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 FE 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 in the usual way on ordered bases. 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. Principal bundles provide a unifying framework for the theory of fiber bundles in the sense that all fiber bundles with structure group G determine a unique principal G-bundle from which the original bundle can be reconstructed.

1 Formal definition
2 Examples
3 Basic properties
- 3.1 Trivializations and cross sections
- 3.2 Characterization of smooth principal bundles
4 Use of the notion
- 4.1 Reduction of the structure group
- 4.2 Associated vector bundles and frames
5 Classification of principal bundles
6 See also
7 References
8 Books

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 and acts freely and transitively on them. This implies that the 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 FM or GL(M). Here the fiber over a point x in M is the set of all frames (i.e. ordered bases) for the tangent space T_xM. 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 $\pi_1(X)/p_{*}(\pi_1(C))$ 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 $\pi_1(X)$ (since the universal cover is simply connected and thus $\pi_1(C)$ 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: S¹ → S¹ given by z ↦ z². This principal Z₂-bundle is the associated bundle of the Möbius strip. Besides the trivial bundle, this is the only principal Z₂-bundle over S¹.

Projective spaces provide some more interesting examples of principal bundles. Recall that the n-sphere Sⁿ is a two-fold covering space of real projective space RPⁿ. The natural action of O(1) on Sⁿ gives it the structure of a principal O(1)-bundle over RPⁿ. Likewise, S²ⁿ⁺¹ is a principal U(1)-bundle over complex projective space CPⁿ and S⁴ⁿ⁺³ is a principal Sp(1)-bundle over quaternionic projective space HPⁿ. We then have a series of principal bundles for each positive n:

$\mbox{O}(1) \to S(\mathbb{R}^{n%2B1}) \to \mathbb{RP}^n$

$\mbox{U}(1) \to S(\mathbb{C}^{n%2B1}) \to \mathbb{CP}^n$

$\mbox{Sp}(1) \to S(\mathbb{H}^{n%2B1}) \to \mathbb{HP}^n.$

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 $\Phi�: \pi^{-1}(U) \to U \times G$ one can define an associated local section $s�: U \to \pi^{-1}(U)$ by

$s(x) = \Phi^{-1}(x,e)\,$

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

$\Phi^{-1}(x,g) = s(x)\cdot g$

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 $\Phi�: \pi^{-1}(U) \to U \times G$ in the form $\Phi(p) = (\pi(p), \phi(p))$ then the map $\phi�: P \to G$ satisfies

$\varphi(p\cdot g) = \varphi(p)g.$

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

$\varphi(s(x)\cdot g) = g.$

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

$t_{ij} = U_i \cap U_j \to G\,$

For any x in U_i ∩ U_j we have

$s_j(x) = s_i(x)\cdot t_{ij}(x).$

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 $P/H$ whose fibers are homeomorphic to the coset space $G/H$ . 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 2n-dimensional real manifold admits an almost-complex structure if the frame bundle on the manifold, whose fibers are $GL(2n,\mathbb{R})$ , can be reduced to the group $GL(n,\mathbb{C}) \subset GL(2n,\mathbb{R})$ .

An n-dimensional real manifold admits a k-plane field if the frame bundle can be reduced to the structure group $GL(k,\mathbb{R}) \subset GL(n,\mathbb{R})$ .

A manifold is orientable if and only if its frame bundle can be reduced to the special orthogonal group, $SO(n) \subset GL(n,\mathbb{R})$ .

A manifold has spin structure if and only if its frame bundle can be further reduced from $SO(n)$ to $Spin(n)$ the Spin group, which maps to $SO(n)$ 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

Classification of principal bundles

Main article: Classifying space

Any topological group G admits a classifying space BG: the quotient of some weakly contractible space EG, i.e. a topological space for which all its homotopy groups are trivial by a free action of G. 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 $EG\longrightarrow BG.$ ^[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.

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

Books

Bleecker, David (1981). Gauge Theory and Variational Principles. Addison-Wesley Publishing. ISBN 0-486-44546-1 (Dover edition).
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.