Principal bundle

In mathematics, a principal bundle[1][2][3][4] is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group . In the same way as with the Cartesian product, a principal bundle is equipped with

  1. An action of on , analogous to for a product space.
  2. A projection onto . For a product space, this is just the projection onto the first factor, .

Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of . Likewise, there is not generally a projection onto generalizing the projection onto the second factor, that exists for the Cartesian product. They may also have a complicated topology that 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 of a vector bundle , which consists of all ordered bases of the vector space attached to each point. The group 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 natural 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 and mathematical gauge theory. They have also found application in physics where they form part of the foundational framework of physical gauge theories.

Formal definitionEdit

A principal  -bundle, where   denotes any topological group, is a fiber bundle   together with a continuous right action   such that   preserves the fibers of   (i.e. if   then   for all  ) and acts freely and transitively (i.e. regularly) on them in such a way that for each   and  , the map   sending   to   is a homeomorphism. In particular each fiber of the bundle is homeomorphic to the group   itself. Frequently, one requires the base space   to be Hausdorff and possibly paracompact.

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

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

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

ExamplesEdit

 
Non-trivial Z/2Z principal bundle over the circle. There is no obvious way to identify which point corresponds to +1 or -1 in each fibre. This bundle is non-trivial as there is no globally defined section of the projection π.
  • The prototypical example of a smooth principal bundle is the frame bundle of a smooth manifold  , often denoted   or  . Here the fiber over a point   is the set of all frames (i.e. ordered bases) for the tangent space  . The general linear group   acts freely and transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principal  -bundle over  .
  • 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  . The example also works for bundles other than the tangent bundle; if   is any vector bundle of rank   over  , then the bundle of frames of   is a principal  -bundle, sometimes denoted  .
  • A normal (regular) covering space   is a principal bundle where the structure group
 
acts on the fibres of   via the monodromy action. In particular, the universal cover of   is a principal bundle over   with structure group   (since the universal cover is simply connected and thus   is trivial).
  • Let   be a Lie group and let   be a closed subgroup (not necessarily normal). Then   is a principal  -bundle over the (left) coset space  . Here the action of   on   is just right multiplication. The fibers are the left cosets of   (in this case there is a distinguished fiber, the one containing the identity, which is naturally isomorphic to  ).
  • Consider the projection   given by  . This principal  -bundle is the associated bundle of the Möbius strip. Besides the trivial bundle, this is the only principal  -bundle over  .
  • Projective spaces provide some more interesting examples of principal bundles. Recall that the  -sphere   is a two-fold covering space of real projective space  . The natural action of   on   gives it the structure of a principal  -bundle over  . Likewise,   is a principal  -bundle over complex projective space   and   is a principal  -bundle over quaternionic projective space  . We then have a series of principal bundles for each positive  :
 
 
 
Here   denotes the unit sphere in   (equipped with the Euclidean metric). For all of these examples the   cases give the so-called Hopf bundles.

Basic propertiesEdit

Trivializations and cross sectionsEdit

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 : PX 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 ({Ui}, {Φi}) of P, we have local sections si on each Ui. On overlaps these must be related by the action of the structure group G. In fact, the relationship is provided by the transition functions

 

By gluing the local trivializations together using these transition functions, one may reconstruct the original principal bundle. This is an example of the fiber bundle construction theorem. For any xUiUj we have

 

Characterization of smooth principal bundlesEdit

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

  •   is a smooth manifold,
  • the natural projection   is a smooth submersion, and
  •   is a smooth principal  -bundle over  .

Use of the notionEdit

Reduction of the structure groupEdit

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   to   . The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of   that is a principal  -bundle. If   is the identity, then a section of   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  -bundle may be rephrased as questions about the admissibility of the reduction of the structure group (from   to  ). For example:

 
The frame bundle   of the Möbius strip   is a non-trivial principal  -bundle over the circle.
  • A  -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  -dimensional real manifold admits a  -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  -dimensional manifold admits   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 framesEdit

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

Classification of principal bundlesEdit

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 EGBG.[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 BBG.

See alsoEdit

ReferencesEdit

  1. ^ Steenrod, Norman (1951). The Topology of Fibre Bundles. Princeton: Princeton University Press. ISBN 0-691-00548-6. page 35
  2. ^ Husemoller, Dale (1994). Fibre Bundles (Third ed.). New York: Springer. ISBN 978-0-387-94087-8. page 42
  3. ^ Sharpe, R. W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9. page 37
  4. ^ Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5. page 370
  5. ^ 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

SourcesEdit