An -orientation of a rank- vector bundle is a coherent choice of generator of for every fiber, equivalently a Thom class with the corresponding fiberwise restriction.
A Thom class of an -oriented rank- vector bundle is a class restricting to the chosen generator on every fiber pair .
The Euler class of an oriented rank- vector bundle is the pullback of its Thom class along the zero section. Over it equals the top Stiefel-Whitney class.
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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.