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 is a weak inverse of under the tensor product of line bundles. Thus the isomorphism classes of line bundles form a group.
which itself represents the generator in the second homotopy group .
The classifying space/Eilenberg-MacLane space carries the circle group-universal principal bundle. The corresponding associated bundle via the canonical action of on is the universal complex line bundle.
|Calabi-Cau n-fold||line n-bundle||moduli of line n-bundles||moduli of flat/degree-0 n-bundles||Artin-Mazur formal group of deformation moduli of line n-bundles||complex oriented cohomology theory||modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory|
|unit in structure sheaf||multiplicative group/group of units||formal multiplicative group||complex K-theory|
|elliptic curve||line bundle||Picard group/Picard scheme||Jacobian||formal Picard group||elliptic cohomology||3d Chern-Simons theory/WZW model|
|K3 surface||line 2-bundle||Brauer group||intermediate Jacobian||formal Brauer group||K3 cohomology|
|Calabi-Yau 3-fold||line 3-bundle||intermediate Jacobian||CY3 cohomology||7d Chern-Simons theory/M5-brane|