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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ... In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

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. In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. ...

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. An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... A Möbius strip made with a piece of paper and tape. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...

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. In mathematics, the Hopf bundle (or Hopf fibration) is a particular fiber bundle with base space S2, total space S3, and fiber S1: S1 → S3 → S2 It was discovered by Heinz Hopf in 1931. ...

Determinant bundles

In general if V is a vector bundle on a space X, with constant fibre dimension n, the n-th exterior power of V taken fibre-by-fibre is a line bundle, called the determinant line bundle. This construction is in particular applied to the tangent bundle of a smooth manifold. The resulting determinant bundle is responsible for the phenomenon of tensor densities, in the sense that for an orientable manifold it has a global section, and its tensor powers with any real exponent may be defined and used to 'twist' any vector bundle by tensor product. In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... Tensor field - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ... This article or section should be merged with Orientability. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

Universal bundles and classifying spaces

From the point of view still of homotopy theory there are universal bundles for real line bundles (respectively, complex line bundles). According to general theory about classifying spaces, we should look for contractible spaces on which there are group actions of the respective groups C₂ and S¹, that are free actions. Those spaces can serve as the universal principal bundles, and the quotients for the actions as the classifying spaces BG. In these cases we can find those explicitly, in the infinite-dimensional analogues of real and complex projective space. In mathematics, a classifying space in homotopy theory of a discrete group G is, roughly speaking, a path connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... This article is about the mathematical concept. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, a projective space is a fundamental construction from any vector space. ...

Therefore the classifying space BC₂ is of the homotopy type of RP^∞, the real projective space given by an infinite sequence of homogeneous coordinates. It carries the universal real line bundle; in terms of homotopy theory that means that any real line bundle L on a CW complex X determines a classifying map from X to RP^∞, making L a bundle isomorphic to the pullback of the universal bundle. This classifying map can be used to define the Stiefel-Whitney class of L, in the first cohomology of X with Z/2Z coefficients, from a standard class on RP^∞. In mathematics, homogeneous co-ordinates, introduced by August Ferdinand Möbius, make calculations possible in projective space just as Cartesian co-ordinates do in Euclidean space. ... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... Stiefel-Whitney classes arise in mathematics as a type of characteristic class associated to real vector bundles . ...

In an analogous way, the complex projective space CP carries a universal complex line bundle. In this case classifying maps give rise to the first Chern class of X, in H²(X) (integral cohomology). In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are a particular type of characteristic class associated to complex vector bundles. ...

There is a further, analogous theory with quaternionic (real dimension four) line bundles. This gives rise to one of the Pontryagin classes, in real four-dimensional cohomology. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the Pontryagin classes are certain characteristic classes. ...

In this way foundational cases for the theory of characteristic classes depend only on line bundles. According to a general splitting principle this can determine the rest of the theory (if not explicitly). In mathematics, the idea of characteristic class is one of the unifying geometric concepts in algebraic topology, differential geometry and algebraic geometry. ...

There are theories of holomorphic line bundles on complex manifolds, and invertible sheaves in algebraic geometry, that work out a line bundle theory in those areas. In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

References

• Michael Murray, Line Bundles, 2002 (PDF web link)