 Orientation of a vector bundle

An $R$-orientation of a rank-$d$ vector bundle is a coherent choice of generator of $H^d(E_b,E_b\setminus \{0\};R)$ for every fiber, equivalently a Thom class with the corresponding fiberwise restriction.

The orientation of a vector bundle is a concept from differential geometry and algebraic topology that is related to the notion of orientability of the fibers of the bundle. A vector bundle \( E \) over a topological space \( X \) consists of a base space \( X \) and, for each point \( x \in X \), a vector space \( E_x \) attached to that point. The vector spaces are called the fibers of the bundle. ### Definition of Orientation 1.