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

$$\array{ && B O \\ & {}^{\mathllap{\hat T X}}\nearrow & \downarrow \\ X &\stackrel{T X}{\to}& B GL }$$

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 $X$

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

One may identify this with the bundle that over each 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 $E : X \to \mathbf{B} O$, the corresponding orientation double cover is the $\mathbb{Z}_2$-bundle classified by

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

Real Hopf fibration

The real Hopf fibration is the non-trivial double cover of the circle by itself.

Spin double cover

• spin group

Related concepts

• orientifold, Jandl gerbe

• orientation

References

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

• geometry of physics.