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A fiber bundle is a space E that locally looks like a product B × F (base × fiber) but may be globally twisted. The projection π : E → B maps each fiber π⁻¹(b) homeomorphically to the standard fiber F, but the fibers may be "glued together" nontrivially across the base. Transition functions encode this twisting and take values in the structure group G ⊂ Aut(F). Fiber bundles unify tangent bundles, vector bundles, principal bundles, and covering spaces into a single framework, and their topology (measured by characteristic classes) is central to differential geometry and physics.
The simplest example of a fiber bundle is a product B × F, where the projection π(b, f) = b maps each "fiber" {b} × F to the base point b. But many natural geometric objects have this local product structure without being globally a product. The Mobius band is locally a product of an interval with a line segment, but globally it has a twist. The tangent bundle TM of a manifold is locally a product U × ℝⁿ (via coordinate charts), but the global structure may be twisted (as for TS²).
A fiber bundle π : E → B with fiber F and structure group G consists of: a total space E, a base space B, a projection π, a typical fiber F, and an open cover {U_α} of B with local trivializations φ_α : π⁻¹(U_α) → U_α × F. On overlaps U_α ∩ U_β, the change of trivialization φ_α ∘ φ_β⁻¹ acts as (b, f) ↦ (b, g_αβ(b) · f) for smooth functions g_αβ : U_α ∩ U_β → G. These transition functions satisfy the cocycle condition g_αβ g_βγ = g_αγ and encode the global twisting of the bundle.
The structure group G is the group of symmetries of the fiber that appears in the transition functions. For vector bundles (fibers are vector spaces), G ⊂ GL(n) acts by linear transformations. For principal bundles (fibers are copies of G itself), the group acts by left or right multiplication. For frame bundles, G = GL(n) or O(n). The structure group encodes what kind of geometry the fibers carry. Reducing the structure group (e.g., from GL(n) to O(n)) corresponds to adding geometric structure (e.g., a metric on the fibers).
Fiber bundles are classified by their transition functions up to equivalence. Two bundles with cohomologous transition functions (related by a coboundary) are isomorphic. This leads to the classification of bundles by Čech cohomology H¹(B; G) — a topological invariant of the base. For more refined invariants, characteristic classes (Chern classes, Pontryagin classes, Euler class, Stiefel-Whitney classes) are cohomology classes of B computed from the curvature of connections on the bundle. These are the primary tools for distinguishing non-isomorphic bundles and for understanding the global topology of geometric structures.