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 one-dimensional vector spaces.

Related Topics:
Mathematics - Tangent - Tangent bundle - Algebraic topology - Differential topology - Vector bundle - Vector space

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There is an evident difference between one-dimensional real line bundles (as just described) and one-dimensional complex line bundles. In fact the topology of the 1×1 invertible real matrices and complex matrices is entirely different: the first of those is a space homotopy equivalent to a discrete two-point space (positive and negative reals contracted down), while the second has the homotopy type of a circle.

Related Topics:
Complex - Invertible - Homotopy - Circle

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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 reminds one of the orientation 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 band corresponds to a double cover of the circle (the Θ → 2Θ mapping) and can be viewed as we wish as having fibre two points, the unit interval or the real line: the data are equivalent.

Related Topics:
Homotopy theory - Fiber bundle - Orientation double cover - Differential manifold - Exterior power - Möbius band - Unit interval

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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.

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