Fiber BundlesEdit

Fiber bundles are a central organizing concept in geometry, topology, and physics. They encode how a space that locally looks like a simple product can be stitched together in a global way that may be considerably more intricate. At their core, fiber bundles separate three ingredients: a total space that contains all the data, a base space that serves as the parameter space, and a fiber that is carried over each point of the base. The way these fibers are glued together is governed by a structure group and transition data, which can create twists and obstructions that prevent the bundle from being globally a product. This framework underlies many constructions in mathematics and in the physical sciences.

One of the main ideas is local triviality: for every point in the base space there exists a neighborhood over which the bundle looks like a simple product. Globally, however, the gluing of these local pictures can produce nontrivial topology. The language of fiber bundles provides a natural setting for describing fields and symmetries in a coordinate-free way, and it connects to powerful invariants such as characteristic classes. The subject sits at the intersection of topology, differential geometry, and algebraic topology, and it has become indispensable in modern mathematics and theoretical physics. For concrete illustrations, consider the Möbius strip as a nontrivial line bundle over the circle, or the Hopf fibration as a nontrivial fiber bundle with fiber circle over a two-sphere.

Definition and basic components

A fiber bundle consists of four primary data: - The total space E, the base space B, the fiber F, and a continuous surjection π: E → B known as the projection. - A structure group G, which acts on the fiber F and encodes how the fibers are glued together. - An atlas of local trivializations: for an open cover {U_i} of B, there are homeomorphisms φi: π^{-1}(U_i) → U_i × F that are fiberwise compatible with the projection. - Transition functions g{ij}: U_i ∩ U_j → G on overlaps, satisfying the cocycle condition on triple overlaps.

In this setup, E locally looks like a product U_i × F, but the global way these local pictures are patched by the g_{ij} can produce nontrivial topology. The pair (E, π) together with F and G is often summarized as a fiber bundle with structure group G and fiber F.

When F is a vector space and G is a subgroup of the general linear group GL(F), one obtains a vector bundle. A classic example is the tangent bundle of a smooth manifold, which assigns to each point the tangent space at that point.
When G itself acts on F and g_{ij} take values in G, one has a principal bundle, encoding the symmetries acting on the fibers.
An associated bundle can be built from a principal bundle by having the fibers carry a representation of G; this broadens the range of bundles encountered in geometry and physics.

Key notions in this language include local triviality, sections (maps s: B → E with π ∘ s = id_B), and the existence of global sections, which can be obstructed by the bundle’s topology. The total space E, base B, and fiber F are related by the projection π, while the atlas and the cocycles g_{ij} describe how the local trivializations are glued together.

Types of bundles and constructions

Vector bundles: fibers are vector spaces, with linear transition functions. Notable examples include the tangent bundle and the cotangent bundle of a manifold.
Principal bundles: fibers are a Lie group G with a right action of G on E that is free and transitive on the fibers; these objects formalize gauge symmetries in physics.
Associated bundles: obtained from a principal bundle by coupling G-action to a representation on a vector space or other fiber type.
General fiber bundles: the fiber F can be any topological space, and the structure group G consists of homeomorphisms of F preserving the relevant structure.

In broad strokes, the way a bundle is constructed often starts with local trivializations and then specifies how to glue them together via the transition functions on overlaps. This data must satisfy consistency conditions (the cocycle condition) to yield a well-defined global object.

Examples and intuitive pictures

Möbius strip: a nontrivial line bundle over the circle, illustrating how a seemingly simple fiber (a line) can twist as one travels around the base.
Tangent bundle: assigns to each point of a smooth manifold its tangent space; sections of this bundle are vector fields.
Hopf fibration: a classical example with total space S^3, base S^2, and fiber S^1, showing a nontrivial fibration with rich topological structure.
Frame bundle: the principal bundle of ordered bases (frames) in the tangent spaces of a manifold; this plays a key role in defining connections and covariant differentiation.
Projective spaces and tautological bundles: these give natural examples of bundles arising from linear algebra and projective geometry, with interesting global twisting.

Triviality, obstructions, and classification

A bundle is called trivial if it is globally a product B × F with the obvious projection. Not all bundles are trivial, and understanding when a bundle is nontrivial is a central concern. Obstructions to triviality are captured by topological invariants known as characteristic classes. For line bundles, the first Chern class provides a complete invariant in reasonable settings; for real vector bundles, Stiefel-Whitney classes play a similar role. The systematic framework for organizing these obstructions uses tools such as Čech cohomology and cohomological classifications of bundles.

Local triviality guarantees that every bundle looks like a product in small neighborhoods, but global twisting can persist.
Classification results connect bundle types to cohomology groups, enabling algebraic descriptions of geometric twisting.
In physics, these invariants correspond to physically meaningful quantities such as quantized charges and topological phases.

Connections, curvature, and physics

A connection on a bundle gives a way to compare fibers at nearby points, leading to the notion of parallel transport and covariant differentiation. In the language of differential geometry, connections can be described by connection 1-forms and curvature 2-forms, whose invariants are captured by characteristic classes via Chern-Weil theory. This machinery underpins gauge theories in physics, where principal bundles encode gauge symmetries and connections represent gauge fields. The geometry of bundles also figures prominently in general relativity through the frame and orthonormal frame bundles, which encode local reference frames in spacetime.

Ehresmann connections provide a broad, coordinate-free viewpoint on connections in fiber bundles.
Gauge theory and general relativity both rely on the bundle formalism to describe fields and interactions in a way that respects the underlying geometry.