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 : EB 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 UB 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 × FU is the natural projection and φ : π−1(U) → U × F is a homeomorphism. The set of all {(Ui, φi)} 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 : EB 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.

Read more about this topic:  Fiber Bundle

Famous quotes containing the words formal and/or definition:

    Good gentlemen, look fresh and merrily.
    Let not our looks put on our purposes,
    But bear it as our Roman actors do,
    With untired spirits and formal constancy.
    William Shakespeare (1564–1616)

    The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.
    Ralph Waldo Emerson (1803–1882)