fiber bundle

From Mathematics wiki

Informally, a fiber bundle is a way of assigning a fiber, $F\,$ , such as a tangent space, to every point on a topological space, $B\,$ (usually a manifold), to form a total space $E\,$ which locally looks like $B \times F\,$ .

For example, let $F\,$ be the line segment $L\,$ and let $B = \mathbb{S}^1\,$ , the circle. Among the possibilities for $E\,$ are the cylinder (which is trivially $\mathbb{S}^1 \times L\,$ ) and the Möbius strip (which is only locally $\mathbb{S}^1 \times L\,$ ).

1 Definition
2 Examples

Definition

A fiber bundle consists of the following:

A topological space $F\,$ called a fiber, a topological space $B\,$ (usually a manifold $\mathcal{M}\,$ ) called a base space, and a topological space $E\,$ called a total space, along with

a continuous surjective map $\pi : E \to B\,$ , called the projection of the bundle. Given $x \in B\,$ , the preimage $\pi^{-1}(x) \equiv F_x\,$ homeomorphic to $F\,$ , and is called the fiber over $x\,$ .
an open covering $\{U_i \}\,$ of $B\,$ and a set of diffeomorphisms $\phi_i : U_i \times F \to \pi^{-1}(U_i)\,$ such that $\pi \circ \phi_i(x,t) = x\,$ . The set $\{\phi_i, U_i\}\,$ is called a local trivialization of the bundle.