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orientation

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Definitions

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

Definition

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

In the case n=0n = 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)(B 2,s 2)(B_1, s_1) \sim (B_2, s_2) iff sgndetI B 2 B 1=s 1/s 2\sgn \det I^{B_1}_{B_2} = s_1/s_2. This is redundant except in dimension 00, 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 ω\omega is an orientation, then there is also an opposite orientation ω-\omega.

A fancier way to say the same is

Definition

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

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

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

Definition

For XX a manifold and VXV \to X a vector bundle of rank nn, an orientation on VV is an equivalence class of trivializations? of the line bundle kV\bigwedge^k V that is obtained by associating to each fiber of VV its kkth alternating power.

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

Definition

For XX a manifold of dimension nn, an orientation of XX is an orientation of the tangent bundle TXT X (or cotangent bundle T *XT^* X).

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

A vector space always has an orientation, but a manifold or bundle may not. If an orientation exists, VV (or XX) is called orientable. If XX is connected space and VV (or XX) is orientable, then there are exactly 22 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 k2^k, where kk is the number of orientable components. (Or we can always say that the number of orientations is 2 k0 m2^k 0^m, where now mm is also the number of nonorientable components.

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)

Last revised on February 28, 2018 at 17:36:34. See the history of this page for a list of all contributions to it.