Line bundle - Imported Knowlpedia Content

Definition

Let $\mathbb F$ be $\mathbb R$ or $\mathbb C$ . An $\mathbb F$ -line bundle over a smooth manifold $M$ is a vector bundle $L\to M$ whose rank over $\mathbb F$ is one. Thus every fiber $L_x$ is a one-dimensional $\mathbb F$ -vector space, and local trivializations identify $L|_U$ with $U\times\mathbb F$ by fiberwise linear maps. Its transition functions take values in $\mathbb F^\times$ . “Real” or “complex” must be specified because the scalar field changes both the structure group and the classification theory.

Basic properties

A line bundle is trivial exactly when it admits a nowhere-vanishing global section: such a section is a basis in every fiber and hence a global frame.

The dual $L^*$ and the tensor product $L\otimes K$ are again line bundles. The evaluation isomorphism $L\otimes L^*\cong M\times\mathbb F$ makes dualization the inverse operation under tensor product. Isomorphism classes of line bundles therefore form an abelian group, often called a Picard group.

Over a paracompact space, complex line bundles are classified by their first Chern class in $H^2(M;\mathbb Z)$ . Real line bundles are classified by their first Stiefel–Whitney class in $H^1(M;\mathbb Z/2)$ Husemoller, chapter 3.

Examples and non-examples

The product $M\times\mathbb F$ is the trivial line bundle. The Möbius bundle over the circle is a nontrivial real line bundle. The tautological bundles over real and complex projective spaces are the canonical nontrivial examples in their respective categories.

A rank-two real vector bundle is not a real line bundle, even if it can be regarded fiberwise as a one-dimensional complex vector space: calling it a complex line bundle requires a smoothly varying complex structure and complex-linear transition functions.

The zero section exists in every vector bundle but does not trivialize a line bundle, because it is nowhere nonzero.

Conventions and scope

In algebraic geometry, “line bundle” usually means a locally free sheaf of rank one or its corresponding algebraic vector bundle. The present definition is smooth. Topological line bundles use continuous rather than smooth local trivializations.

References

D. Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: chapters 1 and 3, vector bundles, line bundles, and classifying constructions.
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton University Press, 1974. DOI record. Relevant: chapter 1, real and complex vector bundles and their basic characteristic classes.