CHAPTER 2 Classification of Bundles In this chapter we prove Steenrod’s classification theorem of principal G - bundles, and the corresponding classification theorem of vector bundles. This theorem states that for every group G, there is a “classifying space” BG with a well defined homotopy type so that the homotopy classes of maps from a space X, [X, BG], is in bijective correspondence with the set of isomorphism classes of principal G - bundles, P rinG(X). We then describe various examples and constructions of these classifying spaces, and use them to study structures on principal bundles, vector bundles, and manifolds. 1. The homotopy invariance of fiber bundles The goal of this section is to prove the following theorem, and to examine certain applications such as the classification of principal bundles over spheres in terms of the homotopy groups of Lie groups. Theorem 2.1. Let p : E B be a fiber bundle with fiber F , and let f : X B and f : X B → 0 → 1 → be homotopic maps.Then the pull - back bundles are isomorphic, f0∗(E) ∼= f1∗(E). The main step in the proof of this theorem is the basic Covering Homotopy Theorem for fiber bundles which we now state and prove. Theorem 2.2. Covering Homotopy theorem. Let p : E B and q : Z Y be fiber 0 → → bundles with the same fiber, F , where B is normal and locally compact. Let h0 be a bundle map h˜ E 0 Z −−−−→ p q B Y " −−−h0−→ " 45 46 2. CLASSIFICATION OF BUNDLES Let H : B I Y be a homotopy of h0 (i.e h0 = H B 0 .) Then there exists a covering of the × → | ×{ } homotopy H by a bundle map ˜ E I H Z × −−−−→ p 1 q × B I Y. "× −−−H−→ " Proof. We prove the theorem here when the base space B is compact. The natural extension is to when B has the homotopy type of a CW - complex. The proof in full generality can be found in Steenrod’s book [39]. The idea of the proof is to decompose the homotopy H into homotopies that take place in local neighborhoods where the bundle is trivial. The theorem is obviously true for trivial bundles, and so the homotopy H can be covered on each local neighborhood. One then must be careful to patch the coverings together so as to obtain a global covering of the homotopy H. Since the space X is compact, we may assume that the pull - back bundle H∗(Z) B I has → × locally trivial neighborhoods of the form U I , where U is a locally trivial covering of B (i.e { α × j} { α} 1 there are local trivializations φ : U F p− (U )), and I , , I is a finite sequence of open α,β α × → α 1 · · · r intervals covering I = [0, 1], so that each Ij meets only Ij 1 and Ij+1 nontrivially. Choose numbers − 0 = t < t < < t = 1 0 1 · · · r so that t I I . We assume inductively that the covering homotopy H˜ (x, t) has been defined j ∈ j ∩ j+1 E [0, t ] so as to satisfy the theorem over this part. × j For each x B, there is a pair of neighborhoods (W, W $) such that for x W , W¯ W $ and ∈ ∈ ⊂ W¯ $ U for some U . Choose a finite number of such pairs (W , W $), (i = 1, , s) covering B. ⊂ α α i i · · · Then the Urysohn lemma implies there is a map u : B [t , t ] such that u (W¯ ) = t and i → j j+1 i i j+1 u (B W $) = t . Define τ (x) = t for x B, and j − i j 0 j ∈ τ (x) = max(u (x), , u (x)), x B, i = 1, , s. i 1 · · · i ∈ · · · Then t = τ (x) τ (x) t (x) = t . j 0 ≤ 1 ≤ · · · ≤ s j+1 Define B to be the set of pairs (x, t) such that t t τ (x). Let E be the part of E I lying i j ≤ ≤ i i × over Bi. Then we have a sequence of total spaces of bundles E t = E E E = E [t , t ]. × j 0 ⊂ 1 ⊂ · · · ⊂ s × j j+1 We suppose inductively that H˜ has been defined on Ei 1 and we now define its extension over Ei. − By the definition of the τ’s, the set Bi Bi 1 is contained in Wi$ [tj, tj+1]; and by the definition − − × of the W ’s, W¯ [t , t ] U I which maps via H to a locally trivial neighborhood, say V , $i × j j+1 ⊂ α × j k 1. THE HOMOTOPY INVARIANCE OF FIBER BUNDLES 47 1 for q : Z Y . Say φ : V F q− (V ) is a local trivialization. In particular we can define → k k × → k 1 ρ : q− (V ) F to be the inverse of φ followed by the projection onto F . We now define k k → k H˜ (e, t) = φk(H(x, t), ρ(H˜ (e, τi 1(x))) − where (e, t) Ei Ei 1 and x = p(e) B. ∈ − − ∈ It is now a straightforward verification that this extension of H˜ is indeed a bundle map on Ei. This then completes the inductive step. ! We now prove theorem 2.1 using the covering homotopy theorem. Proof. Let p : E B, and f ; X B and f : X B be as in the statement of the theorem. → 0 → 1 → Let H : X I B be a homotopy with H = f and G = f . Now by the covering homotopy × → 0 0 1 1 theorem there is a covering homotopy H˜ : f ∗(E) I E that covers H : X I B. By definition 0 × → × → this defines a map of bundles over X I, that by abuse of notation we also call H˜ , × H˜ f ∗(E) I H∗(E) 0 × −−−−→ X I X I. "× −−−=−→ "× This is clearly a bundle isomorphism since it induces the identity map on both the base space and on the fibers. Restricting this isomorphism to X 1 , and noting that since H = f , we get × { } 1 1 a bundle isomorphism H˜ f0∗(E) f1∗(E) −−−∼=−→ X 1 X 1 . ×"{ } −−−=−→ ×"{ } This proves theorem 2.1 ! We now derive certain consequences of this theorem. Corollary 2.3. Let p : E B be a principal G - bundle over a connected space B. Then for → any space X the pull back construction gives a well defined map from the set of homotopy classes of maps from X to B to the set of isomorphism classes of principal G - bundles, ρ : [X, B] P rin (X). E → G 48 2. CLASSIFICATION OF BUNDLES Definition 2.1. A principal G - bundle p : EG BG is called universal if the pull back → construction ρ : [X, BG] P rin (X) EG → G is a bijection for every space X. In this case the base space of the universal bundle BG is called a classifying space for G (or for principal G - bundles). The main goal of this chapter is to prove that universal bundles exist for every group G, and that the classifying spaces are unique up to homotopy type. Applying theorem 2.1 to vector bundles gives the following, which was claimed at the end of chapter 1. Corollary 2.4. If f : X Y and f : X Y are homotopic, they induce the same 0 → 1 → homomorphism of abelian monoids, f ∗ = f ∗ : V ect∗(Y ) V ect∗(X) 0 1 → V ect∗ (Y ) V ect∗ (X) R → R and hence of K theories f ∗ = f ∗ : K(Y ) K(X) 0 1 → KO(Y ) KO(X) → Corollary 2.5. If f : X Y is a homotopy equivalence, then it induces isomorphisms → ∼= f ∗ : P rin (Y ) P rin (X) G −−−−→ G ∼= V ect∗(Y ) V ect∗(X) −−−−→ K(Y ) ∼= K(X) −−−−→ Corollary 2.6. Any fiber bundle over a contractible space is trivial. Proof. If X is contractible, it is homotopy equivalent to a point. Apply the above corollary. ! The following result is a classification theorem for bundles over spheres. It begins to describe why understanding the homotopy type of Lie groups is so important in Topology. Theorem 2.7. There is a bijective correspondence between principal bundles and homotopy groups n P rinG(S ) = πn 1(G) ∼ − 1. THE HOMOTOPY INVARIANCE OF FIBER BUNDLES 49 n 1 where as a set πn 1G = [S − , x0; G, 1 ], which refers to (based) homotopy classes of basepoint − { } n 1 n 1 preserving maps from the sphere S − with basepoint x S − , to the group G with basepoint the 0 ∈ identity 1 G. ∈ Proof. Let p : E Sn be a G - bundle. Write Sn as the union of its upper and lower → hemispheres, n n n n 1 S = D+ S − D . ∪ − n n Since D+ and D are both contractible, the above corollary says that E restricted to each of − these hemispheres is trivial. Morever if we fix a trivialization of the fiber of E at the basepoint n 1 n x S − S , then we can extend this trivialization to both the upper and lower hemispheres. 0 ∈ ⊂ We may therefore write n n E = (D+ G) θ (D G) × ∪ − × n 1 where θ is a clutching function defined on the equator, θ : S − G. That is, E consists of → n n n 1 n the two trivial components, (D+ G) and (D G) where if x S − , then (x, g) (D+ G) × − × ∈ ∈ × is identified with (x, θ(x)g) (Dn G). Notice that since our original trivializations extended a ∈ − × n 1 n 1 common trivialization on the basepoint x S − , then the trivialization θ : S − G maps the 0 ∈ → basepoint x to the identity 1 G.