Fiber bundle
A fiber bundle is a way of gluing space pieces together so that every small piece looks like a simple product, but the whole space may twist in a more complicated way.
- The main ingredients are:
- total space E (the whole shape),
- base space B (the parameter space you’re viewing),
- fiber F (the shape attached at each point of B),
- a projection map π: E → B that tells you which point of B each point of E lies over.
- Local product structure (local triviality): For every point in B, you can find a small neighborhood U where the part of E above U looks like U × F. There is a homeomorphism φ: π^{-1}(U) → U × F that makes π behave like the projection onto U. If this can be done everywhere, the bundle is well-behaved locally even if the global shape is twisted.
- Notation and intuition: E is the total space, B is the base, F is the fiber, and π is the projection. If E is exactly B × F with π(B × F) = B, the bundle is called trivial (no twisting).
- Simple examples:
- Möbius strip: a twisted line segment (fiber) over a circle (base). Locally it looks like a cylinder, but globally it twists.
- Klein bottle: another twisted bundle over a circle.
- Torus (S1 × S1) is a non-twisted example.
- A covering space is a fiber bundle with a discrete fiber.
- Special kinds of fiber bundles:
- Vector bundles: fibers are vector spaces (like the tangent bundle of a manifold).
- Principal bundles: fibers are copies of a group G that acts freely and transitively; these often come with a structure group that describes how local pieces are glued.
- Frame bundle and associated bundles: related constructions built from a given bundle.
- How the twisting is controlled: the way local pieces fit together is described by a structure group G and transition functions t_ij on overlaps of local neighborhoods. These functions tell you how to switch from one local view to another. The compatibility of these switches (the cocycle condition) ensures the whole bundle is consistent.
- Maps and sections:
- A bundle map (morphism) is a pair of functions that respect the base spaces and commute with the projections.
- A section is a choice, for every point in B, of a point in the fiber over that base point. Global sections don’t always exist; sometimes they exist only locally.
- Obstructions to global sections can be described using cohomology, leading to ideas like characteristic classes. A famous example is the hairy ball theorem, which shows certain bundles cannot have nowhere-vanishing sections.
- More features:
- If the group G and the base space have more structure (like smooth manifolds), one talks about smooth or differentiable bundles.
- The unit sphere bundle comes from a vector bundle by taking unit-length vectors in each fiber.
- The Hopf fibration S^3 → S^2 is a classic example of a nontrivial bundle in topology.
- When a quotient space G/H forms a bundle: in many cases (especially with Lie groups and closed subgroups), the projection G → G/H becomes a fiber bundle with fiber H. This framework includes many important examples in geometry and topology.
In short, fiber bundles help us study spaces that look simple in small regions but may have interesting global twists. They are central in differential geometry and topology because they organize how local data (like vectors or frames) change as you move around a space.
This page was last edited on 3 February 2026, at 02:24 (CET).