# nLab double cover

### Context

#### Bundles

bundles

fiber bundles in physics

## Context

• dependent type theory

• ## Classes of bundles

• covering space

• numerable bundle

• ## Presentations

• bundle gerbe

• groupal model for universal principal ∞-bundles

• microbundle

• ## Examples

• trivial vector bundle

• tautological line bundle

• basic line bundle on the 2-sphere?
• Hopf fibration

• canonical line bundle

• ## Constructions

• clutching construction

• inner product on vector bundles

• dual vector bundle

• projective bundle

• #### Cohomology

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

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

## References

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

Last revised on June 2, 2015 at 11:33:37. See the history of this page for a list of all contributions to it.