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Definition

Definition

For XX a manifold and VXV \to X a vector bundle of rank kk, an orientation on VV is an equivalence class of trivializations of the real line bundle kV\wedge^k V that is obtained by associating to each fiber of VV its skew-symmetric kkth tensor power.

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

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

If a trivialization of kV\wedge^k V exists, VV is called orientable.

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

Properties

In terms of lifting through Whitehead tower

An orientation on a Riemannian manifold XX is equivalently a lift g^\hat g of the classifying map g:XO(n)g : X \to \mathcal{B}O(n) of its tangent bundle through the fist step SO(n)O(n)S O(n) \to O(n) in the Whitehead tower of XX:

SO(n) g^ X g O(n). \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 XX that continue with

In terms of orientation in generalized cohomology

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

For R=H()R = H(\mathbb{R}) be the Eilenberg-MacLane spectrum for the discrete abelian group \mathbb{R} of real numbers, orientation in RR-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-groupBNinebrane\mathbf{B}Ninebrane ninebrane structurethird fractional Pontryagin class
\downarrow
fivebrane 6-groupBFivebrane1np 3B 11U(1)\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)fivebrane structuresecond fractional Pontryagin class
\downarrow
string 2-groupBString16p 2B 7U(1)\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)string structurefirst fractional Pontryagin class
\downarrow
spin groupBSpin12p 1B 3U(1)\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)spin structuresecond Stiefel-Whitney class
\downarrow
special orthogonal groupBSOw 2B 2 2\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2orientation structurefirst Stiefel-Whitney class
\downarrow
orthogonal groupBOw 1B 2\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2orthogonal structure/vielbein/Riemannian metric
\downarrow
general linear groupBGL\mathbf{B}GLsmooth manifold

(all hooks are homotopy fiber sequences)

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