fiber bundle FiberBundle 2002-10-31 18:28:50 2003-06-24 16:29:24 Definition trivial bundle local trivializations structure group cocycle condition local trivialization % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{xypic} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $F$ be a topological space and $G$ be a topological group which acts on $F$ on the left. A \emph{fiber bundle} with fiber $F$ and \emph{structure group} $G$ consists of the following data: \begin{itemize} \item a topological space $B$ called the base space, a space $E$ called the total space and a continuous surjective map $\pi:E \to B$ called the projection of the bundle, \item an open cover $\{U_i\}$ of $B$ along with a collection of continuous maps $\{\phi_i: \pi^{-1}U_i \to F\}$ called \emph{local trivializations} and \item a collection of continuous maps $\{g_{ij}: U_i \cap U_j \to G\}$ called \emph{transition functions} \end{itemize} which satisfy the following properties \begin{enumerate} \item the map $\pi^{-1}U_i \to U_i \times F$ given by $e \mapsto (\pi(e),\phi_i(e))$ is a homeomorphism for each $i$, \item for all indices $i,j$ and $e \in \pi^{-1}(U_i \cap U_j)$, $g_{ji}(\pi(e))\cdot \phi_i(e) = \phi_j(e)$ and \item for all indices $i,j,k$ and $b \in U_i \cap U_j \cap U_k$, $g_{ij}(b)g_{jk}(b) = g_{ik}(b)$. \end{enumerate} Readers familiar with \v{C}ech cohomology may recognize condition 3), it is often called the \emph{cocycle condition}. Note, this imples that $g_{ii}(b)$ is the identity in $G$ for each $b$, and $g_{ij}(b) = g_{ji}(b)^{-1}$. If the total space $E$ is homeomorphic to the product $B \times F$ so that the bundle projection is essentially projection onto the first factor, then $\pi : E \to B$ is called a \emph{trivial bundle}. Some examples of fiber bundles are vector bundles and covering spaces. There is a notion of morphism of fiber bundles $E,E'$ over the same base $B$ with the same structure group $G$. Such a morphism is a $G$-equivariant map $\xi:E\to E'$, making the following diagram commute $$\xymatrix{E\ar[rr]^\xi\ar[dr]_\pi& &E'\ar[dl]^{\pi'}\\ &B&}.$$ Thus we have a category of fiber bundles over a fixed base with fixed structure group.