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Classifying Spaces Omar Ortiz Department of Mathematics and Statistics University of Melbourne Parkville, VIC 3010 Australia [email protected] April 26, 2012 1 Principal Bundles Let G be a topological group. A principal G-bundle (over B) is a fibre bundle π : E −→ B with right G-action on E, π equivariant (with G acting trivially on B) and trivialization maps ψα : π −1 (Uα ) −→ Uα × G equivariant (where Uα × G has right G-action given by (u, g)h = (u, gh)). Proposition 1.1 If π : E −→ B is a principal G-bundle, then 1. π −1 (b) = xG, the orbit of x ∈ E if π(x) = b (i.e. G acts transitively on the fibers). 2. G acts freely on each fiber and on E. 3. B ' E/G. Proof 1. Let (Uα , ψα ) be a local trivialization of π and let b ∈ Uα . Restrict the commutative diagram π −1 (Uα ) π φ ∼ = / Uα × G p y Uα to π −1 (b) ⊆ π −1 (Uα ), to obtain φ ∼ = π −1 (b) π {b} p y 1 / {b} × G Let x ∈ π −1 (b) ⊆ E. So ψα (x) = (b, g) for some g ∈ G. Then φ(xG) = φ(x)G = (b, g)G = {b} × G = φ(π −1 (b)). Thus π −1 (b) = xG. 2. Let b ∈ B and π −1 (b) ⊆ E. Assume x ∈ π −1 (b) and h ∈ G are such that xh = x. Then (b, g) = ψα (x) = ψα (xh) = ψα (x)h = (b, g)h = (b, gh), for some g ∈ G. So gh = g, and therefore h = e, showing that G acts freely on the fiber π −1 (b) and consequently, freely on E. 3. We have seen that π −1 (b) = xG ⊆ E if π(x) = b for x ∈ E and b ∈ B. Hence π −1 : B −→ E/G is a well defined, continuous, bijective map with inverse π continuous, i.e. a homeomorphism. The last two properties of principal bundles in the above proposition actually characterise them, i.e. we have the following converse proposition: Proposition 1.2 Let G be a topological group acting freely and continuously on a space E. If the projection map π : E −→ E/G is a fiber bundle, then it is a principal bundle. Proof We have to show that π is an equivariant map where the action of G on E/G is the induced one from E under the quotient, and that there exists an equivariant local trivialization for π. Let g ∈ G and x ∈ E, then π(xg) = xgG = xG = xGg = π(x)g, showing that π is equivariant. Let {(Uα , ψα )}α∈Λ be a local trivialization family of π. Then for α ∈ Λ, the local section σα : Uα −→ π −1 (Uα ) defined by σα (b) = ψα (b, 1) for b ∈ Uα π −1 (Uα ) o D σα ψα ∼ = Uα × G , π y Uα ∼ = enable us to define an equivariant trivializing map φα : Uα × G −→ π −1 (Uα ) by the rule φα (b, g) = σα (b)g for b ∈ Uα and g ∈ G. 2 Examples 1.1 1. π : C2 \ {0} (z, w) / CP 1 is a principal C∗ -bundle: / [z, w] C2 \{0} is a C∗ -space under the action (z, w)λ = (zλ, wλ) for λ ∈ C∗ , (z, w) ∈ C2 \{0}. The local trivializations are Ui = {[z1 , z2 ] ∈ CP 1 | zi 6= 0} with trivialization maps / Ui × C∗ φi : π −1 (Ui ) (z1 , z2 ) / z1 z2 , , zi zi zi , for i = 1, 2. These trivializations are C∗ -equivariant, since for λ ∈ C∗ and (z1 , z2 ) ∈ π −1 (Ui ), z1 z2 z1 z2 φi ((z1 , z2 )λ) = φi ((z1 λ, z2 λ)) = , , zi λ = , , zi λ = φi ((z1 , z2 ))λ. zi zi z i zi The projection π is also C∗ -equivariant, for λ ∈ C∗ and (z, w) ∈ C2 \ {0}, π((z, w)λ) = π((zλ, wλ)) = [zλ, wλ] = [z, w] = [z, w]λ = π(z, w)λ. 2. π : R r / S1 = {eiθ ∈ C | θ ∈ R} is a principal Z-bundle: / ei2πr The group Z acts on R by translations: rn = r + n for n ∈ Z and r ∈ R. A set of local trivializations is U1 = {eiθ | θ ∈ (0, 2π)} and U2 = {eiθ | θ ∈ (−π, π)}, with trivilization maps φ1 : π −1 (U1 ) r =x+n φ2 : π −1 (U2 ) r =x+n / U1 × Z , where x ∈ (0, 1), / (ei2πx , n) / U2 × Z , where x ∈ / (ei2πx , n) The trivializations and the projection π are Z-equivariant. 3 1 1 − , . 2 2 A morphism of principal G-bundles between two principal G-bundles π and π 0 is an equivariant map σ : E −→ E 0 between the total spaces of the bundles. 2 Associated Bundle Let π : E −→ B be a principal G-bundle and let Y be a left G-space (with continuous action). The bundle associated to π and Y is the fiber bundle πY : E ×G Y −→ B with fiber Y where E ×G Y = (E × Y )/G with G acting on E × Y via (x, y)g = (xg, g −1 y) for x ∈ E, y ∈ Y and g ∈ G. And πY ([x, y]) = π(x) for [x, y] ∈ E ×G Y . Note that [xg, y] = [x, gy] in E ×G Y for x ∈ E, y ∈ Y and g ∈ G. To see that the fiber is Y , let b ∈ B, then πY−1 (b) = eG ×G Y = e ×G GY = e ×G Y ∼ =Y. 3 Pullback Bundles If f : B 0 −→ B is a map and π : E −→ B a fibre bundle, the pullback bundle f ∗ (π) is the fibre bundle f ∗ (π) : E 0 −→ B 0 where E 0 is the pullback /E E0 f ∗ (π) B0 f π /B Proposition 3.1 If f : B 0 −→ B is a map and π : E −→ B is a fibre bundle, then • f ∗ (π) has the same fibre as π. • If π is a principal G-bundle, f ∗ (π) is also. Proposition 3.2 If B 0 is paracompact, π : E −→ B is a principal G-bundle and f, g : B 0 −→ B are homotopic, then f ∗ (π) ' g ∗ (π) the pullback bundles are isomorphic as principal G-bundles. Corollary 3.1 If B is paracompact and contractible, every principal G-bundle over B is trivial (isomorphic to B × G). 4 4 Universal Bundles and Classifying Spaces A universal G-bundle π : EG −→ BG is a principal G-bundle with G and BG having the homotopy type of a CW-complex and such that if B is a CW-complex, the map [B, BG] / PG (B) f / f ∗ (π), from homotopy classes of maps B −→ BG to principal G-bundles over B, is a bijection. Theorem 4.1 A principal G-bundle π : E −→ B, with G and B having the homotopy type of a CW-complex, is universal if and only if E is contractible. Corollary 4.1 Let G be a topological group. Then BG is path-connected. Proof EG is contractible ⇒ EG is n-connected ⇒ EG is path-connected ⇒ BG = EG/G is path-connected. Proposition 4.1 If π : EG −→ BG is a universal G-bundle, then • BG is unique up to homotopy equivalence and is called the classifying space of G. • EG is unique up to G-equivariant homotopy equivalence. Theorem 4.2 (Milnor) (Existence of Classifying Spaces) If G is a topological group, there exists a classifying space for G. Proposition 4.2 Let G be a topological group, then • ΩBG ' G. • πn (BG) ' πn−1 (G), n ≥ 1. • If G is discrete, BG = K(G, 1) the Eilenberg MacLane space with universal cover EG. Theorem 4.3 Let G be a topological group, then • If H is a closed subgroup of G then BH −→ BG is a fibration with fibre G/H. • If N is a closed normal subgroup of G then BG −→ B(G/H) is a fibration with fibre BH. 5 Remark 4.1 It is possible to consider arbitrary topological groups G and base spaces BG for universal bundles instead of restricting to the case of both having the homotopy type of a CW-complex, as we have assumed above. In order to obtain this generalisation, one must work only with numerable bundles (bundles with a trivializing cover having partitions of unity), as can be seen on [Do]. 5 Examples • BZ = S1 . • BZn = Tn . • B(Z/2Z) = RP n . • BS1 = CP ∞ . • If H is a closed subgroup of G, BH = EG/H. References [Ben] D. J. Benson. Representations and cohomology II: Cohomology of groups and modules. Sections 2.3 and 2.4. Cambridge studies in advanced mathematics 31, Cambridge University Press. Cambridge, 1991. [Do] A. Dold. Partitions of Unity in the Theory of Fibrations. Annals of Math, Second Series, Vol. 78, No. 2 (Sep. 1963), pp. 223-225. [Mit] S. A. Mitchell. Notes on principal bundles and classifying spaces. August 2001. [Rey] P. Reynolds. Associated Vector Bundles. Notes, June 2010. 6