Editing Principal bundle (section)

{{Short description|Fiber bundle whose fibers are group torsors}}
{{inline|date=June 2016}}
In [[mathematics]], a '''principal bundle'''<ref>{{cite book | last = Steenrod | first = Norman | author-link=Norman Steenrod| title = The Topology of Fibre Bundles | url = https://archive.org/details/topologyoffibreb0000stee | url-access = registration | publisher = [[Princeton University Press]] | location = Princeton | year = 1951 | isbn = 0-691-00548-6}} page 35
</ref><ref>{{cite book | last = Husemoller | first = Dale | author-link=Dale Husemoller| title = Fibre Bundles | publisher = Springer | edition = Third |location = New York | year=1994 | isbn=978-0-387-94087-8}} page 42
</ref><ref>{{cite book | last = Sharpe | first = R. W. | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer | location = New York | year = 1997 | isbn = 0-387-94732-9}} page 37
</ref><ref>{{Cite book | last1=Lawson | first1=H. Blaine | author-link1=H. Blaine Lawson| last2=Michelsohn | first2=Marie-Louise |author2-link=Marie-Louise Michelsohn| title=Spin Geometry | publisher=[[Princeton University Press]] | isbn=978-0-691-08542-5 | year=1989 }} page 370</ref> is a mathematical object that formalizes some of the essential features of the [[Cartesian product]] <math>X \times G</math> of a space <math>X</math> with a [[group (mathematics)|group]] <math>G</math>. In the same way as with the Cartesian product, a principal bundle <math>P</math> is equipped with
# An [[Group action (mathematics)|action]] of <math>G</math> on <math>P</math>, analogous to <math>(x, g)h = (x, gh)</math> for a [[product space]] (where <math>(x, g)</math> is an element of <math>P</math> and <math>h</math> is the group element from <math>G</math>; the group action is conventionally a right action).
# A projection onto <math>X</math>. For a product space, this is just the projection onto the first factor, <math>(x,g) \mapsto x</math>.
Unless it is the product space <math>X \times G</math>, a principal bundle lacks a preferred choice of identity cross-section; it has no preferred analog of <math>x \mapsto (x,e)</math>. Likewise, there is not generally a projection onto <math>G</math> generalizing the projection onto the second factor, <math>X \times G \to G</math> 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]] <math>F(E)</math> of a [[vector bundle]] <math>E</math>, which consists of all ordered [[basis of a vector space|bases]] of the vector space attached to each point.  The group <math>G,</math> in this case, is the [[general linear group]], which acts on the right [[Frame bundle#Definition and construction|in the usual way]]: by [[change of basis|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 (mathematics)|gauge theory]]. They have also found application in [[physics]] where they form part of the foundational framework of physical [[gauge theory|gauge theories]].