G-fibration

fibre bundle with a structure group

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

Let $ G $ be a topological group and $ X $ an effective right $ G $- space, i.e. a topological space with a given right action of $ G $ such that $ x g = x $ for some $ x \in X $, $ g \in G $, implies $ g= 1 $. Let $ X ^ {*} \subset X \times X $ be the subset of those pairs $ ( x , x ^ \prime ) $ for which $ x ^ \prime = x g $ for some $ g \in G $, let $ B = X / G $ be the orbit space, and let $ p : X \rightarrow B $ be the mapping sending each point to its orbit. If the mapping $ X ^ {*} \rightarrow G : ( x , x g ) \mapsto g $ is continuous, then the tuple $ \xi = ( X , p , B ) $ is called a principal fibre bundle with structure group $ G $.

Let $ F $ be a left $ G $- space. The topological space $ X \times F $ admits a right action of $ G $ by $ ( x, f ) g = ( x g , g ^ {-} 1 f ) $, $ f \in F $. The composition $ X \times F \rightarrow ^ { \mathop{\rm pr} _ {X} } X \rightarrow ^ {p} B $ induces a mapping: $ X _ {F} = ( X \times F ) / G \rightarrow ^ {p _ {F} } B $( where $ X _ {F} $ is the orbit space of $ X \times F $ under the action of $ G $). The quadruple $ ( X _ {F} , p _ {F} , B , F ) $ is called a fibre bundle with structure group associated with the principal fibre bundle $ \xi $, and the quadruple $ ( X _ {F} , p _ {F} , F, \xi ) $ is a fibre bundle with fibre $ F $, base $ B $ and structure group $ G $. 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 $ G $- space $ F $.

If $ \xi = ( X , p , B ) $, $ \xi ^ \prime = ( X ^ { \prime } , p ^ \prime , B ^ \prime ) $ are two principal fibre bundles with structure group $ G $, then a morphism $ \xi \rightarrow \xi ^ \prime $ is a mapping of $ G $- spaces $ h : X \rightarrow X ^ { \prime } $. $ h $ induces a mapping $ f : B \rightarrow B ^ \prime $. A principal fibre bundle with structure group is called trivial is it is isomorphic to a fibre bundle of the following type:

$$ ( B \times G , \mathop{\rm pr} _ {B} , B ) ,\ \ ( b , g ) g ^ \prime = ( b , g g ^ \prime ) ,\ \ b \in B ,\ g , g ^ \prime \in G . $$

Let $ ( X , p , B ) $ be a principal fibre bundle and let $ f : B ^ \prime \rightarrow B $ be a continuous mapping of an arbitrary topological space $ B ^ \prime $ into $ B $. Let $ X ^ \prime \subset B ^ \prime \times X $ be the subset of pairs $ ( b , x) $ for which $ f ( b) = p ( x) $. The projection $ \mathop{\rm pr} _ {B ^ \prime } : B ^ \prime \times X \rightarrow B ^ \prime $ induces a mapping $ p ^ \prime : X ^ { \prime } \rightarrow B ^ \prime $. The space $ X ^ { \prime } $ has the natural structure of a right $ G $- space, and the triple $ ( X ^ { \prime } , p , B ) $ is a principal fibre bundle; it is induced by $ f $ and is called an induced fibre bundle. If $ f : B ^ \prime \rightarrow B $ is the inclusion mapping of a subspace, then $ ( X ^ { \prime } , p ^ \prime , B ^ \prime ) $ is called the restriction of $ ( X , p , B ) $ over the subspace $ B ^ \prime $.

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

Let $ ( X _ {F} , p _ {F} , F , \xi ) $, $ ( X _ {F} ^ { \prime } , p _ {F} ^ \prime , F , \xi ^ \prime ) $ be a pair of fibre bundles with the same structure group and the same $ G $- space as fibre. Given a morphism $ h : \xi \rightarrow \xi ^ \prime $ of principal fibre bundles, the mapping $ h \times \mathop{\rm id} : X \times F \rightarrow X ^ { \prime } \times F $ induces a continuous mapping $ \phi : X _ {F} \rightarrow X _ {F} ^ { \prime } $, and the pair $ ( h , \phi ) $ is called a morphism of fibre bundles with structure group, $ ( X _ {F} , p _ {F} , F , \xi ) \rightarrow ( X _ {F} ^ { \prime } , p _ {F} ^ \prime , F , \xi ^ \prime ) $.

A locally trivial fibre bundle $ \eta = ( X _ {F} , p _ {F} , F , \xi ) $ admits the following characterization, which gives rise to another (also generally accepted) definition of a fibre bundle with structure group. Let $ U = \{ u _ \alpha \} $ be an open covering of the base $ B $ such that the restriction of $ \eta $ to $ u _ \alpha $ is trivial for all $ \alpha $. The choice of trivializations and their equality on the intersections $ u _ \alpha \cap u _ \beta $ leads to continuous functions (called transfer functions) $ g _ {\alpha \beta } : u _ \alpha \cap u _ \beta \rightarrow G $. On the intersection of three neighbourhoods $ u _ \alpha \cap u _ \beta \cap u _ \gamma $ one has $ g _ {\alpha \beta } \circ g _ {\beta \gamma } \circ g _ {\gamma \alpha } = 1 \in G $, while the choice of other trivializations over every neighbourhood leads to new functions $ g _ {\alpha \beta } ^ \prime = h _ \alpha g _ {\alpha \beta } h _ \beta ^ {-} 1 $. In this way, the functions $ \{ g _ {\alpha \beta } \} $ form a one-dimensional Aleksandrov–Čech cocycle with coefficients in the sheaf of germs of $ G $- 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)