Fiber Bundle - Formal Definition

Formal Definition

A fiber bundle consists of the data (E, B, π, F), where E, B, and F are topological spaces and π : E → B is a continuous surjection satisfying a local triviality condition outlined below. The space B is called the base space of the bundle, E the total space, and F the fiber. The map π is called the projection map (or bundle projection). We shall assume in what follows that the base space B is connected.

We require that for every x in E, there is an open neighborhood U ⊂ B of π(x) (which will be called a trivializing neighborhood) such that π−1(U) is homeomorphic to the product space U × F, in such a way that π carries over to the projection onto the first factor. That is, the following diagram should commute:

where proj₁ : U × F → U is the natural projection and φ : π−1(U) → U × F is a homeomorphism. The set of all {(U_i, φ_i)} is called a local trivialization of the bundle.

Thus for any p in B, the preimage π−1({p}) is homeomorphic to F (since proj₁-1({p}) clearly is) and is called the fiber over p. Every fiber bundle π : E → B is an open map, since projections of products are open maps. Therefore B carries the quotient topology determined by the map π.

A fiber bundle (E, B, π, F) is often denoted

that, in analogy with a short exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space.

A smooth fiber bundle is a fiber bundle in the category of smooth manifolds. That is, E, B, and F are required to be smooth manifolds and all the functions above are required to be smooth maps.