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There is an evident difference between one-dimensional real line bundles (as just described) and one-dimensional equivalent to a discrete two-point space (positive and negative reals contracted down), while the second has the homotopy type of a Circle . A real line bundle is therefore in the eyes of ) 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. 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 Fibration s of spheres to spheres. 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 . 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 Space s, we should look for Contractible spaces on which there are Group Action s of the respective groups ''C''2 and ''S''1, that are free actions. Those spaces can serve as the universal Principal Bundle s, 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 . Therefore the classifying space ''BC''2 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'''/2'''Z''' coefficients, from a standard class on RP∞. 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). There is a further, analogous theory with Quaternion ic (real dimension four) line bundles. This gives rise to one of the Pontryagin Class es, in real four-dimensional cohomology. In this way foundational cases for the theory of Characteristic Class es depend only on line bundles. According to a general ''splitting principle'' this can determine the rest of the theory (if not explicitly). There are theories of Holomorphic Line Bundle s on Complex Manifold s, and Invertible Sheaves in Algebraic Geometry , that work out a line bundle theory in those areas. REFERENCES
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