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In mathematics, a principal bundle is a mathematical object that 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 Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of $G$. Likewise, there is not generally a projection onto $P$ generalizing the projection onto the second factor, $(x, g)h = (x, gh)$ 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.
 * 1) An action of $X$ on $(x,g) ↦ x$, analogous to $(x,e)$ for a product space.
 * 2) A projection onto $G$. For a product space, this is just the projection onto the first factor, $X × G → G$.

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 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. They have also found application in physics where they form part of the foundational framework of gauge theories.

Formal definition
Let $G$ be a topological group. To define a principal $$G$$-bundle it is instructive to first define several more general structures.

Definition 1. A (right) $G$-space is a topological space $$X$$ with a group action $$X \times G \to X$$, denoted by juxtaposition $$(x,s) \mapsto xs$$, that fulfils the usual requirements of a group action. One could define a left $$G$$-space analogously, but the two concepts are equivalent.

Now define an equivalence relation on $$X$$: Two elements $$x,\, x'$$are $$G$$-equivalent if there is an $$s \in G$$ such that $$xs = x'$$. One can then consider the set of equivalence classes under this relation which form the set $$X \text{ mod } G$$. It is possible to show that $$X$$ with the projection $$\pi : X \to X \text{ mod } G$$is a fiber bundle over $$X \text{ mod } G$$.

Definition 2. A $$G$$-bundle is a fiber bundle that is isomorphic to the bundle $$(X, \pi, X \text{ mod } G)$$.

Definition 3. A $G$-space $$X$$ is called free if the group action acts freely on $$X$$. On a free $G$-space $$X$$ one can define the space $$X^* := \{(x, xs) : x \in X, s \in G\} \subset X \times X$$.

Intuitively, $$X^*$$ is the set of all pairs in $$X$$ that live on the same fiber if one regards $$X$$ as $$G$$-bundle. Additionally, one can find the translation function $$\tau : X^* \to G$$ fulfilling $$x\, \tau (x, x') = x'$$for all $$(x, x') \in X^*$$.

Defintion 4. A principal $G$-space is a free $G$-space whose translation function $$\tau$$ is continuous. A principal $G$-bundle is a $G$-bundle that is also a principal $G$-space.

These definitions imply that a principal $$G$$-bundle is a fiber bundle together with a continuous right action $X × G → X$ 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.