This is a reprint of the paper

Brown’s representability theorem and branched coverings

by Débora María Tejada

published in Rev. Colombiana Mat. 28 (1994), pp. 23–34

BROWN’S REPRESENTABILITY THEOREM

AND BRANCHED COVERINGS

Débora María Tejada^{⋆}

Universidad Nacional de Colombia, Medellín

Abstract. It is well known that Brown’s Representability Theorem has many applications. The proof of the existence of a universal bundle for a topological group (see [4]) and the proof that any space can be approximated by CW-complexes (see [8]) are among those applications. Here we prove the existence of classifying spaces for branched coverings over CW-complexes using Brown’s Theorem as the main tool. Keywords and phrases. Manifolds, branched coverings, classification spaces, homotopy groups, CW –complexes, simplicial complexes, universal bundles, categories and functors. 1991 Mathematics Subject Classification. 57M12, 57N80, 55Q52, 55R10.

1. Introduction

In 1920 Alexander (see [1]) pointed out the connection between the study of manifolds and of branched coverings over spheres. Actually, his results imply that information about the homotopy groups of classifying spaces of branched coverings translates into information about manifolds. Our principal goal in this paper is to prove the existence of classifying spaces for k-fold branched coverings over CWcomplexes for which the branch set is a stratified set. To accomplish this purpose, we will use Brown’s Representability Theorem as the key tool. Unfortunately, our proof is not constructive. In [3] we give an explicit construction of classifying spaces for branched coverings over manifolds.

First of all, we recall some basic definitions and notations. Let X and Y be topological spaces, A ⊂ X and I the unit interval [0, 1]. We say that the pair (X, A) has the homotopy extension property if any continuous map f : A × I → Y admits a continuous extension F : X × I → Y. Now, let C be the category of topological

Typeset by AMS-TEX

⋆ The author was partially supported by COLCIENCIAS, the Colombian Institute of Science.

spaces with base point which admit a CW-complex structure, and the morphisms being the continuous maps preserving base points. A triple (X, X_{1}, X_{2}) will be called a proper triad of C if X = X_{1} ∪ X_{2}, X_{1}, X_{2} and X_{1} ∩ X_{2} are all in C, all have the same base point, and (X_{1}, X_{1} ∩X_{2}) and (X_{2}, X_{1} ∩X_{2}) both have the homotopy extension property. If X and Y belong to C, [X, Y] denotes the set of homotopy classes of maps of X into Y with respect to homotopies which leave the base point of X fixed. [• , Y] will denote the functor from C to S (where S is the category of sets with a distinguished element and maps preserving distinguished elements) which assigns to each X in C the set [X, Y] with the class of the constant map as distinguished element, and to each map f : X → X′, the map F : [X′, Y] ⟶ [X, Y] defined by F [g] = [g ∘ f], where [g] denotes the homotopy class of g.

Now we state Brown’s Theorem (see [4] and [8]), which guarantees the existence of classifying spaces for functors that verify some special conditions.

Brown’s Representability Theorem. If H : C → S is a contravariant functor, and H satisfies the conditions A, B, C, D listed below, there is a space Y in C, unique up to homotopy type, such that the functors [• , Y] and H are naturally equivalent.

A. If f, g : X → Y are homotopic, H(f) = H(g).

B. (1) If p is a point, H(p) contains only one element. (2) Suppose (X, X_{1}, X_{2}) is a proper triad , A = X_{1} ∩ X_{2}, and j_{i} : A → X_{i} and k_{i} : X_{i} → X are the inclusion maps, i = 1, 2. If u_{1} ∈ H(X_{1}) and u_{2} ∈ H(X_{2}) are such that H(j_{1})u_{1} = H(j_{2})u_{2}, then there is a v in H(X) such that H(k_{1})v = u_{1} and H(k_{2})v = u_{2}. Furthermore, if A is a point, v is unique.

C. Suppose S_{α}^{n} is a collection of disjoint n-spheres whose wedge product ∨S_{α}^{n} is in C. Let i_{β} : S_{β}^{n} ⟶ ∨S_{α}^{n} be the inclusion map. Then the map ∏ H(i_{α}) : H(∨S_{α}^{n}) ⟶ ∏ H(S_{α}^{n}) is bijective.

D. Suppose X_{1} ⊂ X_{2} ⊂ ⋯ ⊂ X_{n} ⋯ is a collection of subcomplexes of X = ⋃ X_{n} in C with respect to some CW-complex structure on X such that X_{n}^{n} = X^{n} (where X^{n} is the n-skeleton of X). Let i_{n} : X_{n} → X be the inclusion map. Let \underleftarrow{lim} H(X_{n}) be the inverse limit of H(X_{n}) with respect to the maps induced by the inclusions of X_{n} to X_{m}. Then the function

\underleftarrow{lim} H(i_{n}) : H(X) ⟶ \underleftarrow{lim} H(X_{n})

is an epimorphism.

Section 2 focuses on the basic notions about k-fold branched coverings over manifolds. There we identify some contravariant functors from the category of (pointed) smooth manifolds to the category of sets. In Section 3 we show that the concepts and properties developed in Section 2 give a natural framework for the definitions of k-fold branched coverings over simplicial complexes and over CWcomplexes. They also allow us to construct a functor from the category of CWcomplexes to the category of sets. Then, we prove that this functor verifies all the conditions demanded by Brown’s Representability Theorem. In this way we show the existence of a classifying space for k-fold branched coverings over CWcomplexes.

2. Branched coverings over manifolds

In this section we study k-fold branched coverings over manifolds and some of their properties. The definitions and propositions exposed in this section are essentially the base for the rest of the paper.

Let N^{n} be a smooth manifold (the index n means that N has dimension n), and let l be a natural number. We say that K ⊂ N is a stratified set in N of type l if there is a sequence ∅ = K_{l+1} ⊂ K_{l} ⊂ K_{l−1} ⊂ K_{l−2} ⊂ ⋯ ⊂ K_{3} ⊂ K_{2} = K of closed sets in N such that (K_{j} − K_{j+1}) is a smooth manifold without boundary of codimension j and \overline{K_{j} − K_{j+1}} = K_{j} for every j, j = 2, ⋯, l. The manifolds (K_{j} − K_{j+1}) are called the strata of K.

Given M, N manifolds, K a stratified set in N of type l, and f : M → N a smooth function, we say that f is transverse to K if f is transverse ([6], p. 28) to the strata.

If M^{n}, N^{n} are smooth manifolds and K ⊂ N is a stratified set in N of type l, using induction over the number of strata it is possible to prove that if g_{0} : M → N is any smooth function, there exists g : M → N, homotopically equivalent to g_{0}, such that g is transverse to K ([9], p. 12). Consequently, g^{−1}(K) is a stratified set in M of type l ([9], p. 14).

Now, we define a branched covering over a manifold. Let Ñ^{n}, N^{n} be smooth manifolds, and k, l be natural numbers, and let f : Ñ → N be a smooth function. We say that f is a k-fold branched covering of type l over N if f satisfies:

(i) The branch set K is stratified in N of type l.

(ii) f is transverse to each stratum of K.

(iii) The set f^{−1}(K) is also a stratified set of type l whose strata are the submanifolds f^{−1}(K_{j} − K_{j+1}), j = 2, ⋯, l.

(iv) f|_{f^{−1}(N−K)} is a k-fold covering.

(v) f|_{f^{−1}(K_{l})}, f|_{f^{−1}(K_{l−1}−K_{l})}, …, f|_{f^{−1}(K_{2}−K_{3})} are k_{s}-coverings over their respective components, where k_{s} is less than k.

Let f_{1} : M̃_{1} → M and f_{2} : M̃_{2} → M be k-fold branched coverings of type l over a manifold M. We say that f_{1} and f_{2} are equivalent up to homeomorphism if there is a homeomorphism h : M̃_{1} → M̃_{2} such that f_{2} ∘ h = f_{1}. Now, we say that f_{1} and f_{2} are concordant (of type l) if there is a k-fold branched covering of type l, F : W̃^{n+1} ⟶ M^{n} × I (I is the closed interval [0,1]), such that F |_{F^{−1}(M×{0})} is equivalent up to homeomorphism to f_{1} and F |_{F^{−1}(M×{1})} is equivalent up to homeomorphism to f_{2}. We will denote with B_{k,l}(M) the set of all concordance classes of k-fold branched coverings over M of type l.

A triple (E, B, γ) is called a k-fold universal branched covering of type l if γ : E → B is a k-fold branched covering of type l and for any k-fold branched covering f : Ñ → N (of type l) there is a continuous function c : N → B such that the pullback of γ under c, namely c^{∗}γ, is a k-fold branched covering that is concordant with f. Moreover, it is also required that if c_{1} : N → B, c_{2} : N → B are maps such that the pullbacks c_{1}^{∗}γ and c_{2}^{∗}γ give concordant branched coverings, then the maps c_{1} and c_{2} are homotopic. B is called a classifying space and c is called a classifying function. It is not difficult to see that there is a bijection between the set B_{k,l}(N) of concordance classes of k-fold branched coverings of type l over N and the set [N, B] of homotopic classes of continuous functions from N to B.

Now, we proceed to define a branched covering over a manifold with boundary. Let Ñ, N be smooth manifolds with boundary and f : Ñ → N be a smooth function. We say that f is a k-fold branched covering of type l over a manifold with boundary if

(i) f|_{(Ñ−∂Ñ)} : (Ñ − ∂Ñ) ⟶ (N − ∂N) and f|_{∂Ñ} : ∂Ñ → ∂N are k- fold branched coverings of type l over a manifold without boundary.

(ii) the following diagram is commutative

\begin{matrix}∂Ñ × I & ⟶^{i} & Ñ \\ ^{f|_{∂_{N}}×id}↓ & & ↓^{f} \\ ∂N × I & ⟶_{j} & N \end{matrix}  (2.1)

where i and j are the embeddings obtained by “collaring” ∂Ñ and ∂N in Ñ and N, respectively (see [5]), and I is the closed interval [0,1] in ℝ.

Remarks

(1) If no ambiguity arises, we will omit the words “with boundary” or “without boundary”.

(2) If H is a set in N, f^{−1}(H) will be denoted by H̃: for example, K̃_{j} = f^{−1}(K_{j}).

Roughly speaking, the next proposition says that the pullback of a k-fold branched covering is also a k-fold branched covering.

Proposition 2.1. Let f : Ñ → N be a k-fold branched covering of type l with branched set K. Let M be a smooth manifold. If g : M → N is any smooth map transverse to K, and M̃ is the fiber product in the diagram

\begin{matrix}M̃ & ⟶ & Ñ \\ ^{f_{1}}↓ & & ↓^{f} \\ M & ⟶_{g} & N \end{matrix}  (2.2)

then the pullback g^{∗}(f) = f_{1} : M̃ → M is a k-fold branched covering of type l.

Proof. In fact, since g is transverse to K, g^{−1}(K) is a stratified set in M of type l. Moreover, the restriction of f to the preimages of the strata of K are finite coverings, and their pullbacks under g are also finite coverings.

We have to show that the branch set of f_{1} is just g^{−1}(K). If x belongs to M − g^{−1}(K), then x is not a branched point, because the restriction of f_{1} to f_{1}^{−1}(M − g^{−1}(K)) is a k-covering. Let x ∈ g^{−1}(K). To show that x is a branched point we need only to observe that the cardinality of the set f_{1}^{−1}(x) is different from the cardinality of the set f_{1}^{−1}(x_{0}) when x_{0} is not a branched point. In fact, if z is any element in M, the cardinality of f_{1}^{−1}(z) is equal to the cardinality of f^{−1}(g(z)), because of the definition of the fiber product. □

Recall that B_{k,l}(N) is the set of concordance classes of k-fold branched coverings of type l over the manifold N.

Proposition 2.2. B_{k,l}(•) is a contravariant functor from the category of smooth manifolds to the category of sets.

Proof. Let M, N, Ñ be smooth manifolds (with or without boundary). Let g : M → N be a smooth map and let f : Ñ → N be a k-fold branched covering of type l. Without loss of generality, we may asssume that g is transverse to K, where K is the branch set of f. The pullback f_{1} : M̃ → M = g^{∗}(f) is a k-fold branched covering of type l. Let us define B_{k,l}(f) as the concordance class of f_{1} = g^{∗}(f). We need to show that B_{k,l} is well defined.

First of all, let us prove that if f′ : Ñ′ → N is a branched covering of type l concordant to f : Ñ → N, then B_{k,l}(f) = B_{k,l}(f′). Without loss of generality, we may assume that g is also transverse to K′, the branch set of f′ (actually, we can change g up to homotopy so that it is transverse to K′; see [9], p. 12). Let F : W^{n+1} → N × I be a branched covering (of type l) that makes f and f′ concordant. Consider the pullback of F under g × id. It is clear that this pullback makes the pullback of f concordant with the pullback of f′ under g, i.e., B_{k,l}(f) = B_{k,l}(f′).

Let g_{1}, g_{2} : M → N be homotopic functions (both transverse to K) with pullbacks f_{1} and f_{2}, respectively, under g. Let H : M × I → N be a homotopy from g_{1} to g_{2}. Without loss of generality, we may assume that H is transverse to K. Taking W_{1} as the fiber product in the diagram

\begin{matrix}W_{1} & ⟶ & Ñ \\ ^{F_{1}}↓ & & ↓^{f} \\ M × I & ⟶_{H} & N \end{matrix}  (2.3)

we get the branched covering F_{1} : W_{1} → M × I, which implies the concordance between f_{1} and f_{2}. □

Now we define a pointed branched covering over a manifold and modify the definitions of equivalence up to homeomorphism and concordance according to this new definition. Let M̃, M be manifolds. A pointed branched covering is a k-fold branched covering f : M̃ → M together with a base point ∗ in M − K (where K is the branch set of f) and a one–to–one correspondence of {1, 2, …, k} with the set f^{−1}(∗). This correspondence is denoted by c(f)_{i}, where 1 ≤ i ≤ k. Two pointed branched coverings f_{1} and f_{2} are equivalent up to homeomorphism if there is a branched covering homeomorphism h which preserves the labeling, that is, such that h(c(f_{1})_{i}) = c(f_{2})_{i}. Two pointed k-fold branched coverings are concordant if there is a branched covering concordance F : W̃^{n+1} → M × I such that ∗ × I does not intersect the branch set of F, and c(f_{1})_{i} and c(f_{2})_{i} are in the same components of F^{−1}(∗ × I) when F |_{F^{−1}(M×{0})} and F |_{F^{−1}(M×{1})} are equivalent up to homeomorphism to f_{1} and f_{2}, respectively.

Let us denote with B_{k,l}(M, ∗) the set of all concordance classes of pointed k-fold branched coverings over the manifold M. It is not difficult to see that B_{k,l}(•, ∗) is a contravariant functor from the category of pointed smooth manifolds to the category of sets with a distinguished element.

3. Branched coverings over simplicial complexes

In this section we establish the existence of a functor that operates over the category of CW-complexes and verifies all the hypotheses of Brown’s Theorem. Since every CW-complex is homotopically equivalent to a simplicial complex, we first work on simplicial complexes. Then, at the end of the section, we proceed to generalize the results to CW-complexes.

Let △_{m} be an m-simplex. For every face △_{m−1} of △_{m} there is a one-to-one function sl : △_{m−1} × I → △_{m} such that (△_{m} − sl(△_{m−1} × [0, 1)) is homeomorphic to △_{m}. We call this kind of function a “slice”.

Now, we extend the definition of branched coverings over manifolds to branched coverings over simplicial complexes. Let △_{m} be an m-simplex. We say that f : ˜△_{m} → △_{m} (where ˜△_{m} is not necessarily an m-simplex) is a k-fold branched covering of type l over the simplex △_{m} if for every face △_{s} of △_{m} the function f|_{f^{−1}(int△_{s})} : f^{−1}(int△_{s}) → int△_{s} (where int means interior) is a k-fold branched covering of type l over the manifold int△_{s}, and for each face △_{s−1} of △_{s}, we have the following commutative diagram

\begin{matrix}f^{−1}(△_{s−1}) × I & ⟶ & f^{−1}(△_{s}) \\ ^{f×I}↓ & & ↓^{f} \\ △_{s−1} × I & ⟶_{slice} & △_{s} \end{matrix}  (3.1)

where I is the closed interval [0,1] and f^{−1}(△_{s−1}) × I → f^{−1}(△_{s}) is some embedding.

Let Σ be a simplicial complex and let f : Σ̃ → Σ be a continuous function (where Σ̃ is not necessarily a simplicial complex). We say that f is a k-fold branched covering of type l over Σ, if for every m-simplex △_{m} contained in any subdivision of Σ, the function f : ˜△_{m} → △_{m} is a k-fold branched covering of type l over △_{m}.

The next proposition is in the spirit of Proposition 2.1.

Proposition 3.1. Let Σ_{1}, Σ_{2} be simplicial complexes, f : Σ̃_{2} → Σ_{2} be a k-fold branched covering of type l and g : Σ_{1} → Σ_{2} be a simplicial function. If Σ̃_{1} is the fiber product in the diagram

\begin{matrix}Σ̃_{1} & ⟶ & Σ̃_{2} \\ ^{f_{1}}↓ & & ↓^{f} \\ Σ_{1} & ⟶_{g} & Σ_{2} \end{matrix}  (3.2)

then the pullback f_{1} is a k-fold branched covering of type l over the simplicial complex Σ_{1}.

Proof. Let △_{n} be any n-simplex contained in Σ_{1}. Since g : Σ_{1} → Σ_{2} is a simplicial function, g(△_{n}) is a simplex in Σ_{2}. Moreover, for every △_{s} ⊂ △_{n}, g(int△_{s}) is equal to the interior of the simplex g(△_{s}). The linearity of g in △_{s} now implies that g is transverse to the branch set of

f|_{f^{−1}(int g(△_{s}))} : f^{−1}(int g(△_{s})) → int g(△_{s}),

and by Proposition 2.1, the restriction f_{1}|_{f^{−1}(int△_{s})} : f^{−1}(int△_{s}) → int△_{s} is a k–fold branched covering of type l.

Now, consider the slice : △_{s−1} × I ⊂ △_{s}, for every △_{s−1} ⊂ △_{s}. The linearity of g in △_{n} implies that g(△_{s−1}) × I is also a slice inside g(△_{s}), and for some embedding f_{1}^{−1}(△_{s−1}) × I → f_{1}^{−1}(△_{s}) the following diagram commutes:

\begin{matrix}f_{1}^{−1}(△_{s−1}) × I & ⟶ & f_{1}^{−1}(△_{s}) \\ ^{f_{1}|_{f{−1 \atop 1}(△_{s−1})}×id}↓ & & ↓^{f_{1}} \\ △_{s−1} × I & ⟶_{slice} & △_{s} \end{matrix}  (3.3)

Therefore, f_{1} : Σ̃_{1} → Σ_{1} is a k-fold branched covering of type l over the simplex △_{n}. □

Let us extend the definition of concordance for branched coverings over simplicial complexes. We say that two k-fold branched coverings f_{1}, f_{2} over a simplicial complex Σ are concordant if there is a branched covering F : W → Σ × I over the simplicial complex Σ×I such that F |_{F^{−1}(Σ×{j})} is equivalent up to homeomorphism with f_{j+1}, for j = 0, 1.

The proofs of the following lemmas are not difficult ([9], p. 22, 23).

Lemma 3.2. Let Σ_{1}, Σ_{2} be simplicial complexes and let f_{2} : Σ̃_{2} → Σ_{2} and f′_{2} : Σ̃′_{2} → Σ_{2} be two concordant branched coverings over Σ_{2}, and g : Σ_{1} → Σ_{2} a simplicial function. Then the pullbacks g^{∗}f_{2} and g^{∗}f′_{2} given in the diagrams below are concordant:

\begin{matrix}Σ̃_{1} & ⟶ & Σ̃_{2} \\ ^{g^{∗}f_{2}}↓ & & ↓^{f_{2}} \\ Σ_{1} & ⟶_{g} & Σ_{2} \end{matrix}  (3.4)

\begin{matrix}Σ̃′_{1} & ⟶ & Σ̃′_{2} \\ ^{g^{∗}f′_{2}}↓ & & ↓^{f′_{2}} \\ Σ_{1} & ⟶_{g} & Σ_{2} \end{matrix}  (3.5)

Lemma 3.3. Let Σ_{1}, Σ_{2} be simplicial complexes and let g_{1} : Σ_{1} → Σ_{2} and g_{2} : Σ_{1} → Σ_{2} be homotopic simplicial functions. Also let f_{2} : Σ̃_{2} → Σ_{2} be a branched covering over the simplicial complex Σ_{2}. Then, the branched coverings g_{1}^{∗}f_{2} and g_{2}^{∗}f_{2} are concordant.

Let Σ be a simplicial complex and denote with B_{k}(Σ) the set of all concordance classes of k-fold branched coverings over Σ.

Proposition 3.4. B_{k}(•) is a contravariant functor from the category of simplicial complexes to the category of sets.

Proof. It comes straightfordwardly from Proposition 3.1 and Lemmas 3.2 and 3.3. □

The notions of pointed branched coverings and of concordance for pointed branched coverings over manifolds are easily extended to the case of simplicial complexes. Similarly, B_{k}(Σ, ∗) denotes the set of all concordance classes of pointed k-fold branched coverings over the simplicial complex Σ. In this context, Proposition 3.1 and Lemmas 3.2 and 3.3 extend to pointed branched coverings over simplicial complexes, and the following proposition is straightforward.

Proposition 3.5. B_{k}(•, ∗) is a contravariant functor from the category of pointed simplicial complexes to the category of sets with distinguished element the constant function.

So far we have proved the existence of the functor B_{k}(•, ∗). Now we proceed to verify Conditions A, B, C and D, of Brown’s Theorem for this functor. Actually, the extension of Lemma 3.3 to the case of pointed branched coverings proves that the functor B_{k}(•, ∗) verifies Condition A, and the first part of Condition B is straightforward. The following propositon establishes the second part of Condition B.

Proposition 3.6. Let (Σ, Σ_{1}, Σ_{2}) be a proper triad in the category of pointed simplicial complexes C. Let A = Σ_{1} ∩ Σ_{2} and j_{i} : A → Σ_{i}, k_{i} : Σ_{i} → Σ be the canonical inclusions. If f_{1} ∈ B_{k}(Σ_{1}, ∗) and f_{2} ∈ B_{k}(Σ_{2}, ∗) are such that the pullbacks j_{1}^{∗}f_{1} and j_{2}^{∗}f_{2} are concordant, then there is f ∈ B_{k}(Σ, ∗) such that k_{i}^{∗}f is concordant to f_{i}, i = 1, 2. Furthermore, if A is a point, then f is unique.

Proof. First of all, let us take subdivisions in A, Σ_{1}, Σ_{2} such that every simplex in A is a simplex in Σ_{1} and in Σ_{2}. Let F : W → A × I be such that F |_{F^{−1}(A×{0})} is equivalent up to homeomorphism to j_{1}^{∗}f_{1}|_{(j_{1}^{∗}f_{1})^{−1}(A)} and F |_{F^{−1}(A×{1})} is equivalent up to homeomorphism to j_{2}^{∗}f_{2}|_{(j_{2}^{∗}f_{2})^{−1}(A)}. Without loss of generality, we may assume that F |_{F^{−1}(A×{0})} and F |_{F^{−1}(A×{1})} are exactly the functions j_{1}^{∗}f_{1}|_{(j_{1}^{∗}f_{1})^{−1}(A)} and j_{2}^{∗}f_{2}|_{(j_{2}^{∗}f_{2})^{−1}(A)}, respectively. Consider the quotient space ((A × I) ∪ Σ_{1})⧸∼ (where a ∼ (a, 0), for all a in A). Call this space X_{1}. Since (Σ_{1}, A) has the homotopy extension property, there exists a homotopy H : Σ_{1} × I → X_{1} such that H|_{X_{1}} = id_{X_{1}}, where id_{X_{1}} is the identity function of X_{1}.

If id_{i} : Σ_{i} → Σ_{i} is the identity function, then id_{i}^{∗}f_{i} coincides with j_{i}^{∗}f_{i} in (j_{i}^{∗}f_{i})^{−1}(A), for i = 1, 2. For i = 1, 2, let us denote (j_{i}^{∗}f_{i})^{−1}(A) and (id_{i}^{∗}f_{i})^{−1}(Σ_{i}) by Ã_{i} and Σ̃_{i}, respectively. Now, consider the function G, G : (W ∪ Σ̃_{1}) ⟶ X_{1} defined by G(x) = F (x) if x ∈ W and G(x) = (id_{1}^{∗}f_{1})(x) if x ∈ Σ̃_{1}; G is well defined, because F coincides with id_{1}^{∗}f_{1} in Ã_{1}.

Let H^{∗}G be the pullback in the diagram

\begin{matrix}(H^{∗}G)^{−1}(Σ_{1} × I) & ⟶ & W ∪ Σ̃_{1} \\ _{H^{∗}G}↓ & & ↓^{G} \\ Σ_{1} × I & ⟶_{H} & X_{1} \end{matrix}  (3.8)

Then, H^{∗}G|_{(H^{∗}G)^{−1}(A×{0})} is equivalent up to homeomorphism to j_{1}^{∗}f_{1} and H^{∗}G|_{(H^{∗}G)^{−1}(A×{1})} is equivalent up to homeomorphism to j_{2}^{∗}f_{2}.

Construct the following space: X = ((H^{∗}G)^{−1}(Σ_{1} × {1}) ∪ Σ̃_{2})⧸∼ where we use the homeomorphism that makes H^{∗}G|_{(H^{∗}G)^{−1}(A×{1})} equivalent to j_{2}^{∗}f_{2} to identify the elements in f_{2}^{−1}(A) with those in (H^{∗}G)^{−1}(A × {1}). Now let us define f :

X → ((Σ_{1} × {1}) ∪ Σ_{2})⧸∼ (where a ∼ (a, 1) for every a ∈ A) by f(x) = H^{∗}G(x) if x ∈ (H^{∗}G)^{−1}(Σ_{1} × {1}) and f(x) = id_{2}^{∗}f_{2}(x) if x ∈ Σ̃_{2}. Clearly f is a well defined branched covering over ((Σ_{1} × {1}) ∪ Σ_{2})⧸∼.

Since ((Σ_{1} × {1}) ∪ Σ_{2})⧸∼ is isomorphic to Σ = Σ_{1} ∪ Σ_{2}, we can say that f ∈ B_{k}(Σ, ∗). Moreover, the construction of f implies that k_{i}^{∗}f is concordant to f_{i}, (i = 1, 2). Hence, f is the desired function.

It is also straightforward that if A reduces to a single point the construction of f is unique up to concordance. □

The next proposition verifies Condition D for the functor B_{k}(•, ∗).

Proposition 3.7. Suppose Σ_{1} ⊂ Σ_{2} ⊂ ⋯ ⊂ Σ_{n} ⊂ ⋯ is a collection of simplicial subcomplexes of Σ = ⋃_{n} Σ_{n} ∈ C with respect to some simplicial-complex structure on Σ such that Σ_{n}^{n} = Σ^{n} (where X^{n} is the n-skeleton of X). Let i_{n} : Σ_{n} → Σ be the inclusion map and \underleftarrow{lim} B_{k}(Σ_{n}, ∗) be the inverse limit of B_{k}(Σ_{n}, ∗) with respect to the maps induced by the inclusions ψ_{m}^{n} of Σ_{n} into Σ_{m}. Then \underleftarrow{lim}(i_{n}^{∗}) : B(Σ, ∗) ⟶ \underleftarrow{lim} B(Σ_{n}, ∗) is an epimorphism.

Proof. First of all, we recall that the inverse limit of B_{k}(Σ_{n}, ∗) is

\underleftarrow{lim} B(Σ_{n}, ∗) = \{(f_{n}) ∈ ∏_{n} B_{k}(Σ_{n}, ∗) | f_{n} = (ψ_{m}^{n})^{∗}f_{m}, m ≤ n\}

(see [7]). Let (f_{n}) ∈ \underleftarrow{lim} B_{k}(Σ_{n}, ∗). We want to find f ∈ B_{k}(Σ, ∗) such that \underleftarrow{lim}(i_{n}^{∗})(f) = (f_{n}).

Notice that f_{1} = (ψ_{1}^{2})^{∗}f_{2}; in other words, that f_{1} and the restriction f_{2}|_{Σ_{1}} are concordant. Using this concordance we construct a function f′_{2} ∈ B_{k}(Σ_{2}, ∗) such that f′_{2}|_{(f′_{2})^{−1}(Σ_{1})} = f_{1}. Inductively, we construct f′_{n} ∈ B_{k}(Σ_{n}, ∗) such that f′_{n}|_{(f′_{n})^{−1}(Σ_{n−1})} = f′_{n−1}. Hence, we have a collection of functions f_{1} = f′_{1}, f′_{2}, f′_{3}, … such that f′_{n} is an extension of f′_{n−1}, for all n in ℕ.

Let Σ̃ = ⋃_{n} Σ̃_{n}, where Σ̃_{n} = (f′_{n})^{−1}(Σ_{n}), and define f : Σ̃ → Σ by f(x) = f′_{n}(x) if x ∈ (f′_{n})^{−1}(Σ_{n}). It is obvious that f is well defined. Therefore, f ∈ B_{k}(Σ, ∗). Because of the construction of f′_{n}, we have that f′_{n} is concordant to f_{n}. Therefore, \underleftarrow{lim}(i_{n}^{∗})(f) = (f_{n}), which implies that

\underleftarrow{lim}(i_{n}^{∗}) : B(Σ,^{∗}) ⟶ \underleftarrow{lim} B(Σ_{n}, ∗)

is an epimorphism. □

The following proposition proves condition C for the functor B_{k}(•, ∗). Here we regard the boundary of an (n + 1)–simplex as an n–sphere.

Proposition 3.8. Suppose {S_{α}^{n}} is a collection of disjoint n-spheres whose wedge product ⋁ S_{α}^{n} is in the category of simplicial complexes C. Let i_{β} : S_{β}^{n} → ⋁ S_{α}^{n} be the inclusion map. Then the function

∏ i_{β}^{∗} : B_{k}(⋁ S_{α}^{n}, ∗) → ∏ B_{k}(S_{α}^{n}, ∗)

is bijective.

Proof. Let (f_{α}) ∈ ∏ B_{k}(S_{α}^{n}, ∗), where f_{α} ∈ B(S_{α}^{n}, ∗). Recall that ∗ never is a branched point of f_{α} and that there is a one-to-one correspondence of {1, 2, …, k} with f_{α}^{−1}(∗). This correspondence is denoted by c(f_{α})_{i}, for every α.

Let X = (⋃_{α} f_{α}^{−1}(S_{α}^{n}))⧸∼, where x_{α} ∼ x_{β} if f_{α}(x_{α}) = f_{β}(x_{β}) = ∗ and the corresponding i ∈ {1, 2, …, k} under c(f_{α})_{i} and c(f_{β})_{i} is the same for both elements.

If f : X → ⋁ S_{α}^{n} is such that f(x) = f_{α}(x) for x ∈ f_{α}^{−1}(S_{α}^{n}), f is a well defined branched covering in B_{k}(⋁ S_{α}^{n}, ∗). Moreover, ∏ i_{β}^{∗}(f) = (f_{α}). Therefore ∏ i_{β}^{∗} is surjective.

Let us prove the injectivity. Let f, g ∈ B_{k}(⋁ S_{α}^{n}, ∗) and assume that ∏ i_{β}^{∗}(f) = (f_{α}) and ∏ i_{β}^{∗}(g) = (g_{α}). Here (f_{α}) = (g_{α}) means that f_{α} is concordant to g_{α} for each α. So, there exists a branched covering F_{α} : W_{α} → S_{α}^{n} × I such that F_{α}|_{F_{α}^{−1}(S_{α}^{n}×{1})} is equivalent up to homeomorphism to f_{α} and F_{α}|_{F_{α}^{−1}(S_{α}^{n}×{0})} is equivalent up to homeomorphism to g_{α}. Recall that ∗ × I does not intersect the branch set. Therefore F_{α}^{−1}(∗ × I) contains exactly k copies of I, and c(f_{α})_{i} belongs to the same component of F_{α}^{−1}(∗ × I) that contains c(g_{α})_{i}. Let us denote by F_{α}^{−1}(∗ × I)_{i} the copy of I that contains c(f_{α})_{i}. Without loss of generality, we may assume that if α ≠ β then W_{α} ≠ W_{β}. Let X̃ = (⋃_{α} W_{α})⧸∼, where ∼ is defined in the following way: x ∼ y if there are α, β and i (i = 1, …, k) such that x ∈ F_{α}^{−1}(∗ × I)_{i}, y ∈ F_{β}^{−1}(∗ × I)_{i}, and F_{α}(x) = F_{β}(y). Consider F : X̃ → ⋁ S_{α} × I defined by F (x) = F_{α}(x) if x ∈ W_{α}. Clearly F is a well defined function such that F |_{F^{−1}(⋁ S_{α}^{n}×{1})} is equivalent up to homeomorphism to f and F |_{F^{−1}(⋁ S_{α}^{n}×{0})} is equivalent up to homeomorphism to g. Hence, f is concordant to g, i.e., ∏ i_{β}^{∗} is injective. □

Now recall that every CW-complex is homotopically equivalent to a simplicial complex, i.e., that if X is a CW-complex there exist a simplicial complex Σ and maps h : X → Σ, h′ : Σ → X such that h ∘ h′ is homotopic to the identity of Σ and h′ ∘ h is homotopic to the identity of X. We will say that a function f : X̃ → X is a k-fold branched covering of type l over the CW-complex X, if the function (h′)^{∗}f : Σ̃ → Σ is a k-fold branched covering over the simplicial complex Σ. In a natural way we extend all definitions, Lemmas and propositions given for (resp. pointed) branched coverings over simplicial complexes to (resp. pointed) branched coverings over CW-complexes. Actually, we prove that if C is the category of CWcomplexes (not necessarily finite), the functor B_{k}(•, ∗) satisfies Conditions A, B, C, D of Brown’s Representability Theorem. Hence, the following theorem holds.

Theorem 3.9. Let C be the category of spaces with base point ∗ which has as objects all topological spaces admitting a CW-complex structure. Let S be the category of sets with a distinguished element. Let B_{k}(•, ∗) : C → S be the functor defined earlier. Then, there is a space Y in C, unique up to homotopy type, such that the functors [•, Y] and B_{k}(•, ∗) are naturally equivalent.

In other words, we have obtained the existence of a classifying space Y for k-fold branched coverings over CW-complexes.

ACKNOWLEDGMENT. The author is grateful to Neal Brand for many suggestions that greatly improved this paper.

References

1. J. W. Alexander, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920), 370–372.

2. N. Brand, Homotopy theory of branched coverings, Thesis, Stanford University, 1978.

3. N. Brand and D. M. Tejada, Construction of Universal Branched Coverings, to appear.

4. E. H. Brown Jr., Cohomology theories, Ann. of Math. 75 (1962), 467–485.

5. M. Greenberg and J. R. Harper, Algebraic Topology. A first course, Addison-Wesley Publishing Company, Redwood City, California, 1981.

6. V. Guillemin and A. Pollack, Differential Topology, Prentice Hall, Englewood Cliffs, New Jersey, 1974.

7. J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979.

8. E. Spanier, Algebraic Topology, Springer-Verlag, New York, 1982.

9. D. M. Tejada, Universal Branched Coverings, Ph.D. thesis, University of North Texas, 1993.

(Recibido en enero de 1994)

Débora M. Tejada

Departamento de Matemáticas Universidad Nacional de Colombia

Apartado Aéreo 3840, Medellín, COLOMBIA e–mail: jcossio@sigma.eafit.edu.co