Line Bundle

In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology a line bundle is defined as a vector bundle of rank 1.

One-dimensional real line bundles (as just described) and one-dimensional complex line bundles differ. The topology of the 1×1 invertible real matrices is a space homotopy equivalent to a discrete two-point space (positive and negative reals contracted down), while 1×1 invertible complex matrices have the homotopy type of a circle.

A real line bundle is therefore in the eyes of homotopy theory as good as a fiber bundle with a two-point fiber - a double covering. This is like the orientable double cover on a differential manifold: indeed that's a special case in which the line bundle is the determinant bundle (top exterior power) of the tangent bundle. The Möbius strip corresponds to a double cover of the circle (the θ → 2θ mapping) and can be viewed if we wish as having fibre two points, the unit interval or the real line: the data are equivalent.

In the case of the complex line bundle, we are looking in fact also for circle bundles. There are some celebrated ones, for example the Hopf fibrations of spheres to spheres.