vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A double cover is equivalently
a -principal bundle;
an etale space with local sections the 2-element set.
For a manifold, not necessarily oriented or even orientable, write
for any choice of orthogonal structure. The orientation double cover or orientation bundle of is the -principal bundle classified by the first Stiefel-Whitney class (of the tangent bundle) of
One may identify this with the bundle that over each neighbourhood of a point has as fibers the two different choices of volume forms up to positive rescaling (the two different choices of orientation).
More generally, for any orthogonal group-principal bundle classified by a morphism , the corresponding orientation double cover is the -bundle classified by
The real Hopf fibration is the non-trivial double cover of the circle by itself.
An exposition in a broader context is in the section higher spin structures at
Last revised on February 15, 2019 at 12:08:01. See the history of this page for a list of all contributions to it.