Universal Principal Bundles And Classifying Spaces

Universal Principal Bundles And Classifying Spaces

Mehdi Nabil

Cadi Ayyad University, Morocco

Abstract

The content of this document is intended to be complementary notes to the course of Pr. Abdelhak Abouqateb presented as part of the 2018 international conference "Ecole Mathématique Africaine" (EMA) at Rabat Morocco. In these few notes, we treat the theory of principal bundles from the perspective of algebraic topology.

Keywords:

Universal Bundles, Classifying Spaces, Principal Bundles, Vector Bundles, CW-complex, Grassmannian, Stiefel, Manifolds, ...

1. Introduction

LetX be a compact Hausdorff space. Swan’s theorem [Swa62] states that any rankkvector bundleE −→^π X is a subbundle of the trivial vector bundle X ×Rⁿ for some n ∈ N. There is another way of stating this result in the following manner : For everyp∈X, the fiberE_p is ak-dimensional subspace of Rⁿ, thus we can construct a mapf :X −→Gr_R(k, n)given byf(p) =E_p, we can check that f is continuous. Now if we define :

E_k,n(R) ={(`, v)∈Gr<sub>R</sub>(k, n)×R, v ∈` }.

We can check that Ek,n(R) is a rank k vector bundle over Gr_R(k, n), fur- thermore Swan’s theorem is equivalent to claiming that E ' f^∗(E_k,n(R)).

Now this is far from being a classification result since the integer n depends on many factors, namely the space X, the rank k and the vector bundle E (i.e n=n(k, X, E)), to resolve this it is sufficient to remark that when E is a vector subbundle of X×Rⁿ then it is a subbundle of X×R^m for every

m ≥n. So we define the "infinite" GrassmannianGr_R(k,∞)as the collection of k-dimensional subspaces ofR^∞, and sinceE is a subbundle ofX×R^∞ we define a continuous mapf :X −→Gr_R(k,∞),p7→Ep. Now define the rank k vector bundle E_k(R)over Gr_R(k,∞) by the formula :

E_k(R) = {(`, v)∈Gr<sub>R</sub>(k,∞)×R, v ∈` },

we obtain as before that E ' f^∗(E_k(R)), the complex vector bundle case is exactly the same. Now if Eˆ −→Y is a vector bundle, it is well known that homotopic maps f, g : X −→ Y give rise to isomorphic vector bundles, in summary we get the following statement :

For every k ∈ N there exists a rank k vector bundle Ek(R) −→ Gr_K(k,∞) (K = R or C) satisfying the following property : For every rank k vector bundle E −→ X over a compact Hausdorff space, there exists a homotopy class f :X −→Gr_K(k,∞) such that E ' f^∗(Ek(R)), in other words we get that the map :

[X,Gr_K(k,∞)]−→Vect_k(X,R), [f]7→[f^∗(E_k(R))]

is surjective. We can show in fact that this is a bijective correspondance.

Since there is a bijective correspondance between vector bundles of rank k with structure group G⊂ GL(k,R) and principal G-bundles, the preceding statement can be reformulated as follows :

For every matrix group G, there exists a principal G-bundleEG−→BGsat- isfying the following property : For every principal G-bundleP −→X over a compact Hausdorff space, there exists a unique homotopy class f :X −→BG such that P 'f^∗(EG), in other words we get that the map :

[X, BG]−→Prin_G(X), [f]7→[f^∗(EG)]

is a bijective correspondance.

Now the natural question to ask is whether this classification theorem ex- tends to the case whenGis any topological group. The answer is affirmative for CW-complexes and the rest of this course is about proving this claim, the proof will not rely on vector bundles, in fact the classification theorem for vector bundles will be obtained as a consequence which will be discussed in the final section.

2. Homotopy properties of fiber bundles

We begin by a quick review of the homotopy invariance of the pullback operation, the most important result of the section is the following theorem due to Steenrod :

Theorem 2.0.1 (Covering Homotopy theorem). LetE −→^π B and Eˆ −→^ˆπ Bˆ two fiber bundles with the same fiber type F and consider a bundle map f˜ : E −→ Eˆ over some map f : B −→ Bˆ (i.e πˆ ◦f˜ = f ◦π). Suppose that H : B ×I −→ Bˆ is a homotopy such that H₀ = f, then there exists a homotopy H˜ :E×I −→Eˆ whose induced homotopy is H, i.e :

E×I Eˆ

B×I Bˆ

(π,Id)

H H˜

ˆ π

(1) Moreover H˜₀ = ˜f.

The rest is just a consequence of the Covering homotopy theorem : Theorem 2.0.2. LetE −→^π B be a fiber bundle with fiber typeF and suppose that f₀ :X−→B and f₁ :X −→B are homotopic maps. Then the pullback bundles are isomorphic, i.e f₀^∗(E)'f₁^∗(E).

Proof. Define a homotopyH :X×I −→Bbetweenf₀andf₁. Since the fiber bundles f₀^∗(E) −→^pr2 X and E −→^π B have the same fiber type, the covering homotopy theorem gives that there exists a map H˜ : f₀^∗(E)×I −→E such that π◦H˜ =H◦pr₂, from the universal property of the pullback bundle,H˜ can be seen as a fiber bundle homomorphism :

f₀^∗(E)×I H^∗(E)

X×I X×I

Id H˜

(2)

To check that H˜ : f₀^∗(E)×I −→ H^∗(E) is an isomorphism, it suffices to notice that H˜ induce the identity when restricted to the fibers. To conclude, we remark that H(f˜ ₀^∗(E)× {1}) =f₁^∗(E).

It is worth to mention that when f₀, f₁ : X −→ B are homotopic maps andE −→^π Bis a principalG-bundle and then the pullbacksf₀^∗(E)andf₁^∗(E) are isomorphic as principal G-bundles. Similarly, if E is a vector bundle we get that the pullbacks f₀^∗(E) and f₁^∗(E) are isomorphic as vector bundles.

Corollary 2.0.1. Let E −→^π B be a fiber bundle and suppose that B is a contractible space. Then E is trivial.

3. CW-complexes

We begin by recalling the notion of an adjunction space. Let X, Y be two topological spaces, A⊂X a closed subset andf :A−→Y a continuous map. Define on the disjoint union X tY the equivalence relation : for all x∈X and y∈Y,

x∼y if and only if x∈Aandy=f(x).

The adjunction space of X and Y (viaf) is the topological space XtY /∼ endowed with the quotient topology which we denote by X ∪_f Y. Denote p:XtY −→X∪_f Y the canonical projection.

Proposition 3.0.1. We have the following properties :

1. The canonical projection induce an imbedding p|Y : Y −→ X ∪_f Y of Y onto a closed subspace and an imbedding p|X\A :X\A −→X∪_f Y of X\A onto an open subspace.

2. The adjunction space X∪_f Y is a Hausdorff space whenever X and Y are Hausdorff spaces.

3. If X and Y are normal spaces, then X∪_f Y is a normal space.

We turn now to the notion of CW-complex. Basically, given a family of topological spaces X⁰ ,−→ X¹ ,−→ . . . ,−→ Xⁿ ,−→ . . ., the colimit of such a family is the set X = ∪n∈NXⁿ endowed with the weak topology : A subset U ⊂ X is open if and only if U ∩Xⁿ is open in Xⁿ for alln ∈N. We write X = colim

n→+∞ Xⁿ. Now a CW-complex is just a colimit space X = colim

n→+∞ Xⁿ where the spaces Xⁿ are defined inductively in the following manner :

1. We start with a discrete spaceX⁰.

2. Suppose thatXⁿ⁻¹ is defined, then choose a family of attaching maps ϕ_α :∂D_αⁿ⊂Dⁿ_α −→Xⁿ⁻¹ and put Xⁿ := (t_αDⁿ_α) ∪

ϕα, αXⁿ⁻¹.

For all n ∈ N, let p_n : (t_αDⁿ_α)tXⁿ⁻¹ −→ Xⁿ be the canonical projection.

An n-dimensional cell in X is the embedded imageeⁿ_α= (ι_n◦p_n)(Dⁿ_α\∂D_αⁿ) where ι_n : Xⁿ −→ X is the inclusion of Xⁿ in X. To each cell eⁿ_α we can associate a characteristic map φⁿ_α : D_αⁿ −→ X which is by definition the composition :

Dⁿ_α ,−→^inc (t_βDⁿ_β)tXⁿ⁻¹ −→^pn Xⁿ,−→^ιn X.

This is clearly a continuous map, and it is straightforward to check that it defines a homeomorphism int(Dⁿ_α)−→eⁿ_α and that φⁿ_α|∂Dn

α =ιn−1◦ϕⁿ_α. Proposition 3.0.2. A set A ⊂ X is open (resp. closed) if and only if the set (φⁿ_α)⁻¹(A) is open (resp. closed) in D_αⁿ for every characteristic mapφⁿ_α. Proof. The continuity of the characteristic maps φⁿ_α assures that (φⁿ_α)⁻¹(A) is open for every open set A ⊂ X. Conversely, let A ⊂ X and assume that (φⁿ_α)⁻¹(A) is open in D_αⁿ for every characteristic map φⁿ_α, we will show inductively that A∩X^k is open in X^k for all k ∈N.

For k = 0, A∩X⁰ is immediately open in X⁰ since X⁰ is a discrete space.

Now let k ∈ N^∗ and suppose that A∩X^k−1 is open in X^k−1, now denote p_k : (t_αD^k_α)tX^k−1 −→X^k the canonical projection, then :

p⁻¹_k (A) = (t_α(φ^k_α)⁻¹(A))t(A∩X^k−1)

which is an open set of (t_αD_α^k)tX^k−1 by hypothesis. Thus A∩X^k is open in X^k. We conclude that A is open inX.

Many important spaces admit a CW-complex structure, we state some of the in the following theorem :

Theorem 3.0.1. Any smooth manifold admits a CW-complex structure.

Similarly, any closed topological manifold of dimension 6= 4 admits a CW- complex structure.

In general, an arbitrary topological space may not admit a CW-complex structure and might not even be homotopy equivalent to a CW-complex.

However there is still a way of "approaching" topological spaces of interest

from the point of view of homotopy theory which we now present :

Given a map of topological spaces f : X −→ Y, we say that f is a weak homotopy equivalence if the induced morphism f^∗ : π_n(Y) −→ π_n(X) is an isomorphism (clearly any homotopy equivalence is a weak homotopy equiv- alence). Now a CW-approximation of a space Y is just a weak homotopy equivalence f : X −→ Y such that X is a CW-complex. The existence of CW-approximations is guaranteed by the following theorem :

Theorem 3.0.2. Any topological space admits a CW-approximation.

We call a subcomplex of a CW-complexX any subspace A⊂X which is a unions of cells of X such that the closure of each cell eⁿ_α ⊂A is contained inA. This induce a CW-complex structure onAitself. The obvious example of a subcomplex of X are the n-skeletons Xⁿ, to see this notice that Xⁿ is closed in X thus given a k-dimensional cell e^k_α of X with k ≤n, its closure must be inXⁿ. SinceXⁿ is the union of such cells, it is a subcomplex ofX.

(In fact Xⁿ is the maximal n-dimensional subscomplex of X).

A pair (X, A)consisting to a CW-complex X and a subscomplex A is called a CW-pair. Finally a subcomplex A⊂X is said to be finite if it is the union of a finite number of cells of X.

4. Universal principal bundles and classifying spaces

We begin by recalling some properties of principal bundles. A morphism of principal G-bundles P₁ −→^π1 B₁ and P₂ −→^π2 B₂ is any G-equivariant map φ :P₁ −→P₂. Any morphism of principal G-bundles φ :P₁ −→ P₂ lies over a map φ˜:B₁ −→B₂, i.e the following diagram commutes :

P₁ P₂

B₁ B₂

φ

π1 π2

φ˜

(3)

The map φ˜ : B1 −→ B2 is uniquely defined by φ(b) =˜ π2(φ(p)) where b = π₁(p). It is well-defined since given p, q ∈ π₁⁻¹(b) we have that q = p·g for some g ∈G and thus :

π₂(φ(q)) = π₂(φ(p·g)) =π₂(φ(p)·g) =π₂(φ(p)).

To check that it is continuous, choose an open neighborhood U of b in B₁ over which P₁ −→^π1 B₁ is trivial and a local section s₁ :U −→ P₁. Then we can easily check that φ˜|U =π2◦φ◦s1, hence φ˜is continuous on U and since U was arbitray we obtain thatφ˜is continuous on B₁.

Proposition 4.0.1. Let φ:P₁ −→P₂ be a morphism of principalG-bundles P₁ −→^π1 B and P₂ −→^π2 B lying over the identity Id : B −→B. Then φ is an isomorphism.

Proof. Letp, q ∈P₁ such that φ(p) =φ(q). Since π₂◦φ =π₁ we obtain that π₁(p) =π₁(q)thus q=p·g for some g ∈Gand consequently

φ(q) = φ(p)·g =φ(p).

The group G acts freely on P₂, hence g =e_G and p=q. This shows that φ is injective. For surjectivity, take p₂ ∈ P₂ and put b = π₂(p₂), now choose any p∈π₁⁻¹(b) then we have that :

π₂(φ(p)) =π₁(p) = b=π₂(p₂).

This gives that φ(p) =p₂ ·g for some g ∈ G. Now putting p₁ = p·g⁻¹ we obtain that φ(p₁) = φ(p)·g⁻¹ = p₂ which shows that φ is surjective. It is only left to show that φ⁻¹ : P₂ −→ P₁ is continuous. For this we consider local trivializations ϕ₁ : U ×G −→ π₁⁻¹(U) and ϕ₂ : U ×G −→ π₂⁻¹(U).

Since φ is surjective and π₂◦φ = π₁, we obtain that φ π₁⁻¹(U)

=π₂⁻¹(U).

Thus the composite map :

ϕ⁻¹₂ ◦φ◦ϕ1 :U ×G−→U ×G

is a well-defined bijective morphism of principal G-bundles lying over the identity of U, hence it is automatically of the form :

(ϕ⁻¹₂ ◦φ◦ϕ₁)(x, g) = (x, τ(x, g)) = (x, τ(x, e)·g),

where τ :U ×G−→G is a G-equivariant map. It is now straightforward to check that the inverse map ϕ⁻¹₁ ◦φ⁻¹◦ϕ₂ :U ×G−→U×G is given by :

(ϕ⁻¹₁ ◦φ⁻¹◦ϕ₂)(x, g) = (x, τ(x, e)⁻¹·g),

which is clearly continuous since G −→ G , g 7→ g⁻¹ is continuous. We conclude that φ⁻¹ is continuous onπ₂⁻¹(U), and since U is arbitrarily chosen, it is globally continuous.

Recall that if P −→^π B is a principal G-bundle and F is a left G-space endowed with a topological action ρ:G−→Aut(F), then we can define the associated fiber bundle P ×_GF −→^πF B to P by ρ where π_F([p, f]) = π(p) (the space P ×_G F is the quotient of P × F by the equivalence relation (p, f)∼(p·g, g⁻¹·f)). WhenP −→^π B is trivial, then P×GF −→^πF B is also trivial.

Proposition 4.0.2. Let P −→^π B be a principal G-bundle, F a left G-space and E =P ×_GF the associated bundle. For any open set U ⊂B, there is a bijective correpondance Γ(U, E)←→Map(π⁻¹(U), F)^G.

Proof. Letφ:π⁻¹(U)−→F be aG-equivariant map and defines_φ :U −→E by the formula s_φ(b) = [p, φ(p)]. Notice that s_φ is well defined, since given p, q ∈π⁻¹(b) we can write q=p·g and thus :

[q, φ(q)] = [p·g, φ(p·g)] = [p·g, g⁻¹·φ(p)] = [p, φ(p)].

It is clear that πF ◦sφ= IdU, to check that sφ is continuous we consider an open set V ⊂U over whichP −→^π B is trivial and a local sectionσ_V :V −→

P. Hence for all b ∈ V, s_φ(b) = [σ_V(b),(φ◦σ_V)(b)] which shows that s_φ|V is continuous, since V ⊂U is arbitrary, sφ is continuous on U. In summary, s_φ∈Γ(U, E) and we have constructed a map :

Map(π⁻¹(U), F)^G −→Γ(U, E), φ 7→s_φ.

We show in what follows that this map is bijective by constructing its inverse : Choose s∈Γ(U, E)and define φ_s:π⁻¹(U)−→F by the formula :

φ_s(p) = f whens(π(p)) = [p, f].

To see that this if well-defined write s(π(p)) = [p, f1] = [p, f2], then(p, f1) = (p·g, g⁻¹·f₂) and since the action of G on P is free we obtain that g =e_G and f₁ =f₂. Hence we have that :

s(π(p)) = [p, φ_s(p)].

Furthermore for any g ∈ G and any p ∈ π⁻¹(U) we have that s(π(p·g)) = [p·g, φs(p·g)] and s(π(p)) = [p, φs(p)], sinceπ(p·g) =π(p) we obtain that

[p·g, φ_s(p·g)] = [p, φ_s(p)].

Thus there exists h ∈ G such that (p· g, φ_s(p·g)) = (p· h, h⁻¹ · φ_s(p)), again because the action of G on P is free we obtain that g = h and φ_s(p· g) = g⁻¹ ·φs(p) which gives that φs is G-equivariant. Now we check that φ_s : π⁻¹(U) −→ F is continuous : Consider a local trivialization ϕ : V × G −→ π⁻¹(V) of P −→^π B over V ⊂ U, this induce a local trivialization ϕF :V ×F −→E|V given by :

ϕ_F(b, f) = [ϕ(b, e), f].

Now write ϕ⁻¹(q) = (π_F(q), τ_F(q) where τ_F : E|V −→ F is a G-equivariant map, thus we obtain that :

(b, f) =ϕ⁻¹_F [ϕ(b, e), f] = ((π◦ϕ)(b, e), τ_F[ϕ(b, e), f]) = (b, τ_F[ϕ(b, e), f]).

Thus for every f ∈F we have thatf =τF[ϕ(b, e), f]. Hence we get that : (τ_F ◦s)(b) = (τ_f ◦s◦pr₁)(b, e) = (τ_F ◦s◦π◦ϕ)(b, e)

= τ_F[ϕ(b, e),(φ_s◦ϕ)(b, e)]

= (φs◦ϕ)(b, e).

Thus for every b∈V and g ∈G :

(φ_s◦ϕ)(b, g) =φ_s(ϕ(b, e)·g) =g⁻¹·(φ_s◦ϕ)(b, e) = g⁻¹·(τ_F ◦s)(b).

This gives that φ_s◦ϕ is continuous on V ×G, hence φ_s is continuous on π⁻¹(V). SinceV ⊂U is arbitrary we conclude thatφ_s is globally continuous.

We have thus constructed a map :

Γ(U, E)−→Map(π⁻¹(U), F)^G, s7→φ_s.

It is straightforward to check that it is the inverse of the first map.

An important consequence is the following :

Corollary 4.0.1. Given two principal G-bundles P −→^π B and P^{0 π}

0

−→ B⁰. There is a bijective correspondance Mor_G(P, P⁰)←→Γ(B, P ×_GP⁰).

Proof. It is sufficient to notice that Mor(P, P⁰) = Map(P, P⁰)^G and use the preceding proposition.

We recall that if P −→^π B is a principal G-bundle and f : X −→ B is a map then the pullback bundle f^∗(P) −→ X is a principal G-bundle.

Also, if f0, f1 : X −→ B are homotopic maps, then there is a principal G- bundle isomorphism f₀^∗(P)' f₁^∗(P). Thus, if we denote [X, B] the space of homotopic classes X −→B, there is a well-defined map

[X, B]−→Prin_G(X) [f]7→[f^∗P].

In what follows, we treat the following question :

Question: For which conditions on X and P −→^π B the map [X, B] −→

Prin_G(X) is a bijection ?

Recall that a space F is said to be weakly-contractible when π_n(F) = 0 for all n ∈N.

Definition 4.0.1. A principal G-bundle EG−→^π BG is said to be universal if the total space EG is weakly contractible.

We will show that whenX admits a CW-complex structure andEG−→^π BG is a universal principalG-bundle, the [X, BG] −→Prin_G(X) is a bijec- tion. We start with an important lemma :

Lemma 4.0.1. Let(B, A) be a CW-pair andE −→^π B a fiber bundle with fi- breF. Suppose thatπ_k(F) = 0wheneverB\Acontains a (k+1)-dimensional cell, then every section s ∈ Γ(A, E) can be extended to a global section

˜

s ∈ Γ(B, E). In particular, when A = ∅ and F is weakly contractible, the fiber bundle E −→^π B admits a global section.

Proof. We will show inductively that the section s_|Ak :A^k −→E can be ex- tended to a section ˜s^k:B^k−→E. Fork = 0, it is clear that any extension of s|A⁰ is automatically continuous sinceB⁰ is a discrete space. Let k ∈N and suppose that s_|Ak : A^k −→ E can be extended to a section ˜s^k ∈ Γ(B^k, E).

We denote an arbitrary (k + 1)-dimensional cell in B by e^k+1_α and we put C₁^k+1 ={α, e^k+1_α ⊂A} and C₂^k+1 ={α, e^k+1_α ⊂ B\A}. We distinguish two situations :

1. Either C₂^k+1 = ∅ (there are no (k + 1)-dimensional cells in B \A), then we define sˆ^k+1 : (tαD_α^k+1)tB^k−→E in the following manner :

ˆ

s^k+1 = ˜s^konB^k and sˆ^k+1 =s◦φ^k+1_α onD_α^k+1,

where φ^k+1_α : D_α^k+1 −→ B is the characteristic map of e^k+1_α which satis- fies φ^k+1_α (D^k+1_α ) ⊂ A. It is clear that ˆs^k+1 is continuous, to show that it induce a continuous map on B^k+1 we choose an attaching map ϕ^k+1_α :

∂D_α^k+1 −→ B^k (the map ϕ^k+1_α is just φ^k+1_{α |∂D}^k+1

α ), we easily check that ˆ

s^k+1(x) = ˆs^k+1(ϕ^k+1_α (x))for everyx∈∂D^k+1_α . Thusˆs^k+1induce a continuous map ˜s^k+1 :B^k+1 −→E. It is straightforward to check thatπ◦˜s^k+1 = Idand that s˜^k+1_|Ak+1 =s_|Ak+1, thus s˜^k+1 :B^k+1 −→E is a section that extends s_|Ak+1. 2. Suppose now that C₂^k+1 6= ∅. Let e^k+1_α be a (k + 1)-dimensional cell contained in B \A and denote φ^k+1_α : D^k+1_α −→ B its characteristic map.

Since φ^k+1_α (∂D_α^k+1)⊂B^k, then the composition map :

˜

s^k_α := ˜s^k◦φ^k+1_α :∂D_α^k+1 −→E is well-defined and satisfies π ◦ s˜^k_α = φ^k+1_{α |∂D}k+1

α , thus it defines a section of the pullback bundle (φ^k+1_{α |∂D}^k+1

α )^∗(E) over ∂D_α^k+1. Now since D_α^k+1 is a contractible space, we get that the pullback bundle (φ^k+1_α )^∗(E)−→ D^k+1_α is trivial, i.e (φ^k+1_α )^∗(E)'D_α^k+1×F. Thus (φ^k+1_{α |∂D}k+1

α )^∗(E)−→∂D_α^k+1 is also trivial as shown in the following commutative diagram :

∂D^k+1_α ×F D_α^k+1×F E

∂D^k+1_α D_α^k+1 B

ι

pr₁

pullback

pr₁

ι φα

(4)

Hences˜^k_α can be seen as a mapx7→(x,˜τ_α^k(x))whereτ˜_α^k:∂D^k+1_α −→F. Since by hypothesis π_k(F) = 0, we obtain that˜τ_α^k can be extended to a continuous map τˆ_α^k+1 :D^k+1_α −→F and thus s˜^k_α can be extended to a continuous section ˆ

s^k+1_α :D^k+1_α −→(φ^k+1_α )^∗(E). Finally we define ˆs^k+1 : (tααD_α^k+1)tB^k−→E in the following manner :

ˆ s^k+1 =











s◦φ^k+1_α onD^k+1_α forα∈C₁^k+1 ˆ

s^k+1_α onD^k+1_α forα∈C₂^k+1

˜

s^k onB^k

It is clear that sˆ^k+1 is continuous and we check as before that for every x∈∂D^k+1_α we have ˆs^k+1(x) = ˆs^k+1(ϕ^k+1_α (x)). Thus ˆs^k+1 induce a continuous map s˜^k+1 :B^k+1 −→E and it is straightforward to check that this is indeed a section that extends s_A^k+1 :A^k+1 −→E. This achieves the proof.

We are now ready to prove the main theorem :

Theorem 4.0.1. Suppose thatEG−→^π BGis a universal principalG-bundle and X is a CW-complex. Then the correspondance [X, BG] −→ Prin_G(X) given by [f]7→[f^∗(EG)] is bijective.

Proof. We start by showing that the map is surjective, so consider a principal G-bundle P −→^π X over X, we must construct a map f : X −→ BG such that P ' f^∗(EG). This is equivalent to the construction of G-equivariant map over the identity map, i.e :

P f^∗(EG)

X X

'

πP

Id

(5)

which is in turn equivalent to finding a G-equivariant map φ : P −→ EG and putting f : X −→BG the induced base-map as shown in the following diagram :

P EG

X BG

φ

πP

f

(6)

But this is the same (according to corollary (4.0.1)) as finding a section of the associated bundleP ×GEG−→X, the preceding lemma guarantees the existence of such a section since EG is weakly contractible. This proves the surjectivity.

To prove the injectivity, let f₀, f₁ : X −→ BG be two maps with isomor- phic pullbacks, i.e f₀^∗(EG)' f₁^∗(EG), we will show that f₀ and f₁ must be homotopic. The map f0 : X −→ BG induce a morphism of principal G- bundles via the pullback diagram and the isomorphism f₀^∗(EG) ' f₁^∗(EG) is equivalent to morphism of principal G-bundles φ₁ :f₀^∗(EG) −→EG over f1 :X −→BG, this can be summarized in the following diagrams :

f₀^∗(EG) EG f₀^∗(EG) EG

X BG X BG

φ0

π0 π

φ1

π0 π

f0 f1

(7)

Denote s₀, s₁ : X −→ f₀^∗(EG) ×_G EG the corresponding sections, which according to the proof of proposition (4.0.2) are defined by the relation :

s₀(π₀(Z)) = [z, φ₀(z)] and s₁(π₀(z)) = [z, φ₁(z)]. (8) Put P :=f₀^∗(EG)×I and X_t = X× {t}. We can view s_i as a section s_i ∈ Γ(X_i, P×_GEG)fori= 0,1. Thus we can defines₀ts₁ ∈Γ(X₀tX₁, P×_GEG) by the formula :

(s₀ts₁)_|Xi =s_i, i= 0,1.

Now since (X ×I, X₀ tX₁) is a CW-pair and EG is weakly contractible, we obtain according to the preceding lemma that s₀ ts₁ can be extended to a section s ∈ Γ(X×I, P ×_GEG). Denote φ : P −→ EG the morphism corresponding to s via the relation s(π(z), t) = [(z, t), φ(z, t)] and consider the commutative diagram :

P EG

X×I BG

φ1

(π0,Id) π

F

(9)

Since by definition s(π₀(z),0) = s₀(π₀(z)) and s(π₀(z),1) = s₁(π₀(z)), we obtain from relation (8) that :

φ(z,0) =φ₀(z) and φ(z,1) = φ₁(z).

Thus for all x∈X, we have that F(x,0) =f₀(x) and F(x,1) = f₁(x) which gives that f₀ and f₁ are homotopic maps.

The space BG will be called a classifying space for the group G, and if P −→^π X is a principal G-bundle, any map f : X −→ BG such that P 'f^∗(EG) will be called aclassifying map for P.

Now we address the uniqueness of the universal principal G-bundle. We start with the statement of a popular theorem calledthe long exact sequence of a fibration :

Theorem 4.0.2. Let E −→^π B be a fiber bundle with fiber type F. Then there exists a long exact sequence of homotopy groups :

· · · −→π_n(F)→π_n(E)→π_n(B)→πn−1(F)→ · · · →π₀(F)→π₀(E)→π₀(B).

A first consequence of this theorem is that BG can be taken to have a relatively simple topological structure :

Corollary 4.0.2. The classifying space BG can be taken to have a CW- complex structure.

Proof. Let EG −→^π BG be any universal principal G-bundle and consider a CW-approximation φ : BG⁰ −→ BG of BG. Then we have a pullback diagram :

φ^∗(EG) EG

BG⁰ BG

pr₂

pr₁ π

F

(10)

To be able to show that BG can be replaced by the CW-complex BG⁰, one must show that the principal G-bundles φ^∗(EG) −→^pr1 BG⁰ is universal, i.e φ^∗(EG) is a contractible space. This can be done by considering the long exact homotopy sequences stated in the previous theorem which along with the pullback diagram gives the following commutative diagram :

. . . πn(G) πn(EG) πn(BG) πn−1(G) . . .

. . . π_n(G) π_n(φ^∗EG) π_n(BG⁰) πn−1(G) . . .

' pr^∗₂ φ^∗ ' (11)

Since φ^∗ : π_n(BG) −→ π_n(BG⁰) is an isomorphism, we obtain by the five lemma that πn(EG)'πn(φ^∗EG) for alln ∈N, which shows that φ^∗(EG)is weakly contractible and achieves the proof.

We suppose in what follows that the classifying spaceBGalways admits a CW-complex structure. We now state the "uniqueness" theorem for universal principal G-bundles.

Theorem 4.0.3. Let EG−→BG and E⁰G−→B⁰G be two universal prin- cipal G-bundles. There exists a homotopy equivalence B⁰G −→ BG that is covered by aG-equivariant homotopy equivalenceE⁰G−→EG. In this sense, universal principal G-bundles are unique up to homotopy equivalence.

Proof. Choose two classifying maps f : B⁰G −→ BG and g : BG −→ B⁰G such that E⁰G ' f^∗(EG) and EG ' g^∗(E⁰G). Then the composite map f ◦g :BG−→BG is a classifying map of EG itself since :

(f ◦g)^∗(EG)'g^∗(f^∗(EG))'g^∗(E⁰G)'EG.

Thus by the injectivity of the map [BG, BG]−→Prin_G(BG) we obtain that f◦g must be homotopic to Id_BG. Similarly, we show thatg◦f is homotopic to Id_B⁰_G. Hence f :B⁰G−→BG is a homotopy equivalence.

We thus have a well-defined correspondence G7→BG from the category of topological groups to the category of homotopy classes of CW-complexes, which is functorial according to the following theorem :

Theorem 4.0.4. To each homomorphism of topological groups φ :G−→H is associated a natural homotopy class Bφ ∈ [BG, BH] such that if φ ∈ Hom(G, H) andψ ∈Hom(H, K)then [B(φ◦ψ)] = [Bφ◦Bψ]and BId = Id.

Proof. We begin by the construction of Bφ : BG×BH. The group homo- morphism φ :G−→H induce a topological actionρ_φ :G−→Aut(H)given byg 7→r_φ(g)so that the associated bundleEG×_ρφH is a principalH-bundle over BG and thus corresponds to a classifying map Bφ:BG −→BH, i.e :

(Bφ)^∗(EH)'EG×_ρφH.

Now we check the functoriality of B. If φ = Id then ρ_φ : G −→ Aut(G) is just the trivial action g 7→r_g and thusEG×_ρφG'EG'(Bφ)^∗(EG)hence we get [Bφ] = [IdBG]. Next, select two group homomorphisms φ :G −→H and ψ :H −→K, then the isomorphism :

EG×_ρψ◦φK '(EG×_ρφH)×_ρψ K, gives that :

B(ψ◦φ)^∗(EK)'(Bφ)^∗

(Bψ)^∗(EK)

'(Bψ◦Bφ)^∗(EK).

Thus [B(ψ◦φ)] = [Bψ◦Bφ] and we conclude that B is a functor.

We achieve this section with a result on the universal bundle associated to a subgroup of a given topological group :

Proposition 4.0.3. Let H ,−→^ι G be an inclusion of topological groups such that the canonical projection G−→G/H is a principal H-bundle. Then we can take EH =EG and BH =EG×_G(G/H).

In particular if G is a Lie group and H is a closed subgroup, then the canonical projection G −→ G/H is always a principal H-bundle and thus the preceding result holds in this case.

5. Existence and explicit examples of universal principal bundles We begin by presenting some specific situations where we can determine explicitly the universal principal bundle :

1. The easiest example is that of G = Z since the universal principal Z- bundle is just the universal cover R−→^π S¹ since Ris a contractible space.

2. A non-trivial situation occurs in the case G=Z2 : We begin by defining the infinite unit sphere S^∞= colim

n7→+∞Sⁿ, we have that : Lemma 5.0.1. The infinite unit sphere S^∞ is contractible.

Proof. Denote R^∞ = colimn→∞Rⁿ the vector space of infinite sequencesx= (x_n)_n∈N whose terms vanish starting from a given rank. We can check that R^∞ is a Hilbert space for the norm :

kxk= X

n∈N

x²_n

!¹₂ .

The infinite sphere S^∞ can be viewed as the unit sphere of R^∞, i.e : S^∞ ={x∈R^∞, kxk= 1}.

Pute= (1,0, . . .)∈S^∞. To show thatS^∞ is contractible we must construct a homotopy between IdS^∞ and the constant map e, to do this we start by defining the linear map f :R^∞−→R^∞ given by the formula :

f(x₁, x₂, . . .) = (0, x₁, x₂, . . .).

Since kf(x)k= kxk, we get that f is continuous and that it induces a con- tinuous map f :S^∞−→S^∞. Now f is homotopic to the identity, to see this define the map H :S^∞×I −→S^∞ by the formula :

H(x, t) = tf(x) + (1−t)x ktf(x) + (1−t)xk.

This is well defined since {x, f(x)} is always linearly independent. Further- more it is clear that H is continuous and that H(x,0) = x and H(x,1) = f(x). HenceH is a homotopy betweenf and the identity map. Now the con- cluding remark is thatf is also homotopic toe via the mapHˆ :S^∞×I −→

S^∞ given by :

H(x, t) =ˆ te+ (1−t)f(x) kte+ (1−t)f(x)k.

We thus obtain that Id_S^∞ is homotopic toe which achieves the proof.

Now defineRP^∞= colim

n7→+∞RPⁿ, and consider the action : Z²×S^∞−→S^∞, ([n], x)7→(−1)ⁿx.

This is clearly continuous since its restriction to Z2×Sⁿ is continuous and S^∞ is the colimit of the unit spheres Sⁿ. SinceZ2 is compact, the canonical projection π : S^∞ −→ S^∞/Z² defines a principal Z²-bundle. Let’s inspect further S^∞/Z2, from π(Sⁿ) = RPⁿ we get that S^∞/Z2 = ∪n∈NRPⁿ. More- over, a subset U ⊂ S^∞/Z2 is open if and only if π⁻¹(U)∩Sⁿ is open in Sⁿ which is equivalent to U ∩RPⁿ being open in RPⁿ, thus we obtain that :

S^∞/Z2 = colim

n7→+∞RPⁿ =RP^∞.

In summary we conclude that S^∞ −→^π RP^∞ is a universal principal Z2- bundle.

3. The infinite dimensional sphere S^∞ can also be viewed as the colimit of odd dimensional spheres S²ⁿ⁺¹ ⊂ Cⁿ⁺¹. Define the infinite complex pro- jective space to be the colimit space :

CP^∞= colim

n7→+∞CPⁿ.

We can show as before that the principal S¹-bundles S²ⁿ⁺¹ −→CP^∞ define by taking the colimit a principal S¹-bundleS^∞−→CP^∞ which is universal

since S^∞ is contractible.

4. We proceed to the calculation of the universal principal U(n)-bundle (resp. O(n)-bundle) : Start by defining the infinite dimensionalStiefel mani- fold St_K(k,∞) = colim

n7→+∞St_K(k, n) (K=R orC). Then we have that :

Lemma 5.0.2. The infinite dimensional Stiefel manifold St_K(k,∞) is con- tractible.

Proof. We show this for K=R, the case K=C is similar. Endow R^∞ with the scalar product h, i given by :

hv, wi=X

n∈N

v_nw_n.

Then St_R(k,∞) might be viewed as the space of k-tuples (v₁, . . . , v_k) such that v_i ∈ R^∞ and hv_i, v_ji=δ_ij. Define the shift operator f :R^∞ −→R^∞ as the linear map given by

f(x₁, x₂, . . .) = (0, x₁, x₂, . . .).

This induce a map f : St_R(k,∞) −→ St_R(k,∞). We claim that the iden- tity Id_StR(k,∞) is homotopic to f^k. Indeed, define the continuous map H : St_R(k,∞)×I −→St_R(k,∞)given for v = (v₁, . . . , v_k)by the formula :

H(v, t) = Gram(tf^k(v₁) + (1−t)v₁, . . . , tf^k(v_k) + (1−t)v_k),

where Gram denotes the Gram-Schmidt orthogonalization procedure. It is easily checked that H(v,0) = v and H(v,1) = f^k(v), which gives the de- sired homotopy. Now, f^k is also homotopic to the point (e₁, . . . , e_k) via the homotopy Hˆ : St_R(k,∞)×I −→St_R(k,∞) given by the expression :

H(v, t) = Gram(teˆ 1+ (1−t)f^k(v1), . . . , tek+ (1−t)f^k(vk)).

We conclude that Id_St

R(k,∞) is homotopic to(e₁, . . . , e_k), and thusSt_R(k,∞) is contractible. Essentially the same reasoning shows that St_C(k,∞) is con- tractible.

We define the infinite dimensional Grassamannian by : Gr_K(k,∞) = colim

n7→+∞Gr_K(k, n).

By taking the colimit of the principal U(n)-bundles St_C(k, n)−→^π Gr_C(k, n) we get a principal U(n)-bundle St_C(k,∞)−→^π Gr_C(k,∞) which is universal according to the preceding lemma.

Similarly, for K=Rwe can show thatSt_R(k,∞)is contractible, and thus we obtain that the universal principalO(n)-bundle is just the (colimit) principal O(n)-bundleSt_R(k,∞)−→^π Gr_R(k,∞).

5. Since every compact Lie groupGadmits a faithful representation in some unitary group U(k), the preceding example and proposition (4.0.3) shows that we can take EG = St_C(k,∞) and BG = EU(k)×_U(k)(U(k)/G). In summary, we have shown the existence of the universal principal G-bundle for every compact Lie group G.

6. Universal vector bundles

We know from theorem (2.0.2) that when E −→^π B is a vector bundle and f₀, f₁ :X −→ B are homotopic maps, then we get that f₀^∗(E)' f₁^∗(E) as vector bundles. Thus if we denote Vect_k(X,K) the space of isomorphism classes K-vector bundles over X of rank k, we obtain a well-defined corre- spondance :

[X, B]−→Vect_k(X,K), [f]7→[f^∗E].

So the question that arise naturally at this point is whether there exists a

"universal" vector bundle E −→^π B such that the map the preceding corre- spondance is bijective. We show in what follows that this question admits an affirmative answer when X is a CW-complex. The proof of this claim will use the results of the preceding section which sort of ease the task. Note that the result still holds in the more general case where X is a paracompact Hausdorff space, for a complete proof that is independent of principal bun- dles one might consult [Hat03].

We start we the case K = R, the proof for K = C is analogous. Sup- pose that X is a CW-complex and let G= O(k), we have seen that we can take BG= Gr_R(k,∞) and EG= St_R(k,∞). Recall that the map :

α : [X,Gr_R(k,∞)]−→Prin_O(k)(X), [f]7→[f^∗EG], is a bijective correspondence. Now define the associated bundle

E_k(R) := EG×_O(k)R^k.

This is a rank k (real) vector bundle over Gr_R(k,∞). The main theorem of this part can be stated as follows :

Theorem 6.0.1. Suppose that X is a CW-complex , then the map : β : [X,Gr_R(k,∞)]−→Vect_k(X,R), [f]7→[f^∗(E_k(R))], is a bijective correspondence.

Proof. The key point is to recall that over a paracompact space, the operation of association :

PrinO(k)(X)−→Vectk(X,R), P 7→P ×O(k)R^k,

is bijective. Next, it is straightforward to check that the following diagram is commutative :

[X,Gr_R(k,∞)] Prin_O(k)(X)

Vectk(X,R) α

β '

(12) i.ef^∗(EG×_O(k)R^k)'f^∗(EG)×_O(k)R^k. Thus we conclude thatβ is bijective.

Now for K= C we let G= U(k), we have seen in the preceding section that it is possible to set BG = Gr_C(k,∞) and EG = St_C(k,∞). We put E_k(C) =EG×_U(k)C^k, and by analogy to the former case we get the following statement :

Theorem 6.0.2. Suppose that X is a CW-complex , then the map : β : [X,Gr_C(k,∞)]−→Vect_k(X,C), [f]7→[f^∗(E_k(C))], is a bijective correspondence.

We call E_k(K) −→ Gr_K(k,∞) a universal K-vector bundle of rank k.

Given a rank k vector bundle E −→^πE X, a map f : X −→ Gr_K(k,∞) such that E 'f^∗(E_k(K)) will be called a classifying map for E.

In the final part of this section, we specialize in the case where X is a differ- entiable manifold which we will denote by the letterM. We show that for an important class of differentiable manifolds (finite type manifolds), there is a classification theorem for vector bundles such that the corresponding univer- sal bundle is a finite dimensional differentiable vector bundle. Before delving into this, let us recall some elementary facts about manifolds :

Definition 6.0.1. A good cover of a differentiable manifold M is any open cover whose elements are contractible and such that finite intersections of its elements are also contractible.

A well-known result about good covers is the following :

Theorem 6.0.3. Any differentiable manifold admits a good cover.

We say that a differentiable manifold M is of finite type if it admits a finite good cover. In particular, compact differentiable manifolds are of fi- nite type. We will show that for differentiable manifolds of finite type, the universal vector bundle is a (finite dimensional) differentiable vector bundle.

As usual, we treat the case K=R since the complex case is identical.

Denote V −→^π Gr_R(k, n) the trivial bundle of rank n over the Grassman- nian Gr_R(k, n) , i.e V = Gr_R(k, n)×Rⁿ and define :

E_k,n(R) ={(v, `)∈V, v∈`}.

It is straightforward to check thatEk,n(R)−→^π Gr_R(k, n)is vector subbundle ofV of rankkover the GrassmannianGr_R(k, n)calledthe tautological bundle of rank k. We start with a lemma :

Lemma 6.0.1. Let E −→^πE M be a vector bundle over a differentiable mani- fold of finite type. There exists on M finitely many continuous sections of E which span the fiber at each point.

Proof. Let U₁, . . . , U_r be a finite good cover of M. Since U_i is contractible, we get thatE|U_i is trivial and so we can find ksectionsσ_i,1, . . . , σ_i,k over each U_i which form a basis of the fiber above any point in U_i. Next choose open setsV₁, . . . , V_r ofM such thatV¯_i ⊂U_i and continuous functionsf_i :M −→R such that f_i = 1 on V_i and f_i = 0 outside of U_i (this is possible since M is paracompact), we thus obtain that :

{f_iσ_i,1, . . . , f_iσ_i,k, 1≤i≤r},

is a family of global sections of E which span the fiber at every point.

Proposition 6.0.1. Let E be a rank k (real) vector bundle over a differen- tiable manifold M. Suppose there are n global sectionsσ₁, . . . , σ_n of E which span the fiber at every point. Then there is a mapfE :M −→Gr_R(k, n)such that E 'f_E^∗(E_k,n(R)).

Proof. Endow Rⁿ with its usual scalar product h , i and define for each p∈M the evaluation map, which is the linear map given by :

ev_p :Rⁿ−→E_p, ev_p(e_i) =σ_i(p).

Now writeRⁿ= ker(ev_p)⊕^⊥S_p. Sincerg(ev_p) =k we obtain thatdim(S_p) =k and thus Sp ∈Gr_R(k, n). Next we consider the map :

f :M −→Gr_R(k, n), p7→Sp. We start by checking that f is a continuous map :

Letp∈M, we can suppose without loss of generality that{σ₁(p), . . . , σ_k(p)}

is a basis for Ep. Since the sections σi are continuous, there exists an open neighborhood U of p in M such that {σ₁(q), . . . , σ_k(q)} is a basis for E_q for all q∈U. Now define the local trivialization :

φ:U ×R^k−→E|U, (p, e_i)7→σ_i(q), and write for all k+ 1 ≤i≤n :

φ⁻¹_p (σi(q)) =

k

X

j=1

aij(q)ej,

where a_ij :U −→R are continuous functions. If we viewR^k as the subspace of Rⁿ spanned by the first k vectors of the canonical basis, is becomes clear that ker(ev_q) = kerφ⁻¹_q ◦ev_q and an easy calculation leads to :

(φ⁻¹_q ◦ev_q)(e_i) =

( e_i if1≤i≤k Pk

j=1a_ij(q)e_j ifk+ 1≤i≤n Thus Bq={ei−Pk

j=1aij(q)ej, k+ 1 ≤i≤n} is contained in ker(evq), and since it is clearly a free family of (n−k) elements, it is a basis of ker(e_v)_q. Now using the Gram-Schmidt process, we complete B_q into an orthonormal

basis {f₁(q), . . . , f_k(q)}of S_q, and the so obtained function f_i :U −→Rⁿ are continuous. This gives a continuous function :

g :U −→St_R(k, n), q7→(f₁(q), . . . , f_k(q)), which in turn gives a commutative diagram :

St_R(k, n)

U Gr_R(k, n)

π f_|U

g

(13) This shows that f|U :U −→Gr_R(k, n) is continuous, and sinceU was an ar- bitrary neighborhood we get thatf is globally continuous. Finally, we check that E 'f^∗(E_k,n(R)) :

Notice that since E_k,n(R) −→^π Gr_R(k, n) has fiber over S_p equal to S_p it- self, we obtain that f^∗(E_k,n(R))−→^π M is a vector bundle overM with fiber over p equal to Sp. Next define the map I : f^∗(Ek,n(R)) −→ E such that I|S_p := ev_p|S

p :S_p −→E_p. ThenI satisfies the commutative diagram : E f^∗(E_k,n(R))

M M

I

π_E π

Id_M

(14) Furthermore, I induce an isomorphism on the fiber by its definition. To verify that it is an isomorphism of vector bundles, it is thus sufficient to check that it is continuous.

The map f_E : M −→ Gr_R(k, n) such that f^∗(E_k,n(R)) ' E is called a classifying map for the vector bundle E overM. We now give a restatement of the preceding result in the style of a "classification theorem", to do this we will call a manifold of type rif it admits a good cover with r contractible open sets. The proof of lemma (6.0.1) shows that any rank k vector bundle

E over a typer manifold admits n=kr global sections which span the fiber at every point. Thus we get the following theorem :

Theorem 6.0.4. Let M be a differentiable manifold and type r. Then for every n≥kr, the map :

β_n: [M,Gr_R(k, n)]−→Vect_k(M,R) [f]7→f^∗(E_k,n(R)).

is a surjective correspondence.

One might of course expect based on the previous study that the map βn is injective : isomorphic pullbacks f^∗(Ek,n(R))'g^∗(Ek,n(R)) give rise to homotopic maps f ' g, but this fails to be true. However the statement might still have some truth in it and a weaker version of it is given by the following theorem which can be consulted in [Spi79] and [Osb83] :

Theorem 6.0.5. Let f, g:M −→Gr_R(k, n) be two maps such that : f^∗(E_k,n(R))'g^∗(E_k,n(R)).

Consider the natural inclusion α : Gr_R(k, n) −→ Gr_R(k, m) for m ≥ 2n.

Then f¯=α◦f and g¯=α◦g are homotopic in Gr_R(k, m).

Proposition 6.0.2. LetE −→^πE M be a vector bundle over a manifold a finite type, then any two classifying maps for E are homotopic.

A concluding remark of this section is that the universal vector bundle E_k(K)−→Gr_K(k,∞) of rankk might be taken as the co-limit of the tauto- logical bundles E_k,n(K) −→ Gr_K(k, n) as n 7→ ∞, again one might want to see [Hat03].

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