nLab
orientation

Definition (orientaton on a manifold)

In terms of lifting through Whitehead tower

One way to encode an orientation on a manifold X is as a lift ĝ of the classifying map g:XO(n) of its tangent bundle through the fist step SO(n)O(n) in the Whitehead tower of X:

SO(n) ĝ X g O(n).\array{ && \mathcal{B}S O(n) \\ & {}^{\hat g}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathcal{B} O(n) } \,.

From this perspective a choice of orientation is the first in a series of special structures on X that continue with