In
mathematics, the theory of
fiber bundles with a
structure group
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E an ...
(a
topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from
to
, which are both
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s with a
group action of
. 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
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Aug ...
, for which
is the
cyclic group of order 2,
. We can take as
any of: the real number line
, the interval