# Principal bundle

In mathematics, a **principal bundle**^{[1]}^{[2]}^{[3]}^{[4]} is a mathematical object which formalizes some of the essential features of the Cartesian product *X* × *G* of a space *X* with a group *G*. In the same way as with the Cartesian product, a principal bundle *P* is equipped with

- An action of
*G*on*P*, analogous to (*x*,*g*)*h*= (*x*,*gh*) for a product space. - A projection onto
*X*. For a product space, this is just the projection onto the first factor, (*x*,*g*) ↦*x*.

Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (*x*,*e*). Likewise, there is not generally a projection onto *G* generalizing the projection onto the second factor, *X* × *G* → *G* which exists for the Cartesian product. They may also have a complicated topology, which prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space.

A common example of a principal bundle is the frame bundle F*E* of a vector bundle *E*, which consists of all ordered bases of the vector space attached to each point. The group *G* in this case is the general linear group, which acts on the right in the usual way: by changes of basis. Since there is no preferred way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section.

Principal bundles have important applications in topology and differential geometry. They have also found application in physics where they form part of the foundational framework of gauge theories.

## Formal definition

A principal *G*-bundle, where *G* denotes any topological group, is a fiber bundle *π*:*P* → *X* together with a continuous right action *P* × *G* → *P* such that *G* preserves the fibers of *P* (i.e. if *y* ∈ P_{x} then *yg* ∈ P_{x} for all *g* ∈ *G*) and acts freely and transitively on them. This implies that each fiber of the bundle is homeomorphic to the group *G* itself. Frequently, one requires the base space *X* to be Hausdorff and possibly paracompact.

Since the group action preserves the fibers of *π*:*P* → *X* and acts transitively, it follows that the orbits of the *G*-action are precisely these fibers and the orbit space *P*/*G* is homeomorphic to the base space *X*. Because the action is free, the fibers have the structure of *G*-torsors. A *G*-torsor is a space which is homeomorphic to *G* but lacks a group structure since there is no preferred choice of an identity element.

An equivalent definition of a principal *G*-bundle is as a *G*-bundle *π*:*P* → *X* with fiber *G* where the structure group acts on the fiber by left multiplication. Since right multiplication by *G* on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by *G* on *P*. The fibers of *π* then become right *G*-torsors for this action.

The definitions above are for arbitrary topological spaces. One can also define principal *G*-bundles in the category of smooth manifolds. Here *π*:*P* → *X* is required to be a smooth map between smooth manifolds, *G* is required to be a Lie group, and the corresponding action on *P* should be smooth.

## Examples

The prototypical example of a smooth principal bundle is the frame bundle of a smooth manifold *M*, often denoted F*M* or GL(*M*). Here the fiber over a point *x* ∈ *M* is the set of all frames (i.e. ordered bases) for the tangent space *T*_{x}*M*. The general linear group GL(*n*,**R**) acts freely and transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal GL(*n*,**R**)-bundle over *M*.

Variations on the above example include the orthonormal frame bundle of a Riemannian manifold. Here the frames are required to be orthonormal with respect to the metric. The structure group is the orthogonal group O(*n*). The example also works for bundles other than the tangent bundle; if *E* is any vector bundle of rank *k* over *M*, then the bundle of frames of *E* is a principal GL(*k*,**R**)-bundle, sometimes denoted F(*E*).

A normal (regular) covering space *p*:*C* → *X* is a principal bundle where the structure group

acts on the fibres of *p* via the monodromy action. In particular, the universal cover of *X* is a principal bundle over *X* with structure group *π*_{1}(*X*) (since the universal cover is simply connected and thus *π*_{1}(**C**) is trivial).

Let *G* be a Lie group and let *H* be a closed subgroup (not necessarily normal). Then *G* is a principal *H*-bundle over the (left) coset space *G*/*H*. Here the action of *H* on *G* is just right multiplication. The fibers are the left cosets of *H* (in this case there is a distinguished fiber, the one containing the identity, which is naturally isomorphic to *H*).

Consider the projection *π*:*S*^{1} → *S*^{1} given by *z* ↦ *z*^{2}. This principal **Z**_{2}-bundle is the associated bundle of the Möbius strip. Besides the trivial bundle, this is the only principal **Z**_{2}-bundle over *S*^{1}.

Projective spaces provide some more interesting examples of principal bundles. Recall that the *n*-sphere *S*^{n} is a two-fold covering space of real projective space **RP**^{n}. The natural action of O(1) on *S*^{n} gives it the structure of a principal O(1)-bundle over **RP**^{n}. Likewise, *S*^{2n+1} is a principal U(1)-bundle over complex projective space **CP**^{n} and *S*^{4n+3} is a principal Sp(1)-bundle over quaternionic projective space **HP**^{n}. We then have a series of principal bundles for each positive *n*:

Here *S*(*V*) denotes the unit sphere in *V* (equipped with the Euclidean metric). For all of these examples the *n* = 1 cases give the so-called Hopf bundles.

## Basic properties

### Trivializations and cross sections

One of the most important questions regarding any fiber bundle is whether or not it is trivial, *i.e.* isomorphic to a product bundle. For principal bundles there is a convenient characterization of triviality:

**Proposition**.*A principal bundle is trivial if and only if it admits a global cross section.*

The same is not true for other fiber bundles. For instance, Vector bundles always have a zero section whether they are trivial or not and sphere bundles may admit many global sections without being trivial.

The same fact applies to local trivializations of principal bundles. Let *π* : *P* → *X* be a principal *G*-bundle. An open set *U* in *X* admits a local trivialization if and only if there exists a local section on *U*. Given a local trivialization

one can define an associated local section

where *e* is the identity in *G*. Conversely, given a section *s* one defines a trivialization Φ by

The simple transitivity of the *G* action on the fibers of *P* guarantees that this map is a bijection, it is also a homeomorphism. The local trivializations defined by local sections are *G*-equivariant in the following sense. If we write

in the form

then the map

satisfies

Equivariant trivializations therefore preserve the *G*-torsor structure of the fibers. In terms of the associated local section *s* the map *φ* is given by

The local version of the cross section theorem then states that the equivariant local trivializations of a principal bundle are in one-to-one correspondence with local sections.

Given an equivariant local trivialization ({*U*_{i}}, {Φ_{i}}) of *P*, we have local sections *s*_{i} on each *U*_{i}. On overlaps these must be related by the action of the structure group *G*. In fact, the relationship is provided by the transition functions

For any *x* ∈ *U*_{i} ∩ *U*_{j} we have

### Characterization of smooth principal bundles

If *π* : *P* → *X* is a smooth principal *G*-bundle then *G* acts freely and properly on *P* so that the orbit space *P*/*G* is diffeomorphic to the base space *X*. It turns out that these properties completely characterize smooth principal bundles. That is, if *P* is a smooth manifold, *G* a Lie group and *μ* : *P* × *G* → *P* a smooth, free, and proper right action then

*P*/*G*is a smooth manifold,- the natural projection
*π*:*P*→*P*/*G*is a smooth submersion, and *P*is a smooth principal*G*-bundle over*P*/*G*.

## Use of the notion

### Reduction of the structure group

Given a subgroup *H* of *G* one may consider the bundle whose fibers are homeomorphic to the coset space . If the new bundle admits a global section, then one says that the section is a **reduction of the structure group from G to H **. The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of

*P*which is a principal

*H*-bundle. If

*H*is the identity, then a section of

*P*itself is a reduction of the structure group to the identity. Reductions of the structure group do not in general exist.

Many topological questions about the structure of a manifold or the structure of bundles over it that are associated to a principal *G*-bundle may be rephrased as questions about the admissibility of the reduction of the structure group (from *G* to *H*). For example:

- A 2
*n*-dimensional real manifold admits an almost-complex structure if the frame bundle on the manifold, whose fibers are , can be reduced to the group . - An
*n*-dimensional real manifold admits a*k*-plane field if the frame bundle can be reduced to the structure group . - A manifold is orientable if and only if its frame bundle can be reduced to the special orthogonal group, .
- A manifold has spin structure if and only if its frame bundle can be further reduced from to the Spin group, which maps to as a double cover.

Also note: an *n*-dimensional manifold admits *n* vector fields that are linearly independent at each point if and only if its frame bundle admits a global section. In this case, the manifold is called parallelizable.

### Associated vector bundles and frames

If *P* is a principal *G*-bundle and *V* is a linear representation of *G*, then one can construct a vector bundle with fibre *V*, as the quotient of the product *P*×*V* by the diagonal action of *G*. This is a special case of the associated bundle construction, and *E* is called an associated vector bundle to *P*. If the representation of *G* on *V* is faithful, so that *G* is a subgroup of the general linear group GL(*V*), then *E* is a *G*-bundle and *P* provides a reduction of structure group of the frame bundle of *E* from GL(*V*) to *G*. This is the sense in which principal bundles provide an abstract formulation of the theory of frame bundles.

## Classification of principal bundles

Any topological group *G* admits a **classifying space** *BG*: the quotient by the action of *G* of some weakly contractible space *EG*, *i.e.* a topological space with vanishing homotopy groups. The classifying space has the property that any *G* principal bundle over a paracompact manifold *B* is isomorphic to a pullback of the principal bundle *EG* → *BG*.^{[5]} In fact, more is true, as the set of isomorphism classes of principal *G* bundles over the base *B* identifies with the set of homotopy classes of maps *B* → *BG*.

## See also

- Associated bundle
- Vector bundle
- G-structure
- Reduction of the structure group
- Gauge theory
- Connection (principal bundle)
- G-fibration

## References

- ↑ Steenrod, Norman (1951).
*The Topology of Fibre Bundles*. Princeton: Princeton University Press. ISBN 0-691-00548-6. page 35 - ↑ Husemoller, Dale (1994).
*Fibre Bundles*(Third ed.). New York: Springer. ISBN 978-0-387-94087-8. page 42 - ↑ Sharpe, R. W. (1997).
*Differential Geometry: Cartan's Generalization of Klein's Erlangen Program*. New York: Springer. ISBN 0-387-94732-9. page 37 - ↑ Lawson, H. Blaine; Michelsohn, Marie-Louise (1989).
*Spin Geometry*. Princeton University Press. ISBN 978-0-691-08542-5. page 370 - ↑ Stasheff, James D. (1971), "
*H*-spaces and classifying spaces: foundations and recent developments",*Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970)*, Providence, R.I.: American Mathematical Society, pp. 247–272, Theorem 2

## Sources

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*Gauge Theory and Variational Principles*. Addison-Wesley Publishing. ISBN 0-486-44546-1. - Jost, Jürgen (2005).
*Riemannian Geometry and Geometric Analysis*((4th ed.) ed.). New York: Springer. ISBN 3-540-25907-4. - Husemoller, Dale (1994).
*Fibre Bundles*(Third ed.). New York: Springer. ISBN 978-0-387-94087-8. - Sharpe, R. W. (1997).
*Differential Geometry: Cartan's Generalization of Klein's Erlangen Program*. New York: Springer. ISBN 0-387-94732-9. - Steenrod, Norman (1951).
*The Topology of Fibre Bundles*. Princeton: Princeton University Press. ISBN 0-691-00548-6.