In mathematics, the theory of fiber bundles with a structure group
, which are both topological spaces with a group action of
, the transition functions of the fiber (i.e., the cocycle) in an overlap of two coordinate systems
, the real number line less the point 0, or the two-point set
We could say that more formally in terms of gluing two rectangles
This data can be written down as a patching function, with values in
In general it is enough to explain the transition from a bundle with fiber
Details in terms of data for an open covering are given as a case of descent.
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.
be a specified topological space, equipped with a continuous left action of
with a local trivialization subordinate to the cover
are the same as those obtained from the local trivialization of the original bundle
This definition clearly respects the cocycle condition on the transition functions, since in each case they are given by the same system of
(Using another local trivialization, and passing to a common refinement if necessary, the
has a free and transitive left action on
is a principal homogeneous space for the left action of
Note that, although there is no canonical way to specify a right action on a principal homogeneous space for
-bundle equipped with a right action is often thought of as part of the data specifying a fibre bundle with structure group
Without loss of generality, we can take this action to be effective.
via[3][4] We then identify by this action to obtain the space
Note that Define a projection map
The companion concept to associated bundles is the reduction of the structure group of a
More concretely, this asks whether the transition data for
In other words, we ask to identify the image of the associated bundle mapping (which is actually a functor).
Examples for vector bundles include: the introduction of a metric resulting in reduction of the structure group from a general linear group
; and the existence of complex structure on a real bundle resulting in reduction of the structure group from real general linear group
to complex general linear group
Another important case is finding a decomposition of a vector bundle
One can also express the condition for a foliation to be defined as a reduction of the tangent bundle to a block matrix subgroup - but here the reduction is only a necessary condition, there being an integrability condition so that the Frobenius theorem applies.