In mathematics, the theory of fiber bundles with a structure group ${\displaystyle G}$ (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from ${\displaystyle F_{1}}$ to ${\displaystyle F_{2}}$, which are both topological spaces with a group action of ${\displaystyle G}$. For a fiber bundle ${\displaystyle F}$ with structure group ${\displaystyle G}$, the transition functions of the fiber (i.e., the cocycle) in an overlap of two coordinate systems ${\displaystyle U_{\alpha }}$ and ${\displaystyle U_{\beta }}$ are given as a ${\displaystyle G}$-valued function ${\displaystyle g_{\alpha \beta }}$ on ${\displaystyle U_{\alpha }\cap U_{\beta }}$. One may then construct a fiber bundle ${\displaystyle F'}$ as a new fiber bundle having the same transition functions, but possibly a different fiber.

In general it is enough to explain the transition from a bundle with fiber ${\displaystyle F}$, on which ${\displaystyle G}$ acts, to the associated principal bundle (namely the bundle where the fiber is ${\displaystyle G}$, considered to act by translation on itself). For then we can go from ${\displaystyle F_{1}}$ to ${\displaystyle F_{2}}$, via the principal bundle. Details in terms of data for an open covering are given as a case of descent.

This section is organized as follows. We first introduce the general procedure for producing an associated bundle, with specified fiber, from a given fiber bundle. This then specializes to the case when the specified fiber is a principal homogeneous space for the left action of the group on itself, yielding the associated principal bundle. If, in addition, a right action is given on the fiber of the principal bundle, we describe how to construct any associated bundle by means of a fiber product construction.

Let ${\textstyle \pi :E\to X}$ be a fiber bundle over a topological space ${\displaystyle X}$ with structure group ${\displaystyle G}$ and typical fiber ${\displaystyle F}$. By definition, there is a left action of ${\displaystyle G}$ (as a transformation group) on the fiber ${\displaystyle F}$. Suppose furthermore that this action is faithful. There is a local trivialization of the bundle ${\displaystyle E}$ consisting of an open cover ${\displaystyle U_{i}}$ of ${\displaystyle X}$, and a collection of fiber maps$${\displaystyle \varphi _{i}:\pi ^{-1}(U_{i})\to U_{i}\times F}$$such that the transition maps are given by elements of ${\displaystyle G}$. More precisely, there are continuous functions ${\displaystyle g_{ij}\colon U_{i}\cap U_{j}\to G}$ such that$${\displaystyle \psi _{ij}(u,f):=\varphi _{i}\circ \varphi _{j}^{-1}(u,f)={\big (}u,g_{ij}(u)f{\big )},\quad {\text{for each }}(u,f)\in (U_{i}\cap U_{j})\times F\,.}$$This satisfies the cocycle condition:$${\displaystyle g_{ij}(x)g_{jk}(x)=g_{ik}(x),\quad x\in U_{i}\cap U_{j}\cap U_{k}}$$Now let ${\displaystyle F'}$ be a specified topological space, equipped with a continuous left action of ${\displaystyle G}$. Then the bundle associated with ${\displaystyle E}$ with fiber ${\displaystyle F'}$ is a bundle ${\displaystyle E'}$ with a local trivialization subordinate to the cover ${\displaystyle U_{i}}$ whose transition functions are given by$${\displaystyle \psi '_{ij}(u,f')={\big (}u,g_{ij}(u)f'{\big )},\quad {\text{for each }}(u,f')\in (U_{i}\cap U_{j})\times F'\,,}$$where the ${\displaystyle G}$-valued functions ${\displaystyle g_{ij}(u)}$ are the same as those obtained from the local trivialization of the original bundle ${\displaystyle E}$. This definition clearly respects the cocycle condition on the transition functions, since the functions ${\displaystyle \{g_{ij}\}_{i,j}}$ satisfy the cocycle condition. Hence, by the existence part of the fiber bundle construction theorem, this produces a fiber bundle ${\displaystyle E'}$ with fiber ${\displaystyle F'}$, which is associated with ${\displaystyle E}$ as claimed.

As before, suppose that ${\displaystyle E}$ is a fiber bundle with structure group ${\displaystyle G}$. In the special case when ${\displaystyle G}$ has a free and transitive left action on ${\displaystyle F'}$, so that ${\displaystyle F'}$ is a principal homogeneous space for the left action of ${\displaystyle G}$ on itself, then the associated bundle ${\displaystyle E'}$ is called the principal ${\displaystyle G}$-bundle associated with the fiber bundle ${\displaystyle E}$. If, moreover, the new fiber ${\displaystyle F'}$ is identified with ${\displaystyle G}$ (so that ${\displaystyle F'}$ inherits a right action of ${\displaystyle G}$ as well as a left action), then the right action of ${\displaystyle G}$ on ${\displaystyle F'}$ induces a right action of ${\displaystyle G}$ on ${\displaystyle E'}$. With this choice of identification, ${\displaystyle E'}$ becomes a principal bundle in the usual sense.

By the isomorphism part of the fiber bundle construction theorem, the construction is unique up to isomorphism. That is, between any two constructions, there is a ${\displaystyle G}$-equivariant bundle isomorphism. This is also called a gauge transformation. This allows us to speak of the principal G-bundle associated with a G-bundle. In this way, a principal ${\displaystyle G}$-bundle equipped with a right action is often thought of as part of the data specifying a fiber bundle with structure group ${\displaystyle G}$. One may then, as in the next section, go the other way around and derive any fiber bundle by using a fiber product.

Let ${\displaystyle \pi \colon P\to X}$ be a principal G-bundle. Given a faithful left action ${\displaystyle \rho :G\to \operatorname {Homeo} (F)}$ of ${\displaystyle G}$ on a fiber space ${\displaystyle F}$ (in the smooth category, we should have a smooth action on a smooth manifold), the goal is to construct a G-bundle ${\displaystyle \pi _{\rho }:E\to X}$ of the fiber space ${\displaystyle F}$ over the base space ${\displaystyle X}$ such that it is associated with ${\displaystyle P}$.

Define a right action of ${\displaystyle G}$ on ${\displaystyle P\times F}$ via