nLab
line bundle

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

A line bundle is a vector bundle of rank (or dimension) 11, i.e. a vector bundle whose typical fiber is a 11-dimensional vector space (a line).

For complex vector bundles, complex line bundles are canonically associated bundles of circle group-principal bundles.

Properties

The class of line bundles has a nicer behaviour (in some ways) than the class of vector bundles in general. In particular, the dual? of a line bundle LL is a weak inverse of LL under the tensor product of line bundles. Thus the isomorphism classes of line bundles form a group.

Examples

Example

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.

Revised on February 5, 2013 20:29:39 by Urs Schreiber (131.174.41.0)