nLab
double cover

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

A double cover is equivalently

Examples

Orientation double cover

For XX a manifold, not necessarily oriented or even orientable, write

BO T^X X TX BGL \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 XX is the 2\mathbb{Z}_2-principal bundle classified by the first Stiefel-Whitney class (of the tangent bundle) of XX

w 1(T^X):XT^XBOw 1B 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 xUXx \in U \subset X of a point xx has as fibers the two different choices of volume forms up to positive rescaling (the two different choices of orientation).

More generally, for EXE \to X any orthogonal group-principal bundle classified by a morphism E:XBOE : X \to \mathbf{B} O, the corresponding orientation double cover is the 2\mathbb{Z}_2-bundle classified by

w 1(E):XEBOw 1B 2. w_1(E) : X \stackrel{E}{\to} \mathbf{B} O \stackrel{w_1}{\to} \mathbf{B} \mathbb{Z}_2 \,.

Spin double cover

References

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

Revised on November 20, 2013 07:15:05 by Urs Schreiber (82.169.114.243)