# 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.

How to Cite This Entry:
G-fibration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G-fibration&oldid=47030
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article