# nLab orientation

cohomology

### Theorems

#### Integration theory

integration

analytic integrationcohomological integration
measureorientation in generalized cohomology
Riemann/Lebesgue integration, of differential formspush-forward in generalized cohomology/in differential cohomology

# Contents

## 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 ${\wedge }_{{C}^{\infty }\left(X\right)}^{k}\Gamma \left(V\right)$.

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 $\stackrel{^}{g}$ of the classifying map $g:X\to ℬO\left(n\right)$ of its tangent bundle through the fist step $SO\left(n\right)\to O\left(n\right)$ in the Whitehead tower of $X$:

$\begin{array}{ccc}& & ℬSO\left(n\right)\\ & {}^{\stackrel{^}{g}}↗& ↓\\ X& \stackrel{g}{\to }& ℬO\left(n\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

### 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

For $R=H\left(ℝ\right)$ 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.

smooth ∞-groupWhitehead tower of smooth moduli ∞-stacksG-structure/higher spin structureobstruction
$⋮$
$↓$
fivebrane 6-group$B\mathrm{Fivebrane}$fivebrane structuresecond fractional Pontryagin class
$↓$
string 2-group$B\mathrm{String}\stackrel{\frac{1}{6}{p}_{2}}{\to }{B}^{7}U\left(1\right)$string structurefirst fractional Pontryagin class
$↓$
spin group$B\mathrm{Spin}\stackrel{\frac{1}{2}{p}_{1}}{\to }{B}^{3}U\left(1\right)$spin structuresecond Stiefel-Whitney class
$↓$
special orthogonal group$B\mathrm{SO}\stackrel{{w}_{2}}{\to }{B}^{2}{ℤ}_{2}$orientation structurefirst Stiefel-Whitney class
$↓$
orthogonal group$BO\stackrel{{w}_{1}}{\to }B{ℤ}_{2}$orthogonal structure/vielbein/Riemannian metric
$↓$
general linear group$B\mathrm{GL}$smooth manifold

(all hooks are homotopy fiber sequences)

Revised on February 19, 2013 01:47:06 by Toby Bartels (64.89.53.232)