Math 754 Chapter III: Fiber bundles. Classifying spaces. Applications

Math 754 Chapter III: Fiber bundles. Classiying spaces. Applications Laurențiu Maxim Department o Mathematics University o Wisconsin maxim@math.wisc.edu April 18, 2018 Contents 1 Fiber bundles 2 2 Principle Bundles 9 3 Classiication o principal G-bundles 14 4 Exercises 18 1

1 Fiber bundles Let G be a topological group (i.e., a topological space endowed with a group structure so that the group multiplication and the inversion map are continuous), acting continuously (on the let) on a topological space F. Concretely, such a continuous action is given by a continuous map ρ: G F F, (g, m) g m, which satisies the conditions (gh) m = g (h m)) and e G m = m, or e G the identity element o G. Any continuous group action ρ induces a map Ad ρ : G Homeo(F ) given by g ( g ), with g G, F. Note that Ad ρ is a group homomorphism since (Ad ρ)(gh)() := (gh) = g (h ) = Ad ρ (g)(ad ρ (h)()). Note that or nice spaces F (such as CW complexes), i we give Homeo(F ) the compact-open topology, then Ad ρ : G Homeo(F ) is a continuous group homomorphism, and any such continuous group homomorphism G Homeo(F ) induces a continuous group action G F F. We assume rom now on that ρ is an eective action, i.e., that Ad ρ is injective. Deinition 1.1 (Atlas or a iber bundle with group G and iber F ). Given a continuous map : E B, an atlas or the structure o a iber bundle with group G and iber F on consists o the ollowing data: a) an open cover {U α } α o B, b) homeomorphisms (called trivializing charts or local trivializations) h α : 1 (U α ) U α F or each α so that the diagram 1 (U α ) h α U α F U α pr 1 commutes, c) continuous maps (called transition unctions) g αβ : U α U β G so that the horizontal map in the commutative diagram 1 (U α U β ) (U α U β ) F h α h β h 1 α h β (U α U β ) F is given by (x, m) (x, g βα (x) m). (By the eectivity o the action, i such maps g αβ exist, they are unique.) 2

Deinition 1.2. Two atlases A and B on are compatible i A B is an atlas. Deinition 1.3 (Fiber bundle with group G and iber F ). A structure o a iber bundle with group G and iber F on : E B is a maximal atlas or : E B. Example 1.4. 1. When G = {e G } is the trivial group, : E B has the structure o a iber bundle i and only i it is a trivial iber bundle. Indeed, the local trivializations h α o the atlas or the iber bundle have to satisy h β h 1 α : (x, m) (x, e G m) = (x, m), which implies h β h 1 α = id, so h β = h α on U α U β. This allows us to glue all the local trivializations h α together to obtain a global trivialization h: 1 (B) = E = B F. 2. When F is discrete, Homeo(F ) is also discrete, so G is discrete by the eectiveness assumption. So or the atlas o : E B we have 1 (U α ) = U α F = m F U α {m}, so is in this case a covering map. 3. A locally trivial iber bundle, as introduced in earlier chapters, is just a iber bundle with structure group Homeo(F ). Lemma 1.5. The transition unctions g αβ satisy the ollowing properties: (a) g αβ (x)g βγ (x) = g αγ (x), or all x U α U β U γ. (b) g βα (x) = g 1 αβ (x), or all x U α U β. (c) g αα (x) = e G. Proo. On U α U β U γ, we have: (h α h 1 β ) (h β h 1 γ ) = h α h 1 γ. Thereore, since Ad ρ is injective (i.e., ρ is eective), we get that g αβ (x)g βγ (x) = g αγ (x) or all x U α U β U γ. Note that (h α h 1 β ) (h β h 1 α ) = id, which translates into (x, g αβ (x)g βα (x) m) = (x, m). So, by eectiveness, g αβ (x)g βα (x) = e G or all x U α U β, whence g βα (x) = g 1 αβ (x). Take γ = α in Property (a) to get g αβ (x)g βα (x) = g αα (x). So by Property (b), we have g αα (x) = e G. Transition unctions determine a iber bundle in a unique way, in the sense o the ollowing theorem. Theorem 1.6. Given an open cover {U α } o B and continuous unctions g αβ : U α U β G satisying Properties (a)-(c), there is a unique structure o a iber bundle over B with group G, given iber F, and transition unctions {g αβ }. 3

Proo Sketch. Let Ẽ = α U α F {α}, and deine an equivalence relation on Ẽ by (x, m, α) (x, g αβ (x) m, β), or all x U α U β, and m F. Properties (a)-(c) o {g αβ } are used to show that is indeed an equivalence relation on Ẽ. Speciically, symmetry is implied by property (b), relexivity ollows rom (c) and transitivity is a consequence o the cycle property (a). Let E = Ẽ/ be the set o equivalence classes in E, and deine : E B locally by [(x, m, α)] x or x U α. Then it is clear that is well-deined and continuos (in the quotient topology), and the iber o is F. It remains to show the local triviality o. Let p : Ẽ E be the quotient map, and let p α := p Uα F {α} : U α F {α} 1 (U α ). It is easy to see that p α is a homeomorphism. We deine the local trivializations o by h α := p 1 α. Example 1.7. 1. Fiber bundles with iber F = R n and group G = GL(n, R) are called rank n real vector bundles. For example, i M is a dierentiable real n-maniold, and T M is the set o all tangent vectors to M, then : T M M is a real vector bundle on M o rank n. = More precisely, i ϕ α : U α R n are trivializing charts on M, the transition unctions or T M are given by g αβ (x) = d(ϕ α ϕ 1 β ) ϕ β (x). 2. I F = R n and G = O(n), we get real vector bundles with a Riemannian structure. 3. Similarly, one can take F = C n and G = GL(n, C) to get rank n complex vector bundles. For example, i M is a complex maniold, the tangent bundle T M is a complex vector bundle. 4. I F = C n and G = U(n), we get real vector bundles with a hermitian structure. We also mention here the ollowing act: Theorem 1.8. A iber bundle has the homotopy liting property with respect to all CW complexes (i.e., it is a Serre ibration). Moreover, iber bundles over paracompact spaces are ibrations. Deinition 1.9 (Bundle homomorphism). Fix a topological group G acting eectively on a space F. A homomorphism between bundles E B and E B with group G and iber F is a pair (, ˆ) o continuous maps, with : B B and ˆ : E E, such that: 1. the diagram E ˆ E commutes, i.e., ˆ =. B B 4

2. i {(U α, h α )} α is a trivializing atlas o and {(V β, H β )} β is a trivializing atlas o, then the ollowing diagram commutes: (V β 1 (U α )) F H β 1 (V β 1 (U α )) ˆ 1 (U α ) h α U α F pr 1 V β 1 (U α ) U α pr 1 and there exist unctions d αβ : V β 1 (U α ) G such that or x V β 1 (U α ) and m F we have: h α ˆ H 1 β (x, m) = ((x), d αβ(x) m). An isomorphism o iber bundles is a bundle homomorphism (, ˆ) which admits a map (g, ĝ) in the reverse direction so that both composites are the identity. Remark 1.10. Gauge transormations o a bundle : E B are bundle maps rom to itsel over the identity o the base, i.e., corresponding to continuous map g : E E so that g =. By deinition, such g restricts to an isomorphism given by the action o an element o the structure group on each iber. The set o all gauge transormations orms a group. Proposition 1.11. Given unctions d αβ : V β 1 (U α ) G and d α β : V β 1 (U α ) G as in (2) above or dierent trivializing charts o and resp., then or any x V β V β 1 (U α U α ), we have d α β (x) = g α α((x)) d αβ (x) g ββ (x) (1.1) in G, where g α α are transition unctions or and g ββ are transition unctions or, Proo. Exercise. The unctions {d αβ } determine bundle maps in the ollowing sense: Theorem 1.12. Given a map : B B and bundles E B, E B, a map o bundles (, ˆ) : exists i and only i there exist continuous maps {d αβ } as above, satisying (1.1). Proo. Exercise. Theorem 1.13. Every bundle map ˆ over = id B is an isomorphism. In particular, gauge transormations are automorphisms. Proo Sketch. Let d αβ : V β U α G be the maps given by the bundle map ˆ : E E. So, i d α β : V β U α G is given by a dierent choice o trivializing charts, then (1.1) holds on V β V β U α U α, i.e., d α β (x) = g α α(x) d αβ (x) g ββ (x) (1.2) 5

in G, where g α α are transition unctions or and g ββ are transition unctions or. Let us now invert (1.2) in G, and set d βα (x) = d 1 αβ (x) to get: d β α (x) = g β β(x) d βα (x) g αα (x). So {d βα } are as in Deinition 1.9 and satisy (1.1). Theorem 1.12 implies that there exists a bundle map ĝ : E E over id B. We claim that ĝ is the inverse ˆ 1 o ˆ, and this can be checked locally as ollows: (x, m) ˆ (x, dαβ (x) m) ĝ (x, d βα (x) (d αβ (x) m)) = (x, d βα (x)d αβ (x) m) = (x, m). }{{} e G So ĝ ˆ = id E. Similarly, ˆ ĝ = id E One way in which iber bundle homomorphisms arise is rom the pullback (or the induced bundle) construction. Deinition 1.14 (Induced Bundle). Given a bundle E B with group G and iber F, and a continuous map : X B, we deine E := {(x, e) X E (x) = (e)}, with projections : E X, (x, e) x, and ˆ : E E, (x, e) e, so that the ollowing diagram commutes: E E e X x B (x) is called the induced bundle under or the pullback o by, and as we show below it comes equipped with a bundle map (, ˆ) :. The above deinition is justiied by the ollowing result: Theorem 1.15. (a) : E X is a iber bundle with group G and iber F. (b) (, ˆ) : is a bundle map. 6

Proo Sketch. Let {(U α, h α )} α be a trivializing atlas o, and consider the ollowing commutative diagram: ( ) 1 ( 1 (U α )) 1 (U α ) U α F h α 1 (U α ) U α We have Deine by ( ) 1 ( 1 (U α )) = {(x, e) 1 (U α ) 1 (U α ) (x) = (e)}. }{{} =U α F k α : ( ) 1 ( 1 (U α )) 1 (U α ) F (x, e) (x, pr 2 (h α (e))). Then it is easy to check that k α is a homeomorphism (with inverse kα 1 (x, m) = (x, h 1 α ((x), m)), and in act the ollowing assertions hold: (i) {( 1 (U α ), k α )} α is a trivializing atlas o. (ii) the transition unctions o are g αβ := g αβ, i.e., g αβ (x) = g αβ ((x)) or any x 1 (U α U β ). Remark 1.16. It is easy to see that ( g) = g ( ) and (id B ) =. Moreover, the pullback o a trivial bundle is a trivial bundle. As we shall see later on, the ollowing important result holds: Theorem 1.17. Given a ibre bundle : E B with group G and iber F, and two homotopic maps g : X B, there is an isomorphism = g o bundles over X. (In short, induced bundles under homotopic maps are isomorphic.) As a consequence, we have: Corollary 1.18. A iber bundle over a contractible space B is trivial. Proo. Since B is contractible, id B is homotopic to the constant map ct. Let b := Image(ct) i B, 7

so i ct id B. We have a diagram o maps and induced bundles: ct i E i E E ct i B ct id B i {b} i B Theorem 1.17 then yields: = (id B ) = ct i. Since any iber bundle over a point is trivial, we have that i = {b} F is trivial, hence = ct i = B F is also trivial. Proposition 1.19. I E E B is a bundle map, then = as bundles over B. Proo. Deine h : E E by e ( (e ), (e )) B E. This is well-deined, i.e., h(e ) E, since ( (e )) = ( (e )). It is easy to check that h provides the desired bundle isomorphism over B. B E h E ˆ E B B Example 1.20. We can now show that the set o isomorphism classes o bundles over S n with group G and iber F is isomorphic to n 1 (G). Indeed, let us cover S n with two contractible sets U + and U obtained by removing the south, resp., north pole o S n. Let i ± : U ± S n be the inclusions. Then any bundle over S n is trivial when restricted to U ±, that is, i ± = U ± F. In particular, U ± provides a trivializing cover (atlas) or, and any such bundle is completely determined by the transition unction g ± : U + U S n 1 G, i.e., by an element in n 1 (G). More generally, we aim to classiy iber bundles on a given topological space. Let B (X, G, F, ρ) denote the isomorphism classes (over id X ) o iber bundles on X with group G 8

and iber F, and G-action on F given by ρ. I : X X is a continuous map, the pullback construction deines a map so that (id X ) = id and ( g) = g. : B (X, G, F, ρ) B (X, G, F, ρ) 2 Principle Bundles As we will see later on, the iber F doesn t play any essential role in the classiication o iber bundle, and in act it is enough to understand the set P (X, G) := B (X, G, G, m G ) o iber bundles with group G and iber G, where the action o G on itsel is given by the multiplication m G o G. Elements o P (X, G) are called principal G-bundles. O particular importance in the classiication theory o such bundles is the universal principal G-bundle G EG BG, with contractible total space EG. Example 2.1. Any regular cover p : E X is a principal G-bundle, with group G = 1 (X) p 1 (E). Here G is given the discrete topology. In particular, the universal covering X X is a principal 1 (X)-bundle. Example 2.2. Any ree (right) action o a inite group G on a (Hausdor) space E gives a regular cover and hence a principal G-bundle E E/G. More generally, we have the ollowing: Theorem 2.3. Let : E X be a principal G-bundle. Then G acts reely and transitively on the right o E so that E G = X. In particular, is the quotient (orbit) map. Proo. We will deine the action locally over a trivializing chart or. Let U α be a trvializing open in X with trivializing homeomorphism h α : 1 (U α ) = U α G. We deine a right action on G on 1 (U α ) by 1 (U α ) G 1 (U α ) = U α G (e, g) e g := h 1 α ( (e), pr 2 (h α (e)) g) Let us show that this action can be globalized, i.e., it is independent o the choice o the trivializing open U α. I (U β, h β ) is another trivializing chart in X so that e 1 (U α U β ), we need to show that e g = h 1 β ( (e), pr 2 (h β (e)) g), or equivalently, h 1 α ( (e), pr 2 (h α (e)) g) = h 1 β ( (e), pr 2 (h β (e)) g). (2.1) 9

Ater applying h α and using the transition unction g αβ or (e) U α U β, (2.1) becomes ( (e), pr 2 (h α (e)) g) = h α h 1 β ( (e), pr 2 (h β (e)) g) = ( (e), g αβ ((e)) (pr 2 (h β (e)) g)), (2.2) which is guaranteed by the deinition o an atlas or. It is easy to check locally that the action is ree and transitive. Moreover, E G is locally given as U α G G = Uα, and this local quotient globalizes to X. The converse o the above theorem holds in some important cases. Theorem 2.4. Let E be a compact Hausdor space and G a compact Lie group acting reely on E. Then the orbit map E E/G is a principal G-bundle. Corollary 2.5. Let G be a Lie group, and let H < G be a compact subgroup. Then the projection onto the orbit space : G G/H is a principal H-bundle. Let us now ix a G-space F. We deine a map P (X, G) B (X, G, F, ρ) as ollows. Start with a principal G bundle : E X, and recall rom the previous theorem that G acts reely on the right on E. Since G acts on the let on F, we have a let G-action on E F given by: g (e, ) (e g 1, g ). Let E G F := E F G be the corresponding orbit space, with projection map ω : E G F E G = X itting into a commutative diagram E F (2.3) E pr 1 X ω E F G Deinition 2.6. The projection ω := G F : E G F X is called the associated bundle with iber F. The terminology in the above deinition is justiied by the ollowing result. Theorem 2.7. ω : E G F X is a iber bundle with group G, iber F, and having the same transition unctions as. Moreover, the assignment ω := G F deines a one-to-one correspondence P (X, G) B (X, G, F, ρ). 10

Proo. Let h α : 1 (U α ) U α G be a trivializing chart or. Recall that or e 1 (U α ), F and g G, i we set h α (e) = ((e), h) U α G, then G acts on the right on 1 (U α ) by acting on the right on h = pr 2 (h α (e)). Then we have by the diagram (2.3) that Let us deine by ω 1 (U α ) = 1 (U α ) F (e, ) (e g 1, g ) = U α G F (u, h, ) (u, hg 1, g ). k α : ω 1 (U α ) U α F [(u, h, )] (u, h ). This is a well-deined map since [(u, hg 1, g )] (u, hg 1 g ) = (u, h ). It is easy to check that k a l is a trivializing chart or ω with inverse induced by U α F U α G F, (u, ) (u, id G, ). It is clear that ω and have the same transition unctions as they have the same trivializing opens. The associated bundle construction is easily seen to be unctorial in the ollowing sense. Proposition 2.8. I E X E X is a map o principal G-bundles (so is a G-equivariant map, i.e., (e g) = (e) g), then there is an induced map o associated bundles with iber F, E G F X G id F E G F X Example 2.9. Let : S 1 S 1, z z 2 be regarded as a principal Z/2-bundle, and let F = [ 1, 1]. Let Z/2 = {1, 1} act on F by multiplication. Then the bundle associated to with iber F = [ 1, 1] is the Möbius strip S 1 Z/2 [ 1, 1] = S1 [ 1, 1] (x, t) (a(x), t), with a : S 1 S 1 denoting the antipodal map. Similarly, the bundle associated to with iber F = S 1 is the Klein bottle. Let us now get back to proving the ollowing important result. Theorem 2.10. Let : E Y be a iber bundle with group G and iber F, and let g : X Y be two homotopic maps. Then = g over id X. 11

It is o course enough to prove the theorem in the case o principal G-bundles. The idea o proo is to construct a bundle map over id X between and g : E? g E X So we irst need to understand maps o principal G-bundles, i.e., to solve the ollowing problem: given two principal G-bundles bundles E 1 1 X and E2 2 Y, describe the set maps( 1, 2 ) o bundle maps E 1 1 ˆ E 2 X Y Since G acts on the right o E 1 and E 2, we also get an action on the let o E 2 by g e 2 := e 2 g 1. Then we get an associated bundle o 1 with iber E 2, namely We have the ollowing result: ω := 1 G E 2 : E 1 G E 2 X. Theorem 2.11. Bundle maps rom 1 to 2 are in one-to-one correspondence to sections o ω. Proo. We work locally, so it suices to consider only trivial bundles. Given a bundle map (, ) : 1 2, let U Y open, and V 1 (U) open, so that the ollowing diagram commutes (this is the bundle maps in trivializing charts) 2 V G U G V 1 2 U We deine a section σ in (V G) G (U G) σ ω as ollows. For e 1 V G, with x = 1 (e 1 ) V, we set V σ(x) = [e 1, (e 1 )]. 12

This map is well-deined, since or any g G we have: [e 1 g, (e 1 g)] = [e 1 g, (e 1 ) g] = [e 1 g, g 1 (e 1 )] = [e 1, (e 1 )]. Now, it is an exercise in point-set topology (using the local deinition o a bundle map) to show that σ is continuous. Conversely, given a section o E 1 G E ω 2 X, we deine a bundle by (, ) by (e 1 ) = e 2, where σ( 1 (e 1 )) = [(e 1, e 2 )]. Note that this is an equivariant map because [e 1 g, e 2 g] = [e 1 g, g 1 e 2 ] = [e 1, e 2 ], hence (e 1 g) = e 2 g = (e 1 ) g. Thus descends to a map : X Y on the orbit spaces. We leave it as an exercise to check that (, ) is indeed a bundle map, i.e., to show that locally (v, g) = ((v), d(v)g) with d(v) G and d : V G a continuous unction. The ollowing result will be needed in the proo o Theorem 2.10. Lemma 2.12. Let : E X I be a bundle, and let 0 := i 0 : E 0 X be the pullback o under i 0 : X X I, x (x, 0). Then = (pr 1 ) 0 = 0 id I, where pr 1 : X I X is the projection map. Proo. It suices to ind a bundle map (pr 1, pr 1 ) so that the ollowing diagram commutes î 0 E 0 E pr 1 E 0 0 X i 0 X I pr 1 X By Theorem 2.11, this is equivilant to the existence o a section σ o ω : E G E 0 X I. Note that there exists a section σ 0 o ω 0 : E 0 G E 0 X = X {0}, corresponding to the bundle map (id X, id E0 ) : 0 0. Then composing σ 0 with the top inclusion arrow, we get the ollowing diagram X {0} σ 0 E G E 0 X I s id ω X I Since ω is a ibration, by the homotopy liting property one can extend sσ 0 to a section σ o ω. We can now inish the proo o Theorem 2.10. 0 13

Proo o Theorem 2.10. Let H : X I Y be a homotopy between and g, with H(x, 0) = (x) and H(x, 1) = g(x). Consider the induced bundle H over X I. Then we have the ollowing diagram. E H E Ĥ E g E H X {0} g i 0 X I X {1} Since = H (, 0), we get = i 0H. By Lemma 2.12, H = pr 1 ( ) = pr 1 (g ), and thus = i 0H = i 0 pr 1 g = g. We conclude this section with the ollowing important consequence o Theorem 2.11 Corollary 2.13. A principle G-bundle : E X is trivial i and only i has a section. Proo. The bundle is trivial i and only i = ct, with ct : X point the constant map, and : G point the trvial bundle over a point space. This is equivalent to saying that there is a bundle map E G X i 1 ct point or, by Theorem 2.11, to the existence o a section o the bundle ω : E G G X. On the other hand, ω =, since E G G X looks locally like H pr 1 Y X 1 (U α ) G = U α G G (u, g1, g 2 ) (u, g 1 g 1, gg 2 ) = Uα G, with the last homeomorphism deined by [(u, g 1, g 2 )] (u, g 1 g 2 ). Altogether, is trivial i and only i : E X has a section. 3 Classiication o principal G-bundles Let us assume or now that there exists a principal G-bundle G : EG BG, with contractible total space EG. As we will see below, such a bundle plays an essential role in the classiication theory o principal G-bundles. Its base space BG turns out to be unique up to homotopy, and it is called the classiying space or principal G-bundles due to the ollowing undamental result: 14

Theorem 3.1. I X is a CW-complex, there exists a bijective correspondence Φ : P(X, G) = [X, BG] G Proo. By Theorem 2.10, Φ is well-deined. Let us next show that Φ is onto. Let P(X, G), : E X. We need to show that = G or some map : X BG, or equivalently, that there is a bundle map (, ) : G. By Theorem 2.11, this is equivalent to the existence o a section o the bundle E G EG X with iber EG. Since EG is contractible, such a section exists by the ollowing: Lemma 3.2. Let X be a CW complex, and : E X B (X, G, F, ρ) with i (F ) = 0 or all i 0. I A X is a subcomplex, then every section o over A extends to a section deined on all o X. In particular, has a section. Moreover, any two sections o are homotopic. Proo. Given a section σ 0 : A E o over A, we extend it to a section σ : X E o over X by using induction on the dimension o cells in X A. So it suices to assume that X has the orm X = A φ e n, where e n is an n-cell in X A, with attaching map φ : e n A. Since e n is contractible, is trivial over e n, so we have a commutative diagram e n σ 0 1 (e n ) e n h pr 1 = e n F σ with h : 1 (e n ) e n F the trivializing chart or over e n, and σ to be deined. Ater composing with h, we regard the restriction o σ 0 over e n as given by σ 0 (x) = (x, τ 0 (x)) e n F, with τ 0 : e n = S n 1 F. Since n 1 (F ) = 0, τ 0 extends to a map τ : e n F which can be used to extend σ 0 over e n by setting σ(x) = (x, τ(x)). Ater composing with h 1, we get the desired extension o σ 0 over e n. Let us now assume that σ and σ are two sections o. To ind a homotopy between σ and σ, it suices to construct a section Σ o id I : E I X I. Indeed, i such Σ exists, then Σ(x, t) = (σ t (x), t), and σ t provides the desired homotopy. Now, by regarding σ as a section o id I over X {0}, and σ as a section o id I over X {1}, the question reduces to constructing a section o id I, which extends the section over X {0, 1} deined by (σ, σ ). This can be done as in the irst part o the proo. 15

In order to inish the proo o Theorem 3.1, it remains to show that Φ is a one-to-one map. I 0 = G = g G = 1, we will show that g. Note that we have the ollowing commutative diagrams: E 0 = E G 0 X = X {0} E G G B G E 0 = E1 = g E G E G 0 X = X {1} ĝ G g B G where we regard ĝ as deined on E 0 via the isomorphism 0 = 1. By putting together the above diagrams, we have a commutative diagram E 0 I 0 Id α=(,0) (ĝ,1) E 0 {0, 1} E G 0 {0,1} G X I X {0, 1} α=(,0) (g,1) B G Thereore, it suices to extend (α, α) to a bundle map (H, Ĥ) : 0 Id G, and then H will provide the desired homotopy g. By Theorem 2.11, such a bundle map (H, Ĥ) corresponds to a section σ o the iber bundle ω : (E 0 I) G E G X I. On the other hand, the bundle map (α, α) already gives a section σ 0 o the iber bundle ω 0 : (E 0 {0, 1}) G E G X {0, 1}, which under the obvious inclusion (E 0 {0, 1}) G E G (E 0 I) G E G can be regarded as a section o ω over the subcomplex X {0, 1}. Since EG is contractible, Lemma 3.2 allows us to extend σ 0 to a section σ o ω deined on X I, as desired. Example 3.3. We give here a more conceptual reasoning or the assertion o Example 1.20. By Theorem 3.1, we have B(S n, G, F, ρ) = P(S n, G) = [S n, BG] = n (BG) = n 1 (G), where the last isomorphism ollows rom the homotopy long exact sequence or G, since EG is contractible. Back to the universal principal G-bundle, we have the ollowing 16

Theorem 3.4. Let G be a locally compact topological group. Then a universal principal G- bundle G : EG BG exists (i.e., satisying i (EG) = 0 or all i 0), and the construction is unctorial in the sense that a continuous group homomorphism µ : G H induces a bundle map (Bµ, Eµ) : G H. Moreover, the classiying space B G is unique up to homotopy. Proo. To show that BG is unique up to homotopy, let us assume that G : E G B G and G : E G B G are universal principal G-bundles. By regarding G as the universal principal G-bundle or G, we get a map : B G B G such that G = G, i.e., a bundle map: E G G B G ˆ E G G B G Similarly, y regarding G as the universal principal G-bundle or G, there exists a map g : B G B G such that G = g G. Thereore, G = g G = g G = ( g) G. On the other hand, we have G = (id BG ) G, so by Theorem 3.1 we get that g id BG. Similarly, we get g id B G, and hence : B G B G is a homotopy equivalence. We will not discuss the existence o the universal bundle here, instead we will indicate the universal G-bundle, as needed, in speciic examples. Example 3.5. Recall that we have a iber bundle O(n) V n (R ) G n (R ), (3.1) with V n (R ) contractible. In particular, the uniqueness part o Theorem 3.4 tells us that BO(n) G n (R ) is the classiying space or rank n real vector bundles. Similarly, there is a iber bundle U(n) V n (C ) G n (C ), (3.2) with V n (C ) contractible. Thereore, BU(n) G n (C ) is the classiying space or rank n complex vector bundles. Beore moving to the next example, let us mention here without proo the ollowing useul result: Theorem 3.6. Let G be an abelian group, and let X be a CW complex. There is a natural bijection T : [X, K(G, n)] H n (X, G) [] (α) where α H n (K(G, n), G) = Hom(H n (K(G, n), Z), G) is given by the inverse o the Hurewicz isomorphism G = n (K(G, n)) H n (K(G, n), Z). 17

Example 3.7 (Classiication o real line bundles). Let G = Z/2 and consider the principal Z/2-bundle Z/2 S RP. Since S is contractible, the uniqueness o the universal bundle yields that BZ/2 = RP. In particular, we see that RP classiies the real line (i.e., rank-one) bundles. Since we also have that RP = K(Z/2, 1), we get: P(X, Z/2) = [X, BZ/2] = [X, K(Z/2, 1)] = H 1 (X, Z/2) or any CW complex X, where the last identiication ollows rom Theorem 3.6. Let now be a real line bundle on a CW complex X, with classiying map : X RP. Since H (RP, Z/2) = Z/2[w], with w a generator o H 1 (RP, Z/2), we get a well-deined degree one cohomology class w 1 () := (w) = called the irst Stieel-Whitney class o. The bijection P(X, Z/2) H 1 (X, Z/2) is then given by w 1 (), so real line bundles on X are classiied by their irst Stieel-Whitney classes. Example 3.8 (Classiication o complex line bundles). Let G = S 1 and consider the principal S 1 -bundle S 1 S CP. Since S is contractible, the uniqueness o the universal bundle yields that BS 1 = CP. In particular, as S 1 = GL(1, C), we see that CP classiies the complex line (i.e., rank-one) bundles. Since we also have that CP = K(Z, 2), we get: P(X, S 1 ) = [X, BS 1 ] = [X, K(Z, 2)] = H 2 (X, Z) or any CW complex X, where the last identiication ollows rom Theorem 3.6. Let now be a complex line bundle on a CW complex X, with classiying map : X CP. Since H (CP, Z) = Z[c], with c a generator o H 2 (CP, Z), we get a well-deined degree two cohomology class c 1 () := (c) called the irst Chern class o. The bijection P(X, S 1 = ) H 2 (X, Z) is then given by c 1 (), so complex line bundles on X are classiied by their irst Chern classes. Remark 3.9. I X is any orientable closed oriented surace, then H 2 (X, Z) = Z, so Example 3.8 shows that isomorphism classes o complex line bundles on X are in bijective correspondence with the set o integers. On the other hand, i X is a non-orientable closed surace, then H 2 (X, Z) = Z/2, so there are only two isomorphism classes o complex line bundles on such a surace. 4 Exercises 1. Let p : S 2 RP 2 be the (oriented) double cover o RP 2. Since RP 2 is a non-orientable surace, we know by Remark 3.9 that there are only two isomorphism classes o complex line bundles on RP 2 : the trivial one, and a non-trivial complex line bundle which we denote by 18

: E RP 2. On the other hand, since S 2 is a closed orientable surace, the isomorphism classes o complex line bundles on S 2 are in bijection with Z. Which integer corresponds to complex line bundle p : p E S 2 on S 2? 2. Consider a locally trivial iber bundle S 2 E S 2. Recall that such can be regarded as a iber bundle with structure group G = Homeo(S 2 ) = SO(3). By the classiication Theorem 3.1, SO(3)-bundles over S 2 correspond to elements in [S 2, BSO(3)] = 2 (BSO(3)) = 1 (SO(3)). (a) Show that 1 (SO(3)) = Z/2. (Hint: Show that SO(3) is homeomorphic to RP 3.) (b) What is the non-trivial SO(3)-bundle over S 2? 3. Let : E X be a principal S 1 -bundle over the simply-connected space X. Let a H 1 (S 1, Z) be a generator. Show that c 1 () = d 2 (a), where d 2 is the dierential on the E 2 -page o the Leray-Serre spectral sequence associated to, i.e., E p,q 2 = H p (X, H q (S 1 )) H p+q (E, Z). 4. By the classiication Theorem 3.1, (isomorphism classes o) S 1 -bundles over S 2 are given by [S 2, BS 1 ] = 2 (BS 1 ) = 1 (S 1 ) = Z and this correspondence is realized by the irst Chern class, i.e., c 1 (). (a) What is the irst Chern class o the Hop bundle S 1 S 3 S 2? (b) What is the irst Chern class o the sphere (or unit) bundle o the tangent bundle T S 2? (c) Construct explicitely the S 1 -bundle over S 2 corresponding to n Z. (Hint: Think o lens spaces, and use the above Exercise 3.) 19