(see also Chern-Weil theory, parameterized homotopy theory)
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A line bundle is a vector bundle of rank (or dimension) $1$, i.e. a vector bundle whose typical fiber is a $1$-dimensional vector space (a line).
For complex vector bundles, complex line bundles are canonically associated bundles of circle group-principal bundles.
The class of line bundles has a nicer behaviour (in some ways) than the class of vector bundles in general. In particular, the dual vector bundle of a line bundle $L$ is a weak inverse of $L$ under the tensor product of line bundles. Thus the isomorphism classes of line bundles form a group.
The Möbius strip is the unique, up to isomorphism, non-trivial real line bundle over the circle.
Over any manifold there is canonically the density line bundle which is the associated bundle to the principal bundle underlying the tangent bundle by the determinant homomorphism.
Similarly:
Every orientable complex manifold carries a comple line bundle of top-degree holomorphic differential forms. This is called its canonical line bundle.
The line bundle on the 2-sphere whose first Chern class is a generator of $H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z}$ is the pullback bundle of the universal complex line bundle (example ) along the map
which itself represents the generator in the second homotopy group $\pi_2(S^2) \simeq \mathbb{Z}$.
Beware that this is not the “canonical line bundle” from example , but “half” of it, its theta characteristic. See at geometric quantization of the 2-sphere for more on this.
The classifying space/Eilenberg-MacLane space $B U(1) \simeq B^2 \mathbb{Z}$ carries the circle group-universal principal bundle. The corresponding associated bundle via the canonical action of $U(1)$ on $\mathbb{C}$ is the universal complex line bundle.
The product of any space $X$ with the moduli stack $Pic_X$ of line bundles over it (its Picard stack) carries a tautological line bundle. This is called the Poincaré line bundle of $X$.
moduli spaces of line n-bundles with connection on $n$-dimensional $X$
Last revised on December 11, 2022 at 17:12:03. See the history of this page for a list of all contributions to it.