bundles

cohomology

# Contents

## Definition

A double cover is equivalently

## Examples

### Orientation double cover

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

$\begin{array}{ccc}& & BO\\ & {}^{\stackrel{^}{T}X}↗& ↓\\ X& \stackrel{TX}{\to }& B\mathrm{GL}\end{array}$\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 ${ℤ}_{2}$-principal bundle classified by the first Stiefel-Whitney class (of the tangent bundle) of $X$

${w}_{1}\left(\stackrel{^}{T}X\right):X\stackrel{\stackrel{^}{T}X}{\to }BO\stackrel{{w}_{1}}{\to }B{ℤ}_{2}\phantom{\rule{thinmathspace}{0ex}}.$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 ${ℤ}_{2}$-bundle classified by

${w}_{1}\left(E\right):X\stackrel{E}{\to }BO\stackrel{{w}_{1}}{\to }B{ℤ}_{2}\phantom{\rule{thinmathspace}{0ex}}.$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

Revised on January 9, 2013 01:08:40 by Urs Schreiber (89.204.138.146)