nLab
orientation

Definition

Definition

For $X$ a manifold and $V\to X$ a vector bundle of rank $k$, an orientation on $V$ is an equivalence class of trivializations of the real line bundle ${\wedge }^{k}V$ that is obtained by associating to each fiber of $V$ its skew-symmetric $k$th tensor power.

Equivalently for a smooth manifold this is an equivalence class of an everywhere non-vanishing element of ${\wedge }_{C^{\mathrm{\infty }}(X)}^k\Gamma (V)$.

An orientation of the tangent bundle $TX$ or cotangent bundle $T^{*}X$ is called an orientation of the manifold. This is equivalently a choice of no-where vanishing differential form on $X$ of degree the dimension of $X$.

If a trivialization of ${\wedge }^{k}V$ exists, $V$ is called orientable.

For $\omega$ an orientation, $-\omega$ is the opposite orientation.

Properties

In terms of lifting through Whitehead tower

An orientation on a Riemannian manifold $X$ is equivalently 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

• orientation

• spin structure

• string structure, differential string structure

• fivebrane structure, differential fivebrane structure

In terms of orientation in generalized cohomology

For $E$ an E-∞ ring spectrum, tthere is a general notion of $R$-orientation of vector bundles. This is described at

• orientation in generalized cohomology.

For $R=H(ℝ)$ be the Eilenberg-MacLane spectrum for the discrete abelian group $ℝ$ of real numbers, orientation in $R$-cohomology is equivalent to the ordinary notion of orientation described above.

Related concepts

• orientation double cover

smooth ∞-group	Whitehead tower of smooth moduli ∞-stacks	G-structure/higher spin structure	obstruction
	$⋮$
	$\downarrow$
fivebrane 6-group	$B\mathrm{Fivebrane}$	fivebrane structure	second fractional Pontryagin class
	$\downarrow$
string 2-group	$B\mathrm{String}\stackrel{\frac{1}{6}{p}_2}{\to }{B}^{7}U(1)$	string structure	first fractional Pontryagin class
	$\downarrow$
spin group	$B\mathrm{Spin}\stackrel{\frac{1}{2}{p}_1}{\to }{B}^{3}U(1)$	spin structure	second Stiefel-Whitney class
	$\downarrow$
special orthogonal group	$B\mathrm{SO}\stackrel{w_2}{\to }{B}^2{\mathbb{Z}}_2$	orientation structure	first Stiefel-Whitney class
	$\downarrow$
orthogonal group	$BO\stackrel{w_1}{\to }B{\mathbb{Z}}_2$	orthogonal structure/vielbein/Riemannian metric
	$\downarrow$
general linear group	$B\mathrm{GL}$	smooth manifold

(all hooks are homotopy fiber sequences)