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 F with structure group G, the transition functions of the fiber (i.e., the cocycle) in an overlap of two coordinate systems U_α and U_β are given as a G-valued function g_αβ on U_α∩U_β. One may then construct a fiber bundle F′ as a new fiber bundle having the same transition functions, but possibly a different fiber.

An example

A simple case comes with the Möbius strip, for which ${\displaystyle G}$ is the cyclic group of order 2, ${\displaystyle \mathbb {Z} _{2}}$. We can take as ${\displaystyle F}$ any of: the real number line ${\displaystyle \mathbb {R} }$, the interval ${\displaystyle [-1,\ 1]}$, the real number line less the point 0, or the two-point set ${\displaystyle \{-1,\ 1\}}$. The action of ${\displaystyle G}$ on these (the non-identity element acting as ${\displaystyle x\ \rightarrow \ -x}$ in each case) is comparable, in an intuitive sense. We could say that more formally in terms of gluing two rectangles ${\displaystyle [-1,\ 1]\times I}$ and ${\displaystyle [-1,\ 1]\times J}$ together: what we really need is the data to identify ${\displaystyle [-1,\ 1]}$ to itself directly at one end, and with the twist over at the other end. This data can be written down as a patching function, with values in G. The associated bundle construction is just the observation that this data does just as well for ${\displaystyle \{-1,\ 1\}}$ as for ${\displaystyle [-1,\ 1]}$.

Construction

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 fibre, from a given fibre bundle. This then specializes to the case when the specified fibre 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 fibre of the principal bundle, we describe how to construct any associated bundle by means of a fibre product construction.^[1]

Associated bundles in general

Let ${\textstyle \pi :E\to X}$ be a fiber bundle over a topological space X with structure group G and typical fibre F. By definition, there is a left action of G (as a transformation group) on the fibre F. Suppose furthermore that this action is effective.^[2]

There is a local trivialization of the bundle E consisting of an open cover U_i of X, and a collection of fibre maps

$${\displaystyle \varphi _{i}:\pi ^{-1}(U_{i})\to U_{i}\times F}$$

such that the transition maps are given by elements of G. More precisely, there are continuous functions g_ij : (U_i ∩ U_j) → G such that

$${\displaystyle \psi _{ij}(u,f):=\varphi _{i}\circ \varphi _{j}^{-1}(u,f)={\big (}u,g_{ij}(u)f{\big )},\quad {\text{for each }}(u,f)\in (U_{i}\cap U_{j})\times F\,.}$$

Now let F′ be a specified topological space, equipped with a continuous left action of G. Then the bundle associated with E with fibre F′ is a bundle E′ with a local trivialization subordinate to the cover 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 G-valued functions g_ij(u) are the same as those obtained from the local trivialization of the original bundle E.

This definition clearly respects the cocycle condition on the transition functions, since in each case they are given by the same system of G-valued functions. (Using another local trivialization, and passing to a common refinement if necessary, the g_ij transform via the same coboundary.) Hence, by the fiber bundle construction theorem, this produces a fibre bundle E′ with fibre F′ as claimed.

Reduction of the structure group

The companion concept to associated bundles is the reduction of the structure group of a ${\displaystyle G}$-bundle ${\displaystyle B}$. We ask whether there is an ${\displaystyle H}$-bundle ${\displaystyle C}$, such that the associated ${\displaystyle G}$-bundle is ${\displaystyle B}$, up to isomorphism. More concretely, this asks whether the transition data for ${\displaystyle B}$ can consistently be written with values in ${\displaystyle H}$. In other words, we ask to identify the image of the associated bundle mapping (which is actually a functor).

See also

• Spinor bundle

References

Books

• Steenrod, Norman (1951). The Topology of Fibre Bundles. Princeton: Princeton University Press. ISBN 0-691-00548-6.
• 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.