trivializable

Trivializable refers to the property of a fiber bundle being globally equivalent to a product bundle. More precisely, a fiber bundle E → B with typical fiber F is trivializable if there exists a bundle isomorphism E ≅ B × F that preserves the projection to B. In this case E is called the trivial bundle B × F. Local triviality is a standard assumption for fiber bundles, but trivializability imposes a global product structure.

For a vector bundle E over B, trivializability is equivalent to the existence of a global frame: a set of sections e1, ..., en that are pointwise a basis of the fiber at each base point. Then E ≅ B × R^n (or C^n for complex vector bundles). For a principal G-bundle P → B, trivializability means P ≅ B × G as a G-bundle, equivalently there exists a global section.

Cohomological viewpoint: The isomorphism class of a principal G-bundle is classified by a cohomology class in the appropriate set (non-abelian H^1(B, G)); it is trivializable precisely when this class is the identity. A vector bundle is classified by the classifying map to BG (or BU(n) for complex rank-n bundles); trivializability corresponds to a null-homotopic classifying map.

Examples: The Möbius strip is a nontrivial real line bundle over S^1, hence not trivializable. The tangent bundle of S^2 is nontrivial, while the tangent bundle of the torus T^2 is trivial. The product bundle B × F is the canonical trivializable form.