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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.
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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.
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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).
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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.
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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].
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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.
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