H/Z-space.

Proof – The next diagram clarifies the situation:

[figure: Z; \overline{i}; G; \overline{p}; G/Z]

[figure: i_{Z}; i_{G}; i_{G/Z}]

[figure: H; \overline{j}; E; \overline{q}; B]

[figure: p_{Z}; p_{G}; p_{G/Z}]

[figure: H/Z; \overline{ĵ}; E/G; \overline{q̂}; B/(G/Z)]

By lemma 7.2 there is an action of H/Z on E with orbit space B/(G/Z). We must check that this action is principal. Let τ be the continuous translation function determined by the principal action of G on E. We are going to show that the composite map

[figure: E^{∗}; \overline{τ}; H; H/Z]

factors through (E/Z)^{∗}. In fact, the composition of the inclusion of E^{∗} into E × E followed by the projection onto E/Z × E/Z factors through the orbit space determined by the action of Z × Z on E × E. Lemma 7.1 then produces an inclusion of E^{∗}/(Z × Z) into E/Z × E/Z and E^{∗}/(Z × Z) ≅ (E/Z)^{∗}; the required map τ̂ : (E/Z)^{∗} H/Z is now easily defined. Next, we factor τ̂ through (E/G)^{∗}; this is done by extending the action of G on E to an action of G/Z on E/Z and showing that (E/Z)^{∗}/(G/Z) ≅ (E/G)^{∗}, thereby obtaining

our continuous translation (E/G)^{∗}  H/Z.  □

Let X be a G-space with action ϕ : X × G X; we define an action ϕ_{I} : (X × I) × G X × I simply by setting ϕ_{I}((x, t), g) = (ϕ(x, g), t) for all x ∈ X, t ∈ I, and g ∈ G.

A pair (X, A) of G-spaces in wHk(T op) is a G-equivariant closed cofibration if it satisfies the G-equivariant homotopy extension property, that is to say, if (X, A) is a G-neighborhood deformation retract (cf. [13]); alternatively, we shall say that the inclusion map i : A X is a G-equivariant closed cofibration to indicate that (X, A) is a G-equivariant closed cofibration. The proof of the following lemma is a routine exercise.

Lemma 7.4 Let X and A be G-spaces with A closed in X; denote by i : A X the inclusion map. Then the induced map

î: A/G X/G

is a closed cofibration ⟺ i is a G-equivariant closed cofibration.

Lemma 7.5 Let X be a space with a principal G-action ϕ : X × G → X; suppose that there exists x_{o} ∈ X such that the restriction of ϕ to {x_{o}} × G is a closed inclusion i : G → X (up to isomorphism) which is also a closed Gcofibration (up to isomorphism). Then the quotient map q : X → X/G is locally trivial.

Proof – Follows the same lines as [12, Theorem 8.3]. □

Proposition 7.6 Let G be a weak Hausdorff k-group with non-degenerate unit and let N be a normal subgroup of G such that i : N G is an N-equivariant closed cofibration. Then (E_{G}, E_{N}) is a E_{N}-equivariant closed cofibration; hence, (E_{G}, E_{q}, E_{G/N}) is a principal E_{N}-bundle.

Proof – E_{G} is a weak Hausdorff k-group (5.1) with non-degenerate unit (4.3); furthermore, E_{N} is a normal subgroup of E_{G} (6.2). The non-degenerate unit u_{G} of G is also a non-degenerate unit of N and the class N is non-degenerate in G/N. We also know (from Theorem 4.3) that (E_{G}, E_{N}) is a closed cofibration, but this is not enough for our purposes. So, we take the commutative diagram

E_{N}  \overline{E_{i}} \overline{}E_{G}

[figure: E_{q}; E_{q}]

E_{N}/E_{N} \overline{Ê_{i}}\overline{}E_{G}/E_{N}

and observe that Ê_{i} : E_{N}/E_{N} E_{G}/E_{N} is a closed cofibration; then E_{i} : E_{N} E_{G} is an E_{N}-equivariant closed cofibration in virtue of Lemma 7.4.

The last statement follows from Lemma 7.5. □

Theorem 7.7 Let G be a weak Hausdorff k-group with non-degenerate unit. Let Z be a central subgroup of G such that (G, Z) is a Z-equivariant closed cofibration, and let q : G G/Z be the quotient map. Then

B_{q} : B_{G} B_{G/Z}

is a locally trivial principal B_{Z}-bundle.

Proof – By Lemma 7.2 there is an action of B_{Z} on B_{G} with orbit space B_{G/Z}. But E_{G}^{∗} is closed in E_{G} × E_{G} because the inverse image by E_{i} × E_{i} of the diagonal of E_{G} × E_{G} is closed (see [4, A.1, pag. 243]); then B_{G} is a principal B_{Z}-space (see Proposition 7.3). Finally, because of Theorem 4.3, Lemma 6.8 and Lemma 7.5, we conclude that B_{q} is locally trivial. □

A final remark: if G is compact Hausdorff, then B_{G/Z} is paracompact (see the end of Section 4) and so, (B_{G}, B_{q}, B_{G/Z}) is a Hurewicz fibration.

References

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Università di Milano Dipartimento di Matematica via Saldini, 50 20133 Milano (Italia)