G-fibration

fibre bundle with a structure group

A generalization of the concept of the direct product of two topological spaces.

Let be a topological group and an effective right -space, i.e. a topological space with a given right action of such that for some , , implies . Let be the subset of those pairs for which for some , let be the orbit space, and let be the mapping sending each point to its orbit. If the mapping is continuous, then the tuple is called a principal fibre bundle with structure group .

Let be a left -space. The topological space admits a right action of by , . The composition induces a mapping: (where is the orbit space of under the action of ). The quadruple is called a fibre bundle with structure group associated with the principal fibre bundle , and the quadruple is a fibre bundle with fibre , base and structure group . Thus, a principal fibre bundle with a given structure group is a part of the structure of any fibre bundle with (that) structure group, and it uniquely determines the fibre bundle for any left -space .

If , are two principal fibre bundles with structure group , then a morphism is a mapping of -spaces . induces a mapping . A principal fibre bundle with structure group is called trivial is it is isomorphic to a fibre bundle of the following type:

Let be a principal fibre bundle and let be a continuous mapping of an arbitrary topological space into . Let be the subset of pairs for which . The projection induces a mapping . The space has the natural structure of a right -space, and the triple is a principal fibre bundle; it is induced by and is called an induced fibre bundle. If is the inclusion mapping of a subspace, then is called the restriction of over the subspace .

A principal fibre bundle with structure group is called locally trivial if its restriction to some neighbourhood of any point of the base is trivial. For a wide class of cases, the requirement of local triviality is unnecessary (e.g. if is a compact Lie group and a smooth -manifold). Hence, the term "fibre bundle" with structure group is often used in the sense of a locally trivial fibre bundle (or fibration).

Let , be a pair of fibre bundles with the same structure group and the same -space as fibre. Given a morphism of principal fibre bundles, the mapping induces a continuous mapping , and the pair is called a morphism of fibre bundles with structure group, .

A locally trivial fibre bundle admits the following characterization, which gives rise to another (also generally accepted) definition of a fibre bundle with structure group. Let be an open covering of the base such that the restriction of to is trivial for all . The choice of trivializations and their equality on the intersections leads to continuous functions (called transfer functions) . On the intersection of three neighbourhoods one has , while the choice of other trivializations over every neighbourhood leads to new functions . In this way, the functions form a one-dimensional Aleksandrov–Čech cocycle with coefficients in the sheaf of germs of -valued functions (the coefficients are non-Abelian), and a locally trivial fibre bundle determines this cocycle up to a coboundary.

References

[1]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
[2]	N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951)