a fiber bundle

Picture a hairbrush. Over every point of the flat handle there stands a bristle, and the whole brush is the handle with all its bristles together. A fiber bundle makes this idea precise: you have a base space B, and over every point of B you attach an identical copy of some fixed space F (the fiber). Locally the result looks like a boring product, a slab B-patch times F, but as you walk around B the slabs may be glued together with a twist, so the whole object E need not be the global product B times F. A Mobius band is the classic twisted example: locally it is a strip (an interval times an interval), but going all the way around flips the fiber.

Formally a fiber bundle is a smooth surjection pi: E -> B (E is the total space, B the base) together with a fiber F, such that every point of B has a neighborhood U with a diffeomorphism phi: pi^{-1}(U) -> U times F that commutes with projection — that is, phi sends the fiber pi^{-1}(b) onto {b} times F. This is the local triviality condition: zoom in and the bundle is just a product. The maps phi are local trivializations, and where two of them overlap the recipe for re-gluing fibers is a transition function. A bundle is trivial if a single trivialization works over all of B at once.

Why care? Fiber bundles are the language for any structure that varies smoothly from point to point: tangent spaces over a manifold (the tangent bundle), spinning frames (the frame bundle), the field configurations of physics (gauge fields). The whole point is that local triviality is cheap but global triviality is not — the obstructions to untwisting a bundle are exactly the characteristic classes. A common beginner error is to assume a bundle that is locally a product is globally a product; the Mobius band shows that is false, and detecting the failure is most of the subject.