nLab
double cover

Definition

A double cover is equivalently

• a ${\mathbb{Z}}_2$-principal bundle;

• an etale space with local sections the 2-element set.

Examples

Orientation double cover

For $X$ a manifold, not necessarily oriented or even orientable, write

$$\begin{array}{ccc}& & BO\\ & {}^{\hat{T}X}\nearrow & \downarrow \\ X& \stackrel{TX}{\to }& B\mathrm{GL}\end{array}$$\array{ && B O \\ & {}^{\mathllap{\hat T X}}\nearrow & \downarrow \\ X &\stackrel{T X}{\to}& B GL }

for any choice of orthogonal structure. The orientation double cover or orientation bundle of $X$ is the ${\mathbb{Z}}_2$-principal bundle classified by the first Stiefel-Whitney class (of the tangent bundle) of $X$

$$w_1(\hat{T}X):X\stackrel{\hat{T}X}{\to }BO\stackrel{w_1}{\to }B{\mathbb{Z}}_2.$$w_1(\hat T X) : X \stackrel{\hat T X}{\to} B O \stackrel{w_1}{\to} B \mathbb{Z}_2 \,.

One may identify this with the bundle that over eachh neighbourhood $x\in U\subset X$ of a point $x$ has as fibers the two different choices of volume forms up to positive rescaling (the two different choices of orientation).

More generally, for $E\to X$ any orthogonal group-principal bundle classified by a morphism $E:X\to BO$, the corresponding orientation double cover is the ${\mathbb{Z}}_2$-bundle classified by

$$w_1(E):X\stackrel{E}{\to }BO\stackrel{w_1}{\to }B{\mathbb{Z}}_2.$$w_1(E) : X \stackrel{E}{\to} \mathbf{B} O \stackrel{w_1}{\to} \mathbf{B} \mathbb{Z}_2 \,.

Related concepts

• orientifold, Jandl gerbe

• orientation

References

An exposition in a broader context is in the section higher spin structures at

• geometry of physics.