nLab
orientation

Definition (orientaton on a manifold)

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In terms of lifting through Whitehead tower

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

$$\begin{array}{ccc}& & ℬSO(n)\\ & {}^{\hat{g}}\nearrow & \downarrow \\ X& \stackrel{g}{\to }& ℬO(n)\end{array}.$$\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

• Spin structure

• String structure

• Fivebrane structure