# nLab double cover

Contents

### Context

#### Bundles

bundles

fiber bundles in physics

cohomology

# Contents

## Definition

A double cover is equivalently

## 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.

## References

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

Last revised on February 15, 2019 at 07:08:01. See the history of this page for a list of all contributions to it.