nLab orientation

Definitions

The basic concept is for vector spaces, and the remainder are defined in terms of that.

Definition

(orientation of a vector space) Given an ordered field $K$ and a vector space $V$ over $K$ of dimension $n$ (a natural number), an orientation of $V$ is a choice of one of the two equivalence classes of ordered bases? of $V$, where two bases are considered equivalent if the transformation matrix? from one to the other has positive determinant.

In the case $n = 0$, the only ordered basis is the empty list, but we still declare there to be two orientations by fiat, usually called positive and negative. We can make the definition seamless by taking the elements of the equivalence class to be pairs consisting of an ordered basis and a nonzero sign (positive or negative), with $(B_1, s_1) \sim (B_2, s_2)$ iff $\sgn \det I^{B_1}_{B_2} = s_1/s_2$. This is redundant except in dimension $0$, where now each equivalence class has a single element, $(*, +)$ for the positive orientation and $(*, -)$ for the negative orientation (where $*$ is the empty list).

In any case, this ensures that if $\omega$ is an orientation, then there is also an opposite orientation $-\omega$.

Another way to say the same is

Definition

(orientation of a vector space) For $V$ a vector space of dimension $n$, an orientation of $V$ is an equivalence class of nonzero elements of the line $\bigwedge^n V$, the $n$th alternating power of $V$, where two such elements are considered equivalent when either (hence each) is a positive multiple of the other.

Note that by both definitions, an orientation of a line (with $n = 1$) is an equivalence class of nonzero elements.

Assuming that $K$ is the field of real numbers or something like it, we can generalize from vector spaces to vector bundles:

Definition

(orientation of a vector bundle) 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 line bundle $\bigwedge^k V$ that is obtained by associating to each fiber of $V$ its $k$th alternating power.

Equivalently for a smooth manifold this is an equivalence class of an everywhere non-vanishing element of $\bigwedge^k_{C^\infty(X)} \Gamma(V)$, which may be considered the sign of the element.

Definition

(orientation of a manifold) For $X$ a manifold of dimension $n$, an orientation of $X$ is an orientation of the tangent bundle $T X$ (or cotangent bundle $T^* X$).

This is equivalently a choice of everywhere non-vanishing differential form on $X$ of degree $n$; the orientation may be considered the sign of the $n$-form (and the $n$-form's absolute value is a pseudo-$n$-form).

A vector space always has an orientation, but a manifold or bundle may not. If an orientation exists, $V$ (or $X$) is called orientable. If $X$ is a connected space and $V$ (or $X$) is orientable, then there are exactly $2$ orientations; more generally, the entire bundle is orientable iff the restriction to each connected component is orientable, and then the number of orientations is $2^k$, where $k$ is the number of orientable components. (Or we can always say that the number of orientations is $2^k 0^m$, where now $m$ is also the number of nonorientable components.)

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 $g : X \to \mathcal{B}O(n)$ of its tangent bundle through the fist step $S O(n) \to O(n)$ in the Whitehead tower of $X$:

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

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

• orientation in generalized cohomology.

For $R = H(\mathbb{R})$ be the Eilenberg-MacLane spectrum for the discrete abelian group $\mathbb{R}$ 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
	$\vdots$
	$\downarrow$
ninebrane 10-group	$\mathbf{B}Ninebrane$	ninebrane structure	third fractional Pontryagin class
	$\downarrow$
fivebrane 6-group	$\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)$	fivebrane structure	second fractional Pontryagin class
	$\downarrow$
string 2-group	$\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)$	string structure	first fractional Pontryagin class
	$\downarrow$
spin group	$\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)$	spin structure	second Stiefel-Whitney class
	$\downarrow$
special orthogonal group	$\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2$	orientation structure	first Stiefel-Whitney class
	$\downarrow$
orthogonal group	$\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2$	orthogonal structure/vielbein/Riemannian metric
	$\downarrow$
general linear group	$\mathbf{B}GL$	smooth manifold

(all hooks are homotopy fiber sequences)