Fiber Bundles — Mathematics Knowledge System

Overview

A fiber bundle is a space that locally looks like a product but may be globally twisted. Formally, a fiber bundle (E, B, pi, F) consists of a total space E, base space B, projection pi: E -> B, and fiber F, such that B has an open cover over which E is homeomorphic to a product. Fiber bundles generalize covering spaces (discrete fiber), vector bundles (vector space fiber), and principal bundles (group fiber), and are fundamental to modern geometry and physics.

Intuition

Picture a Möbius strip: locally it looks like (open interval) × (open interval), but globally the strip is twisted so the total space is not a product. A fiber bundle formalizes this idea: attach a copy of a fiber F to each point of the base B in a locally product-like way that may be globally non-trivial. The non-triviality is captured by the transition functions between local trivializations.

Formal Definition

Definition

A fiber bundle with structure group G is a tuple (E, B, pi, F, G) where pi: E -> B is a continuous surjection, F is the fiber, G is a topological group acting on F, and there exists an open cover {U_alpha} of B with homeomorphisms phi_alpha: pi^{-1}(U_alpha) -> U_alpha × F such that on overlaps phi_alpha ∘ phi_beta^{-1}: (U_alpha ∩ U_beta) × F -> (U_alpha ∩ U_beta) × F is of the form (x, f) |-> (x, g_{alpha beta}(x) · f) for continuous transition functions g_{alpha beta}: U_alpha ∩ U_beta -> G.

F \hookrightarrow E \xrightarrow{\pi} B

Fiber bundle sequence

\varphi_\alpha \circ \varphi_\beta^{-1}(x, f) = (x,\, g_{\alpha\beta}(x)\cdot f)

Transition function condition

g_{\alpha\beta}\cdot g_{\beta\gamma} = g_{\alpha\gamma} \quad \text{on } U_\alpha\cap U_\beta\cap U_\gamma

Cocycle condition

Properties

Homotopy Lifting Property

A fiber bundle \pi: E \to B has the homotopy lifting property: for any homotopy H: X\times[0,1]\to B and lift of H_0, there exists a homotopy \tilde H: X\times[0,1]\to E lifting H.

Long Exact Sequence of a Fibration

For a fibration F \to E \to B, there is a long exact sequence \cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \cdots \to \pi_0(E) \to \pi_0(B).

Worked Examples

Cover S^1 by two open arcs U_1 and U_2 each homeomorphic to an interval, with U_1 ∩ U_2 consisting of two intervals.
Over each U_i, the bundle is trivial: pi^{-1}(U_i) ≅ U_i × [0,1].
On one component of U_1 ∩ U_2 the transition function is the identity; on the other it is the flip f |-> 1 - f. This non-constant transition data makes the total space non-homeomorphic to S^1 × [0,1].
One can verify: a global section would require a continuous function s: S^1 -> [0,1] satisfying s(antipodal) = 1 - s(point) — impossible by the intermediate value theorem (applied to s(x) - s(-x) on S^1).

Answer: The Möbius strip is a nontrivial [0,1]-bundle over S^1 because its transition functions cannot all be taken to be the identity.

Practice Problems

Difficulty 6/10

Show that the tangent bundle TS^1 of the circle is trivial (i.e., isomorphic to S^1 × R).

Difficulty 8/10

Prove the cocycle condition: if {g_{alpha beta}} are transition functions for a bundle, then g_{alpha alpha} = id and g_{alpha beta} ∘ g_{beta gamma} = g_{alpha gamma} on triple overlaps.

Difficulty 7/10

Use the long exact sequence of the Hopf fibration S^1 -> S^3 -> S^2 to show that pi_2(S^3) = 0.

Common Mistakes

Common Mistake

Every fiber bundle is a product space.

A bundle is locally a product, but may be globally twisted. The Möbius strip and Hopf fibration are classic examples of nontrivial bundles.

Common Mistake

The fiber is unique up to homeomorphism for a given bundle.

While fibers over different base points are homeomorphic (in a fiber bundle), the way they fit together globally can vary — that is precisely what the structure group encodes.

Quiz

Summary

A fiber bundle (E, B, pi, F) is a space E that locally looks like B × F but may be globally twisted.
Local trivializations are related by transition functions g_{alpha beta}: U_alpha ∩ U_beta -> G satisfying the cocycle condition.
Examples include the Möbius strip (fiber [0,1] over S^1), the Hopf fibration (S^1 over S^2), and all vector bundles.
Fiber bundles (more generally, fibrations) satisfy the homotopy lifting property, yielding the long exact sequence of homotopy groups.
The classification of fiber bundles with structure group G over a base B is given by homotopy classes [B, BG] where BG is the classifying space.

References

BookHusemoller, D. — Fibre Bundles. Springer, 1994.
WebsiteWikipedia — Fiber bundle