Basics of Fiber Bundles and Principal Bundles

1. Introduction

In this note we aim to provide a basic introduction to the theory of fiber bundles, and the special role principal bundles play in their classification. These notes are primarily adapted from Rui's Math 519 course notes in Spring 2025 at UIUC, with additional perspectives from Jake Rasmussen's Math 526 in Fall 2024.

2. Fiber Bundles

To begin we'll start with the theory of fiber bundles and locally trivial fibrations. Throughout let denote the ambient category of topological objects (e.g. topological spaces, compactly generated weak Hausdorff spaces, possibly infinite dimensional smooth manifolds, etc.).

Locally Trivial Fibration

A locally trivial fibration with fiber type is a map such that for every , there exists an open neighborhood of of and an isomorphism in .

Note that if is a trivializing cover for a fiber bundle with fiber type , then the transition function determines a cocycle (which is a map in if it is a cartesian closed category, which we assume moving forward).

If in addition is a group object such that we have an effective group action (i.e. the induced map is a monomorphism), then we can add the requirement that the cocycle takes values in the image of . This leads to the notion of fiber bundle.

Fiber Bundle

A fiber bundle with fiber and structure group consists of a locally trivial fibration with fiber type , an effective left -action , and a collection of locally trivializing charts such that

  2. There exists in , such that
  3. The collection is maximal in the sense that if is a trivializing chart such that there exists with , then .

When is a vector space and , this definition recovers the notion of a vector bundle. When , with the free transitive action by left translation, this definition encodes the notion of a -principal bundle.

A morphism of -fiber bundles is a map in such that for all , the induced map given by pulling back along fibers is a map of -spaces. In particular, if and are trivializations for and , with , then being -equivariant implies that for each , the induced map must satisfy , so that we get an induced map such that .

Now, we say that -cocycles and for covers of are equivalent if there exists a common refinement for the open covers together with maps such that for all ,
If denotes the set of equivalence classes of -cocycles, then we have a one-to-one correspondence
between elements of and isomorphism classes of -fiber bundles over with fiber fixed. Explicitly, given a -cocycle for a cover of , the associated -fiber bundle is

Construction: Reduction of Structure Group

Given a closed subgroup , we say that the structure group of a -fiber bundle can be reduced to if there exists a cocycle for with values in .

Reduction Examples
  1. A -fiber bundle is trivial (i.e. isomorphic to ) if and only if the structure group can be reduced to .
  2. For a vector bundle, the structure group can be reduced to
    1. if and only if it is orientable
    2. if and only if it admits a metric
    3. if and only if it is orientable and admits a metric

Notice that the cocycle description does not reference the fiber . As a consequence of the previous correspondence, we then have the following:

  • To every -fiber bundle with fiber , there is associated a -principal bundle (defined up to isomorphism)

Alternatively, to every -principal bundle, , if is a space with effective -action , we have an associated -fiber bundle with fiber . Further, and is a -cocycle associated to the -principal bundle , so that , then

so that corresponds to the same cocycle as .

Theorem: Fiber Bundles Via Principal Bundles Via Cocycles

Let be agroup, let be a space with an effective -action, and let . Then the composite of bijections
can be given explicitly by sending a -principal bundle to the -fiber bundle .

This perspective plays an essential role due to the simplicity with which -Principal bundles can be described, and the number of classifying space results associated with them.

2.1. Principal -Bundles

First, let's recall some constructions of -principal bundles when is or .

Theorem: Slice Theorem

Let be a proper action. For each there exists an embedded submanifold containing and satisfying

  1. is -invariant
  2. , where is the orbit at in
  3. For and such that , .

Proof.
TBL
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Theorem: Smooth Structure on Quotient

If is a free and proper action, then the quotient space has a unique (smooth) structure such that the quotient map is a submersion.

Proof.
TBL
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Theorem: Quotient Principal Bundle

Let be a free and proper right action. Then is a -principal bundle. Further, every -principal bundle is of this form.

Proof.
TBL
***

Thus, we can identify -principal bundles with free and proper right actions , so that for any left action , we get another free and proper action , where , so that quotienting by this action gives the associated bundle .

Associated Bundles
  1. The trivial -principal bundle is , with associated bundle for any action the trivial -bundle .
  2. Homogeneous spaces: For a closed subgroup, is a principal -bundle
  3. Covering Spaces: A universal covering is a principal -bundle . More generally, if is a normal subgroup, is a covering space with group of deck transformations , which is a principal -bundle.
  4. Frame Bundle For any vector bundle with fiber a vector space , it's associated frame bundle is the principal -bundle The associated bundle is .

If instead one uses the induced actions or , then we get and

Note that if is a principal -bundle and is a space with an effective -action, then a section of is equivalent to a map such that . In other words, . More generally, if is an open subset, written as for open, then .

Note: In general, a -fiber bundle does not need to have any sections. In particular, a -principal bundle has a section if and only if we have a -equivariant map in the sense that .

In the case where is a -fiber bundle with -contractible, we always have sections, and any two sections are homotopic. Indeed, if is -contractible, so that we have a -equivariant homotopy for , where is necessarily a -fixed point, then gives a -equivariant map, and hence a section, while any other -equivariant map is -homotopic to this one, and hence induces homotopic sections.

-Structure for a Manifold

Given a closed subgroup and a (smooth) manifold , a reduction of is called a -structure. The reduction is called integrable if it locally has sections consisting of commuting vector fields (equivalently if has local charts with induced local frames determining local sections of the reduced -Principal bundle ).

Reduction -Structure Integrable -Structure
Riemannian structure Flat Riemannian structure
Almost symplectic structure ( with ) Symplectic structure (, and )
Almost complex structure (, ) Complex Structure (, )
Almost Hermitian structure Kahler structure

Now, note that the pullback of -fiber bundles with structure group are again -fiber bundles with structure group . In particular, if is a principal -bundle, then for , so is , and for an effective -space , . Further, homotopic maps induce isomorphic principal -bundles, and hence isomorphic -fiber bundles with structure group .

Trivial Bundles over Products with Intervals

Let be a fiber bundle with fiber and structure group . If and are trivial, so is .

Proof.
Let and be trivializations. Then, let be given by

which is continuous. Further, this can be made smooth by using bump functions if and are smooth.
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Open trivializing Cover

If is a fiber bundle with fiber and structure group , then there exists an open cover of such that is trivial.

Proof.
For all and , we have and trivializations . As is compact, for all we have such that is trivial. Let . Then if , the fiber bundle

is trivial for all and all . Applying the lemma on gluing trivializations inductively we have that is trivial, and hence the give the desired cover.
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Isomorphic fibers for Fiber Bundle over Interval

Let be a fiber bundle with fiber and structure group , where is paracompact. Then we have a bundle isomorphism

Proof.
Let be an open cover of such that is trivial for all , as guaranteed in the lemma on open trivializing covers. Let be a partition of unity subordinate to a countable sub-cover of this open cover, with . Then define by

Then let be given by , and consider the fiber bundle given by pulling back along . Explicitly,

Now, by our choice of cover we have trivializations . If and , then

where . Now, we define a bundle isomorphism by

Note that on the overlap we have , so this is continuous and well-defined (in fact if the bundle was smooth and the partition of unity was chosen to be smooth, then this would also be smooth). Thus, for all we have an isomorphism

Taking the colimit we get a bundle isomorphism , but here and , so this completes the proof.
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Isomorphic Bundles from Homotopic Pullbacks

Let be a paracompact Hausdorff space, a fiber bundle with fiber and structure group , and homotopic maps. Then as fiber bundles over .

Recall that metrizable spaces are paracompact Hausdorff, along with CW complexes, so this result applies in a wide-variety of contexts, and in particular on the level of homotopy types.

Proof.
Let be the homotopy (which can be chosen to be smooth if the maps are smooth). Consider the pullback , so that and for the natural inclusions into the top and bottom of the cylinder. Then by the isomorphism for bundles over a product with an interval we have bundle isomorphisms

as desired.
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An important consequence of this result is that fiber bundles over contractible spaces are necessarily trivial (this can be seem by consider the homotopy between the identity map and the constant map at a point).

Bundles over Contractible Spaces are Trivial

If is a fiber bundle with fiber and structure group , and if is contractible and paracompact Hausdorff, then is a trivial bundle.

Bundles over Product via Product Bundles

Suppose is a compact Hausdorff space and is a fiber bundle with fiber and structure group . Then where .

Proof.
Let be a compact Hausdorff space and let be a vector bundle over . Let be the pullback bundle along the inclusion . Define a continuous homotopy by

   

Since is compact and Hausdorff, so is the product , as is compact and Hausdorff, so in particular the product is paracompact. Thus, by homotopy invariance of pullbacks we have that
as vector bundles. Note that is the iterated pullback

I claim . For this observe that by iterating pullbacks we have a pullback square

However, under the canonical isomorphism , the bottom map is the identity since for all . Thus, since identity maps pullback to identity maps, up to isomorphism, we have a unique homeomorphism making the diagram

commute. In particular, , and so for any fixed, maps into by sending to . Since the vector space structure on the fibers of are induced by those on the fibers of , induces linear isomorphisms on fibers, and thus is a vector bundle isomorphism since it is also a homeomorphism.

On the other hand, I claim . First note that we iterated pullbacks

which by iterating pullbacks gives a pullback

Thus, by the universal property of pullbacks to show that it suffices to show that the diagram

is a pullback where maps to , and is the inclusion of . Consider a space and maps and making the outer diagram in

commute. Note that for and . Then the outer square commuting means that for any , In particular, this implies that factors through , while It follows that the map makes the diagram

commute, and further the diagram commuting uniquely specifies . Therefore, by the universal property of the pullback we have a unique homeomorphism making the diagram

commute. Explicitly, , so when restricted to a fiber , with , we have that is a linear isomorphism onto the fiber , being the identity on the vector component. Therefore, as is a homeomorphism which is a linear isomorphism on fibers, is a vector bundle isomorphism over . Therefore, composing isomorphisms of vector bundles we have the isomorphism of vector bundles

as desired.
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It follows that we have the classification result:

Universal -Fiber Bundle with Structure Group

Let with universal -principal bundle . Then if is any effective -space, and is any numerable space, then pullback induces a natural bijection
In particular,

Connections and Monodromy

Recall from linear connections we have a notion of (linear) connection on a general vector bundle over a smooth manifold . We can generalize this to principal bundles using the notion of distribution.

Distributions

For a rank vector bundle on , a distribution is a section of the bundle where is the frame bundle, and is the Grassmannian for -hyperplanes ().

In other words, a distribution is a (smooth) family of hyperplanes in .

Connection on a Principal Bundle

A connection on a -principal bundle is a distribution such that it is

  • (i) Horizontal:
  • (ii) -Invariance: for all and , where is the right action by .

We write for the vertical distribution. Since is horizontal restricts to a linear isomorphism . In this setting we have that horizontal lift:

Principal Connection on Frame Bundle

Every connection on a vector bundle induces a principal connection on where for and , , if we define

then the associated horizontal distribution is given by

From the horizontal space for linear connections this defines a horizontal distribution, and -invariance follows from the linearity of the action of on fibers.

Equivalently, a connection on a principal -bundle corresponds to a connection -form . Here we use that the local trivializations give us isomorphisms

at , where is mapped to . Then the connection -form is defined at by if . The horizontal distribution for the connection can then be recovered as .

Properties of Connection -Form

For a -principal bundle with principal connection , its connection -form satisfies:

  • (i) for all
  • (ii) for all .

Conversely, any satisfying (i) and (ii) is the connection -form of a unique principal connection .

Proof.
(i) is definitional. For (ii) let . Then writing , we have

for some , so that

where by invariance , and so by uniqueness

as desired.

Conversely, if satisfying (i) and (ii), then is horizontal by (i) and -invariant by (ii).
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Corollary

Any -principal bundle admits a connection.

Proof.
A trivial -principal bundle admits a connection given by . So using a trivialization we have connection -forms , so we can use a partition of unity subordinate to the trivialization to obtain

which satisfies (i) and (ii).
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Exterior Covariant Derivative

The exterior covariant derivative operator of a -principal bundle with connection is:

Note: In general .

Principal Curvature -form

For a -principal bundle with connection , the curvature -form of the connection is

Given a trivialization of , we obtain local sections such that . Then the local connection 1-forms and local curvature 2-forms are given by and . On , if is the cocycle, then

so that

Eqn

where is the left Maorer-Cartan form on , which is given by

Also, we have

Eqn
Example

Fix a vector bundle with linear connection . Then the frame bundle has connection -form . Fixing local sections over for gives the usual local connection -forms and curvature tensors:

and

The curvature -form satisfies the Cartan structure equations as well as Bianchi's identity . Here if , is defined by

Properties of Connection 1-form and Curvature 2-form

For a principal -bundle with connection , if is the connection 1-form and is the curvature 2-form, then we have the following:

  • (i) Cartan's Structure Equation:
  • (ii) Bianchi's Identity:

Proof.
By definition, , so that , while is defined at by if . Now,

Recall that
while . From the properties of the connection 1-form we have that

so

TBC
***

2.1.1. References