cohomology

integration

# 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 $\wedge^k_{C^\infty(X)} \Gamma(V)$.

An orientation of the tangent bundle $T X$ 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 $g : X \to \mathcal{B}O(n)$ 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

### 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 $R = H(\mathbb{R})$ 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.

smooth ∞-groupWhitehead tower of smooth moduli ∞-stacksG-structure/higher spin structureobstruction
$\vdots$
$\downarrow$
ninebrane 10-group$\mathbf{B}Ninebrane$ninebrane structurethird fractional Pontryagin class
$\downarrow$
fivebrane 6-group$\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)$fivebrane structuresecond fractional Pontryagin class
$\downarrow$
string 2-group$\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)$string structurefirst fractional Pontryagin class
$\downarrow$
spin group$\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)$spin structuresecond Stiefel-Whitney class
$\downarrow$
special orthogonal group$\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2$orientation structurefirst 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)

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