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Notes on principal bundles and classifying spaces
Stephen A. Mitchell
June 2011
1 Introduction
Consider a real nplane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, kframe bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)bundle” Pξ, which is just the bundle of orthonormal nframes. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnRbundle; this is the bundle of linearly independent nframes.
More generally, if G is any topological group, a principal Gbundle is a locally trivial free Gspace with orbit space B (see below for the precise definition). For example, if G is discrete then a principal Gbundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal Gbundles over X are in natural bijective correspondence with [X,BG]. The correspondence is given by pulling back a universal principal Gbundle over BG. When G is discrete, BG is an EilenbergMaclane space of type (G, 1). When G is either GLnR or O(n), BG is homotopy equivalent to the infinite Grassmanian GnR∞. The homotopy classification theorem for vector bundles then emerges as a special case of the homotopy classification theorem for principal bundles.
As these examples begin to suggest, the concept principal bundle acts as a powerful unifying force in algebraic topology. Classifying spaces also play a central role; indeed, much of the research in homotopy theory over the last fifty years involves analyzing the homotopy type of BG for interesting groups G. There are also many applications in differential geometry, involving connections, curvature, etc. In these notes we will study principal bundles and classifying spaces from the homotopytheoretic point of view.
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2 Definitions and basic properties
Let G be a topological group. A left Gspace is a space X equipped with a continuous left Gaction G×X −→ X. If X and Y are Gspaces, a Gequivariant map is a map φ : X −→ Y such that φ(gx) = gφ(x) for all g ∈ G, x ∈ X. Synonymous terms include equivariant (if the group G is understood) and Gmap (for short). This makes left Gspaces into a category. A Ghomotopy (or Gequivariant homotopy, or equivariant homotopy) between Gmaps φ, ψ is a homotopy F : X × I −→ Y in the usual sense, with the added condition that F be G equivariant (here G acts trivially on the I coordinate). This yields the Ghomotopy category of left Gspaces. Similar definitions apply to right Gspaces.
Now let B be a topological space. Suppose that P is a right Gspace equipped with a Gmap π : P −→ B, where G acts trivially on B (in other words, π factors uniquely through the orbit space P/G). We say that (P, π) is a principal Gbundle over B if π satisfies the following local triviality condition:
B has a covering by open sets U such that there exist Gequivariant homeomorphisms φU : π
−1U −→ U ×G commuting in the diagram
π−1U U ×G
U
φU
?
� �
� �
��
Here U ×G has the right Gaction (u, g)h = (u, gh). Note this condition implies that G acts freely on P, and that π factors through a homeomorphism π : P/G −→ B (thus B “is” the orbit space of P). Summarizing: A principal Gbundle over B consists of a locally trivial free Gspace with orbit space B.
A morphism of principal bundles over B is an equivariant map σ : P −→ Q over the identity of B (i.e., inducing the identity map on the orbit space). This makes the collection of all principal Gbundles over B into a category. The set of isomorphism classes of principal Gbundles over B will be denote PGB. A principal Gbundle is trivial if it is isomorphic to the product principal bundle B × G −→ B. Every principal bundle is locally trivial, by definition.
Note that (P, π) is in particular a local product over B with fibre G. To be a principal G bundle, however, is a far stronger condition. Here are two striking and important properties that illustrate this claim:
Proposition 2.1 Any morphism of principal Gbundles is an isomorphism.
Proof: Let σ : P −→ Q be a morphism. Suppose first that P = Q = B × G. Then σ(b, g) = (b, f(b)g) for some function f : B −→ G; clearly f is continuous. Hence σ is an isomorphism with σ−1(b, g) = (b, f(b)−1g). This proves the proposition in the case when P and Q are trivial. Since every principal bundle is locally trivial, the general case follows immediately.
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Proposition 2.2 A principal Gbundle π : P −→ B is trivial if and only if it admits a section.
Proof: If P is trivial, then there is a section; this much is trivially true for any local product. Conversely, suppose s : B −→ P is a section. Then the map φ : B × G −→ P given by φ(b, g) = s(b)g is a morphism of principal bundles, and is therefore an isomorphism by Proposition 2.1.
The difference between a principal Gbundle and a runofthemill local product with fibre G can be illustrated further in terms of transition functions. Suppose π : E −→ B is a local product with fibre G, and U,V are open sets over which π is trivial, with U ∩ V nonempty. Comparing the two trivializations leads to a homeomorphism (U ∩ V )×G −→ (U ∩ V )×G of the form (x, g) 7→ (x, φ(x)g), where the transition function φ is a map from U ∩V into the set of homeomorphisms from G to itself. In a principal Gbundle, each φ(x) is left translation by an element of G, and φ : U ∩ V −→ G is continuous.
Given a principal Gbundle P over B and a map f : B′ −→ B, we can form the pullback P ′ ≡ f ∗P ≡ B′ ×B P ; the pullback inherits a natural structure of principal Gbundle over B′ from P. The reader should note the following two simple and purely categorical facts: First, if Q is a principal Gbundle over B′, then bundle maps Q −→ f ∗P are in bijective correspondence with commutative squares
Q P
B′ B

? ? 
in which the top arrow is a Gequivariant map. Second, sections of the pullback bundle f ∗P are in bijective correspondence with lifts in the diagram
P
B′ B ?p p p p p
p p p� 
f
We conclude this section with an interesting special case of the pullback construction. We can pull P back over itself:
P ×B P P
P B
π ′′
?
π′
?
π
 π
Here π′ is projection on the lefthand factor, and defines a principal Gbundle structure in which the Gaction on P ×B P is on the righthand factor.
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Proposition 2.3 π′ : P ×B P −→ P is a trivial principal Gbundle over P.
Proof: The diagonal map P −→ P ×B P is a section; now apply Proposition 2.2.
Note that the trivialization obtained is the map P × G −→ P ×B P given by (p, g) 7→ (p, pg). By symmetry, a similar result holds for π′′, with the roles of the left and right factors reversed.
3 Balanced products and fiber bundles with structure
group
Note that any left Gaction on a space X can be converted to right action—and viceversa— by setting xg = g−1x, x ∈ X.
If W is a right Gspace and X is a left Gspace, the balanced product W×GX is the quotient space W ×X/ ∼, where (wg, x) ∼ (w, gx). Equivalently, we can simply convert X to a right Gspace as above, and take the orbit space of the diagonal action (w, x)g = (wg, g−1x); thus W ×G X = (W ×X)/G. The following special cases should be noted:
(i) If X = ∗ is a point, W ×G ∗ = W/G.
(ii) If X = G with the left translation action, the right action of G on itself makes W ×G G into a right Gspace, and the action map W × G −→ W induces a Gequivariant homeomorphism W ×G G
∼=−→ W .
Let G,H be topological groups. A (G,H)space is a space Y equipped with a left Gaction and right Haction, such that the two actions commute: (gy)h = g(yh). If we convert (say) the right Haction to a left action, this is the same thing as a left G×Haction. Note that if Y is a (G,H)space and X is a right Gspace, X ×G Y receives a right Haction defined by [x, y]h = [x, yh]; similarly Y ×H Z has a left Gaction if Z is a left Hspace.
Proposition 3.1 The balanced product is associative up to natural isomorphism: Let X be a right Gspace, Y a (G,H)space, and Z a left Hspace. Then there is a natural homeomor phism
(X ×G Y )×H Z ∼= X ×G (Y ×H Z)
Thus we can write X ×G Y ×H Z without fear of ambiguity. The proof is left as an exercise. The homeomorphism in question takes an equivalence class [x, y, z] on the left to the equivalence class [x, y, z] on the right; the only problem is to show that this map and its inverse are continuous. Here it is important to note the following trivial but useful lemma:
Lemma 3.2 Let X be any Gspace, π : X −→ X/G the