Real line bundle ($L\to X$)

A real line bundle is a rank-one real vector bundle. It is orientable exactly when it is trivial.

Real line bundles over a paracompact space are classified up to isomorphism by their first Stiefel–Whitney class in $H^1(X;{\mathbb{F}}_2)$. Tensor product corresponds to addition of classes.

For a real line bundle, the mod-two Euler class is its top Stiefel–Whitney class, namely $w_1(L)\in H^1(X;{\mathbb{F}}_2)$. Its evaluation on a closed curve records whether parallel transport reverses the fiber orientation.