Orientation of a vector bundle

A complex vector bundle is oriented in a canonical way. The notion of an orientation of a vector bundle generalizes an orientation of a differentiable manifold: an orientation of a differentiable manifold is an orientation of its tangent bundle. In particular, a differentiable manifold is orientable if and only if its tangent bundle is orientable as a vector bundle. (note: as a manifold, a tangent bundle is always orientable.) == Operations ==

To give an orientation to a real vector bundle E of rank n is to give an orientation to the (real) determinant bundle \operatorname{det} E = \wedge^n E of E. Similarly, to give an orientation to E is to give an orientation to the unit sphere bundle of E. Just as a real vector bundle is classified by the real infinite Grassmannian, oriented bundles are classified by the infinite Grassmannian of oriented real vector spaces. == Thom space ==

From the cohomological point of view, for any ring Λ, a Λ-orientation of a real vector bundle E of rank n means a choice (and existence) of a class :u \in H^n(T(E); \Lambda) in the cohomology ring of the Thom space T(E) such that u generates \tilde{H}^*(T(E); \Lambda) as a free H^*(E; \Lambda)-module globally and locally: i.e., :H^*(E; \Lambda) \to \tilde{H}^*(T(E); \Lambda), x \mapsto x \smile u is an isomorphism (called the Thom isomorphism), where "tilde" means reduced cohomology, that restricts to each isomorphism :H^*(\pi^{-1}(U); \Lambda) \to \tilde{H}^*(T(E|_U); \Lambda) induced by the trivialization \pi^{-1}(U) \simeq U \times \mathbf{R}^n. One can show, with some work, that the usual notion of an orientation coincides with a Z-orientation. == See also ==