orientation in nLab

Contents

Contents

Definitions

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

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.

Another way to say the same is

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:

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.

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 a 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:X→ℬO(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:

ℬSO(n) g^↗ ↓ X →g ℬO(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

• orientation

• spin structure

• string structure, differential string structure

• fivebrane structure, differential fivebrane structure

In terms of orientation in generalized cohomology

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

• orientation in generalized cohomology.

For R=H(ℝ)R = H(\mathbb{R}) be the Eilenberg-MacLane spectrum for the discrete abelian group ℝ\mathbb{R} of real numbers, orientation in RR-cohomology is equivalent to the ordinary notion of orientation described above.

• orientation double cover