High Quality Content by WIKIPEDIA articles! 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. 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 1x1 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.

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