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 Let X be a compact Hausdorff space. Swan’s theorem [Swa62] states that any rank k vector bundle E −→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 every p ∈ X, the fiber Ep is a k-dimensional subspace n of R , thus we can construct a map f : X −→ GrR(k, n) given by f(p) = Ep, we can check that f is continuous. Now if we define :

Ek,n(R) = {(`, v) ∈ GrR(k, n) × R, v ∈ ` }.

We can check that Ek,n(R) is a rank k vector bundle over GrR(k, n), fur- ∗ thermore Swan’s theorem is equivalent to claiming that E ' f (Ek,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 × Rm for every m ≥ n. So we define the "infinite" Grassmannian GrR(k, ∞) as the collection of k-dimensional subspaces of R∞, and since E is a subbundle of X × R∞ we define a continuous map f : X −→ GrR(k, ∞), p 7→ Ep. Now define the rank k vector bundle Ek(R) over GrR(k, ∞) by the formula :

Ek(R) = {(`, v) ∈ GrR(k, ∞) × R, v ∈ ` }, ∗ we obtain as before that E ' f (Ek(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) −→ GrK(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 −→ GrK(k, ∞) such that E ' f (Ek(R)), in other words we get that the map :

∗ [X, GrK(k, ∞)] −→ Vectk(X, R), [f] 7→ [f (Ek(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-bundle EG −→ BG satisfying the following property : For every principal G-bundle P −→ 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] −→ PrinG(X), [f] 7→ [f (EG)] is a bijective correspondance.

Now the natural question to ask is whether this classification theorem extends to the case when G is 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 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). Let E −→π B and Eˆ −→πˆ 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 ˆ that H : B × I −→ B is a homotopy such that H0 = f, then there exists a homotopy H˜ : E × I −→ Eˆ whose induced homotopy is H, i.e :

H˜ E × I Eˆ

(π, Id) πˆ

B × I Bˆ H (1) ˜ ˜ Moreover H0 = f. The rest is just a consequence of the Covering homotopy theorem :

Theorem 2.0.2. Let E −→π B be a ﬁber bundle with ﬁber type F 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 homotopy H : X×I −→ B between f0 and f1. Since the fiber ∗ pr2 π bundles f0 (E) −→ X and E −→ 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 :

˜ ∗ H ∗ f0 (E) × I H (E)

X × I X × I Id (2)

3 ˜ ∗ ∗ 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(f0 (E) × {1}) = f1 (E).

It is worth to mention that when f0, f1 : X −→ B are homotopic maps π ∗ ∗ and E −→ B is a principal G-bundle and then the pullbacks f0 (E) and f1 (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 and f : A −→ Y a continuous map. Deﬁne on the disjoint union X t Y the equivalence relation : for all x ∈ X and y ∈ Y ,

x ∼ y if and only if x ∈ A and y = f(x).

The adjunction space of X and Y (via f) is the topological space X t Y/ ∼ endowed with the quotient topology which we denote by X ∪f Y . Denote p : X t Y −→ 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 n family is the set X = ∪n∈NX endowed with the weak topology : A subset U ⊂ X is open if and only if U ∩ Xn is open in Xn for all n ∈ N. We write X = colim Xn. Now a CW-complex is just a colimit space X = colim Xn n→+∞ n→+∞ where the spaces Xn are deﬁned inductively in the following manner :

4 1. We start with a discrete space X0. 2. Suppose that Xn−1 is defined, then choose a family of attaching maps n n n−1 n n n−1 ϕα : ∂Dα ⊂ Dα −→ X and put X := (tαDα) ∪ X . ϕα, α n n−1 n For all n ∈ N, let pn :(tαDα) t X −→ X be the canonical projection. n n n An n-dimensional cell in X is the embedded image eα = (ιn ◦ pn)(Dα \ ∂Dα) n n n where ιn : X −→ X is the inclusion of X in X. To each cell eα we can n n associate a characteristic map φα : Dα −→ X which is by definition the composition : n inc n n−1 pn n ιn Dα ,−→ (tβDβ ) t X −→ X ,−→ X. This is clearly a continuous map, and it is straightforward to check that it n n n n defines a homeomorphism int(D ) −→ e and that φ n = ιn−1 ◦ ϕ . α α α|∂Dα α Proposition 3.0.2. A set A ⊂ X is open (resp. closed) if and only if the n −1 n n set (φα) (A) is open (resp. closed) in Dα for every characteristic map φα.

n n −1 Proof. The continuity of the characteristic maps φα assures that (φα) (A) is open for every open set A ⊂ X. Conversely, let A ⊂ X and assume n −1 n n that (φα) (A) is open in Dα for every characteristic map φα, we will show inductively that A ∩ Xk is open in Xk 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 ∩ Xk−1 is open in Xk−1, now denote k k−1 k pk :(tαDα) t X −→ X the canonical projection, then :

−1 k −1 k−1 pk (A) = (tα(φα) (A)) t (A ∩ X )

k k−1 k which is an open set of (tαDα) t X by hypothesis. Thus A ∩ X is open in Xk. We conclude that A is open in X. 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

5 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 : πn(Y ) −→ πn(X) is an isomorphism (clearly any homotopy equivalence is a weak homotopy equivalence). 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-complex X any subspace A ⊂ X which is n a unions of cells of X such that the closure of each cell eα ⊂ A is contained in A. This induce a CW-complex structure on A itself. The obvious example of a subcomplex of X are the n-skeletons Xn, to see this notice that Xn is k closed in X thus given a k-dimensional cell eα of X with k ≤ n, its closure must be in Xn. Since Xn is the union of such cells, it is a subcomplex of X. (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 π1 π2 of principal G-bundles P1 −→ B1 and P2 −→ 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 :

φ P1 P2

π1 π2 (3) φ˜ B1 B2 ˜ ˜ The map φ : B1 −→ B2 is uniquely deﬁned by φ(b) = π2(φ(p)) where b = −1 π1(p). It is well-deﬁned since given p, q ∈ π1 (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)).

6 To check that it is continuous, choose an open neighborhood U of b in B1 π1 over which P1 −→ 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 that φ is continuous on B1.

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

Proof. Let p, 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 ∈ G and consequently φ(q) = φ(p) · g = φ(p).

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

π2(φ(p)) = π1(p) = b = π2(p2).

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

−1 ϕ2 ◦ 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 :

−1 (ϕ2 ◦ 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 −1 −1 check that the inverse map ϕ1 ◦ φ ◦ ϕ2 : U × G −→ U × G is given by :

−1 −1 −1 (ϕ1 ◦ φ ◦ ϕ2)(x, g) = (x, τ(x, e) · g), which is clearly continuous since G −→ G , g 7→ g−1 is continuous. We −1 −1 conclude that φ is continuous on π2 (U), and since U is arbitrarily chosen, it is globally continuous.

7 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 πF associated ﬁber bundle P ×G F −→ B to P by ρ where πF ([p, f]) = π(p) (the space P ×G F is the quotient of P × F by the equivalence relation −1 π πF (p, f) ∼ (p · g, g · f)). When P −→ B is trivial, then P ×G F −→ 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 ×G F the associated bundle. For any open set U ⊂ B, there is a bijective correpondance Γ(U, E) ←→ Map(π−1(U),F )G.

−1 Proof. Let φ : π (U) −→ F be a G-equivariant map and deﬁne sφ : U −→ E by the formula sφ(b) = [p, φ(p)]. Notice that sφ 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 π open set V ⊂ U over which P −→ B is trivial and a local section σV : V −→

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

−1 G Map(π (U),F ) −→ Γ(U, E), φ 7→ sφ.

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

φs(p) = f when s(π(p)) = [p, f].

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

s(π(p)) = [p, φs(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)], since π(p · g) = π(p) we obtain that

[p · g, φs(p · g)] = [p, φs(p)].

8 −1 Thus there exists h ∈ G such that (p · g, φs(p · g)) = (p · h, h · φs(p)), again because the action of G on P is free we obtain that g = h and φs(p · −1 g) = g · φs(p) which gives that φs is G-equivariant. Now we check that −1 φs : π (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 :

ϕF (b, f) = [ϕ(b, e), f].

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

−1 (b, f) = ϕF [ϕ(b, e), f] = ((π ◦ ϕ)(b, e), τF [ϕ(b, e), f]) = (b, τF [ϕ(b, e), f]).

Thus for every f ∈ F we have that f = τF [ϕ(b, e), f]. Hence we get that :

(τF ◦ s)(b) = (τf ◦ s ◦ pr1)(b, e) = (τF ◦ s ◦ π ◦ ϕ)(b, e)

= τF [ϕ(b, e), (φs ◦ ϕ)(b, e)]

= (φs ◦ ϕ)(b, e).

Thus for every b ∈ V and g ∈ G :

−1 −1 (φs ◦ ϕ)(b, g) = φs(ϕ(b, e) · g) = g · (φs ◦ ϕ)(b, e) = g · (τF ◦ s)(b).

This gives that φs ◦ ϕ is continuous on V × G, hence φs is continuous on −1 π (V ). Since V ⊂ U is arbitrary we conclude that φs is globally continuous. We have thus constructed a map :

−1 G Γ(U, E) −→ Map(π (U),F ) , s 7→ φs.

It is straightforward to check that it is the inverse of the ﬁrst map. An important consequence is the following :

0 Corollary 4.0.1. Given two principal G-bundles P −→π B and P 0 −→π B0. 0 0 There is a bijective correspondance MorG(P,P ) ←→ Γ(B,P ×G P ). Proof. It is suﬃcient to notice that Mor(P,P 0) = Map(P,P 0)G and use the preceding proposition.

9 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] −→ PrinG(X)[f] 7→ [f P ]. In what follows, we treat the following question :

π Question: For which conditions on X and P −→ B the map [X,B] −→ PrinG(X) is a bijection ?

Recall that a space F is said to be weakly-contractible when πn(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 when X admits a CW-complex structure and EG −→π BG is a universal principal G-bundle, the [X,BG] −→ PrinG(X) is a bijection. We start with an important lemma : Lemma 4.0.1. Let (B,A) be a CW-pair and E −→π B a fiber bundle with fibre F . Suppose that πk(F ) = 0 whenever B\A contains 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.

k Proof. We will show inductively that the section s|Ak : A −→ E can be extended to a section s˜k : Bk −→ E. For k = 0, it is clear that any extension of 0 s|A0 is automatically continuous since B is a discrete space. Let k ∈ N and k k k suppose that s|Ak : A −→ E can be extended to a section s˜ ∈ Γ(B ,E). k+1 We denote an arbitrary (k + 1)-dimensional cell in B by eα and we put k+1 k+1 k+1 k+1 C1 = {α, eα ⊂ A} and C2 = {α, eα ⊂ B \ A}. We distinguish two situations :

k+1 1. Either C2 = ∅ (there are no (k + 1)-dimensional cells in B \ A), then k+1 k+1 k we deﬁne sˆ :(tαDα ) t B −→ E in the following manner : k+1 k k k+1 k+1 k+1 sˆ =s ˜ on B and sˆ = s ◦ φα on Dα ,

10 k+1 k+1 k+1 where φα : Dα −→ B is the characteristic map of eα which satis- k+1 k+1 k+1 ﬁes φα (Dα ) ⊂ A. It is clear that sˆ is continuous, to show that k+1 k+1 it induce a continuous map on B we choose an attaching map ϕα : k+1 k k+1 k+1 ∂D −→ B (the map ϕ is just φ k+1 ), we easily check that α α α |∂Dα k+1 k+1 k+1 k+1 k+1 sˆ (x) =s ˆ (ϕα (x)) for every x ∈ ∂Dα . Thus sˆ induce a continuous map s˜k+1 : Bk+1 −→ E. It is straightforward to check that π ◦ s˜k+1 = Id and k+1 k+1 k+1 that s˜|Ak+1 = s|Ak+1 , thus s˜ : B −→ E is a section that extends s|Ak+1 .

k+1 k+1 2. Suppose now that C2 6= ∅. Let eα be a (k + 1)-dimensional cell k+1 k+1 contained in B \ A and denote φα : Dα −→ B its characteristic map. k+1 k+1 k Since φα (∂Dα ) ⊂ B , then the composition map : k k k+1 k+1 s˜α :=s ˜ ◦ φα : ∂Dα −→ E k k+1 is well-defined and satisfies π ◦ s˜ = φ k+1 , thus it defines a section α α |∂Dα k+1 ∗ k+1 k+1 of the pullback bundle (φ k+1 ) (E) over ∂D . Now since D is a α |∂Dα α α k+1 ∗ k+1 contractible space, we get that the pullback bundle (φα ) (E) −→ Dα is k+1 ∗ k+1 k+1 ∗ k+1 trivial, i.e (φ ) (E) ' D × F . Thus (φ k+1 ) (E) −→ ∂D is also α α α |∂Dα α trivial as shown in the following commutative diagram :

k+1 ι k+1 pullback ∂Dα × F Dα × F E

pr1 pr1 (4)

k+1 ι k+1 φα ∂Dα Dα B

k k k k+1 Hence s˜α can be seen as a map x 7→ (x, τ˜α(x)) where τ˜α : ∂Dα −→ F . Since k by hypothesis πk(F ) = 0, we obtain that τ˜α can be extended to a continuous k+1 k+1 k map τˆα : Dα −→ F and thus s˜α can be extended to a continuous section k+1 k+1 k+1 ∗ k+1 k+1 k sˆα : Dα −→ (φα ) (E). Finally we deﬁne sˆ :(tααDα ) t B −→ E in the following manner :

 k+1 k+1 k+1  s ◦ φα on Dα for α ∈ C1  k+1 k+1 k+1 k+1 sˆ = sˆα on Dα for α ∈ C2   s˜k on Bk It is clear that sˆk+1 is continuous and we check as before that for every k+1 k+1 k+1 k+1 k+1 x ∈ ∂Dα we have sˆ (x) =s ˆ (ϕα (x)). Thus sˆ induce a continuous map s˜k+1 : Bk+1 −→ E and it is straightforward to check that this is indeed k+1 a section that extends sAk+1 : A −→ E. This achieves the proof.

11 We are now ready to prove the main theorem :

Theorem 4.0.1. Suppose that EG −→π BG is a universal principal G-bundle and X is a CW-complex. Then the correspondance [X,BG] −→ PrinG(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)

πP (5) X Id X 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 : φ P EG

πP (6) f X BG But this is the same (according to corollary (4.0.1)) as ﬁnding a section of the associated bundle P ×G EG −→ 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 :

∗ φ0 ∗ φ1 f0 (EG) EG f0 (EG) EG (7) π0 π π0 π f f X 0 BG X 1 BG

12 ∗ 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 Xt = X × {t}. We can view si as a section si ∈ Γ(Xi,P ×GEG) for i = 0, 1. Thus we can deﬁne s0ts1 ∈ Γ(X0tX1,P ×GEG) by the formula :

(s0 t s1)|Xi = si, i = 0, 1.

Now since (X × I,X0 t X1) is a CW-pair and EG is weakly contractible, we obtain according to the preceding lemma that s0 t s1 can be extended to a section s ∈ Γ(X × I,P ×G EG). 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 :

φ P 1 EG

(π0,Id) π (9) X × I F BG

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 a classifying map for P .

Now we address the uniqueness of the universal principal G-bundle. We start with the statement of a popular theorem called the 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 :

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

13 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 : pr φ∗(EG) 2 EG (10) pr1 π BG0 F BG To be able to show that BG can be replaced by the CW-complex BG0, one pr must show that the principal G-bundles φ∗(EG) −→1 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) ...

∗ ∗ ' pr2 φ ' (11)

∗ 0 ... πn(G) πn(φ EG) πn(BG ) πn−1(G) ...

∗ 0 Since φ : πn(BG) −→ πn(BG ) is an isomorphism, we obtain by the ﬁve ∗ ∗ lemma that πn(EG) ' πn(φ EG) for all n ∈ N, which shows that φ (EG) is weakly contractible and achieves the proof. We suppose in what follows that the classifying space BG always 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 principal G-bundles. There exists a homotopy equivalence B0G −→ BG that is covered by a G-equivariant homotopy equivalence E0G −→ EG. In this sense, universal principal G-bundles are unique up to homotopy equivalence.

14 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] −→ PrinG(BG) we obtain that f ◦ g must be homotopic to IdBG. Similarly, we show that g ◦ f is homotopic 0 to IdB0G. Hence f : B G −→ BG is a homotopy equivalence. We thus have a well-deﬁned correspondence G 7→ 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) and ψ ∈ Hom(H,K) then [B(φ ◦ ψ)] = [Bφ ◦ Bψ] and BId = Id.

Proof. We begin by the construction of Bφ : BG × BH. The group homomorphism φ : G −→ H induce a topological action ρφ : G −→ Aut(H) given by g 7→ rφ(g) so that the associated bundle EG×ρφ H is a principal H-bundle over BG and thus corresponds to a classifying map Bφ : BG −→ BH, i.e :

∗ (Bφ) (EH) ' EG ×ρφ H.

Now we check the functoriality of B. If φ = Id then ρφ : G −→ Aut(G) is ∗ just the trivial action g 7→ rg and thus EG ×ρφ G ' 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◦φ K ' (EG ×ρφ H) ×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 :

15 ι Proposition 4.0.3. Let H ,−→ 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- π bundle is just the universal cover R −→ S1 since R is a contractible space.

2. A non-trivial situation occurs in the case G = Z2 : We begin by defining the infinite unit sphere S∞ = colimSn, we have that : n7→+∞ Lemma 5.0.1. The infinite unit sphere S∞ is contractible.

∞ n Proof. Denote R = colimn→∞R the vector space of inﬁnite sequences x = (xn)n∈N whose terms vanish starting from a given rank. We can check that R∞ is a Hilbert space for the norm :

1 ! 2 X 2 kxk= xn . n∈N

The inﬁnite sphere S∞ can be viewed as the unit sphere of R∞, i.e :

∞ ∞ S = {x ∈ R , kxk= 1}. Put e = (1, 0,... ) ∈ S∞. To show that S∞ 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,... ).

16 Since kf(x)k= kxk, we get that f is continuous and that it induces a continuous map f : S∞ −→ S∞. Now f is homotopic to the identity, to see this deﬁne the map H : S∞ × I −→ S∞ by the formula : tf(x) + (1 − t)x H(x, t) = . 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). Hence H is a homotopy between f and the identity map. Now the concluding remark is that f is also homotopic to e via the map Hˆ : S∞ × I −→ S∞ given by : te + (1 − t)f(x) Hˆ (x, t) = . kte + (1 − t)f(x)k

We thus obtain that IdS∞ is homotopic to e which achieves the proof. Now deﬁne P ∞ = colim P n, and consider the action : R n7→+∞R

∞ ∞ n Z2 × S −→ S , ([n], x) 7→ (−1) x.

n This is clearly continuous since its restriction to Z2 × S is continuous and ∞ n S is the colimit of the unit spheres S . Since Z2 is compact, the canonical ∞ ∞ projection π : S −→ S /Z2 deﬁnes a principal Z2-bundle. Let’s inspect ∞ n n ∞ n further S /Z2, from π(S ) = RP we get that S /Z2 = ∪n∈NRP . More- ∞ −1 n n over, a subset U ⊂ S /Z2 is open if and only if π (U) ∩ S is open in S which is equivalent to U ∩ RP n being open in RP n, thus we obtain that : ∞ n ∞ S / 2 = colim P = P . Z n7→+∞R R

∞ π ∞ In summary we conclude that S −→ 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 projective space to be the colimit space : P ∞ = colim P n. C n7→+∞C

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

17 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 manifold St (k, ∞) = colimSt (k, n) ( = or ). Then we have that : K n7→+∞ K K R C

Lemma 5.0.2. The inﬁnite dimensional Stiefel manifold StK(k, ∞) is contractible.

Proof. We show this for K = R, the case K = C is similar. Endow R∞ with the scalar product h , i given by : X hv, wi = vnwn. n∈N

Then StR(k, ∞) might be viewed as the space of k-tuples (v1, . . . , vk) such ∞ ∞ ∞ that vi ∈ R and hvi, vji = δij. Deﬁne the shift operator f : R −→ R as the linear map given by

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

This induce a map f : StR(k, ∞) −→ StR(k, ∞). We claim that the identity Id is homotopic to f k. Indeed, deﬁne the continuous map H : StR(k,∞) StR(k, ∞) × I −→ StR(k, ∞) given for v = (v1, . . . , vk) by the formula :

k k H(v, t) = Gram(tf (v1) + (1 − t)v1, . . . , tf (vk) + (1 − t)vk), 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- k sired homotopy. Now, f is also homotopic to the point (e1, . . . , ek) via the ˆ homotopy H : StR(k, ∞) × I −→ StR(k, ∞) given by the expression : ˆ k k H(v, t) = Gram(te1 + (1 − t)f (v1), . . . , tek + (1 − t)f (vk)).

We conclude that Id is homotopic to (e , . . . , e ), and thus St (k, ∞) StR(k,∞) 1 k R is contractible. Essentially the same reasoning shows that StC(k, ∞) is contractible. We deﬁne the inﬁnite dimensional Grassamannian by :

Gr (k, ∞) = colimGr (k, n). K n7→+∞ K

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

Similarly, for K = R we can show that StR(k, ∞) is contractible, and thus we obtain that the universal principal O(n)-bundle is just the (colimit) principal π O(n)-bundle StR(k, ∞) −→ GrR(k, ∞).

5. Since every compact Lie group G admits a faithful representation in some unitary group U(k), the preceding example and proposition (4.0.3) shows that we can take EG = StC(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 Vectk(X, K) the space of isomorphism classes K-vector bundles over X of rank k, we obtain a well-defined correspondance : ∗ [X,B] −→ Vectk(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 correspondance 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 bundles 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 = GrR(k, ∞) and EG = StR(k, ∞). Recall that the map : ∗ α :[X, GrR(k, ∞)] −→ PrinO(k)(X), [f] 7→ [f EG], is a bijective correspondence. Now deﬁne the associated bundle

k Ek(R) := EG ×O(k) R .

19 This is a rank k (real) vector bundle over GrR(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, GrR(k, ∞)] −→ Vectk(X, R), [f] 7→ [f (Ek(R))], is a bijective correspondence. Proof. The key point is to recall that over a paracompact space, the operation of association :

k PrinO(k)(X) −→ Vectk(X, R),P 7→ P ×O(k) R , is bijective. Next, it is straightforward to check that the following diagram is commutative : α [X, GrR(k, ∞)] PrinO(k)(X)

' β

Vectk(X, R) (12)

∗ k ∗ k i.e f (EG×O(k) R ) ' f (EG)×O(k) R . Thus we conclude that β 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 = GrC(k, ∞) and EG = StC(k, ∞). We put k Ek(C) = EG×U(k) C , 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, GrC(k, ∞)] −→ Vectk(X, C), [f] 7→ [f (Ek(C))], is a bijective correspondence.

We call Ek(K) −→ GrK(k, ∞) a universal K-vector bundle of rank k. πE Given a rank k vector bundle E −→ X, a map f : X −→ GrK(k, ∞) such ∗ that E ' f (Ek(K)) will be called a classifying map for E.

20 In the final part of this section, we specialize in the case where X is a differentiable manifold which we will denote by the letter M. 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 universal 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 finite 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.

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

Ek,n(R) = {(v, `) ∈ V, v ∈ `}. π It is straightforward to check that Ek,n(R) −→ GrR(k, n) is vector subbundle of V of rank k over the Grassmannian GrR(k, n) called the 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 manifold of finite type. There exists on M finitely many continuous sections of E which span the fiber at each point.

Proof. Let U1,...,Ur be a finite good cover of M. Since Ui is contractible, we get that E|Ui is trivial and so we can find k sections σi,1, . . . , σi,k over each Ui which form a basis of the fiber above any point in Ui. Next choose open ¯ sets V1,...,Vr of M such that Vi ⊂ Ui and continuous functions fi : M −→ R such that fi = 1 on Vi and fi = 0 outside of Ui (this is possible since M is paracompact), we thus obtain that :

{fiσi,1, . . . , fiσi,k, 1 ≤ i ≤ r}, is a family of global sections of E which span the ﬁber at every point.

21 Proposition 6.0.1. Let E be a rank k (real) vector bundle over a differentiable manifold M. Suppose there are n global sections σ1, . . . , σn of E which span the fiber at every point. Then there is a map fE : M −→ GrR(k, n) such ∗ that E ' fE(Ek,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 :

n evp : R −→ Ep, evp(ei) = σi(p).

⊥ n Now write R = ker(evp)⊕Sp. Since rg(evp) = k we obtain that dim(Sp) = k and thus Sp ∈ GrR(k, n). Next we consider the map :

f : M −→ GrR(k, n), p 7→ Sp. We start by checking that f is a continuous map :

Let p ∈ M, we can suppose without loss of generality that {σ1(p), . . . , σk(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), . . . , σk(q)} is a basis for Eq for all q ∈ U. Now deﬁne the local trivialization :

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

k −1 X φp (σi(q)) = aij(q)ej, j=1

k where aij : U −→ R are continuous functions. If we view R as the subspace of Rn spanned by the ﬁrst k vectors of the canonical basis, is becomes clear −1 that ker(evq) = ker φq ◦ evq and an easy calculation leads to : ( ei if 1 ≤ i ≤ k (φ−1 ◦ ev )(e ) = q q i Pk j=1 aij(q)ej if k + 1 ≤ i ≤ n

Pk Thus Bq = {ei − j=1 aij(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(ev)q. Now using the Gram-Schmidt process, we complete Bq into an orthonormal

22 n basis {f1(q), . . . , fk(q)} of Sq, and the so obtained function fi : U −→ R are continuous. This gives a continuous function :

g : U −→ StR(k, n), q 7→ (f1(q), . . . , fk(q)), which in turn gives a commutative diagram :

StR(k, n) g π

U Gr (k, n) f R |U (13)

This shows that f|U : U −→ GrR(k, n) is continuous, and since U was an arbitrary neighborhood we get that f is globally continuous. Finally, we check ∗ that E ' f (Ek,n(R)) :

π Notice that since Ek,n(R) −→ GrR(k, n) has fiber over Sp equal to Sp it- ∗ π self, we obtain that f (Ek,n(R)) −→ M is a vector bundle over M with fiber ∗ over p equal to Sp. Next define the map I : f (Ek,n(R)) −→ E such that I := ev : S −→ E . Then I satisfies the commutative diagram : |Sp p|Sp p p

I ∗ E f (Ek,n(R))

πE π

M M IdM (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 fE : M −→ GrR(k, n) such that f (Ek,n(R)) ' E is called a classifying map for the vector bundle E over M. 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 r if it admits a good cover with r contractible open sets. The proof of lemma (6.0.1) shows that any rank k vector bundle

23 E over a type r 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 :

∗ βn :[M, GrR(k, n)] −→ Vectk(M, R)[f] 7→ f (Ek,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 −→ GrR(k, n) be two maps such that :

∗ ∗ f (Ek,n(R)) ' g (Ek,n(R)).

Consider the natural inclusion α : GrR(k, n) −→ GrR(k, m) for m ≥ 2n. ¯ Then f = α ◦ f and g¯ = α ◦ g are homotopic in GrR(k, m). Proposition 6.0.2. Let E −→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

Ek(K) −→ GrK(k, ∞) of rank k might be taken as the co-limit of the tautological bundles Ek,n(K) −→ GrK(k, n) as n 7→ ∞, again one might want to see [Hat03].

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