Editing Line bundle (section)

==Characteristic classes, universal bundles and classifying spaces==
The first [[Stiefel–Whitney class]] classifies smooth real line bundles; in particular, the collection of (equivalence classes of) real line bundles are in correspondence with elements of the first cohomology with <math>\mathbb{Z}/2\mathbb{Z}</math> coefficients; this correspondence is in fact an isomorphism of abelian groups (the  group operations being tensor product of line bundles and the usual addition on cohomology).  Analogously, the first [[Chern class]] classifies smooth complex line bundles on a space, and the group of line bundles is isomorphic to the second cohomology class with integer coefficients.  However, bundles can have equivalent [[smooth structure]]s (and thus the same first Chern class) but different holomorphic structures.  The Chern class statements are easily proven using the [[exponential sequence]] of [[Sheaf (mathematics)|sheaves]] on the manifold.

One can more generally view the classification problem from a homotopy-theoretic point of view.  There is a universal bundle for real line bundles, and a universal bundle for complex line bundles. According to general theory about [[classifying space]]s, the heuristic is to look for [[contractible]] spaces on which there are [[Group action (mathematics)|group action]]s of the respective groups <math>C_2</math> and <math>S^1</math>, that are free actions. Those spaces can serve as the universal [[principal bundle]]s, and the quotients for the actions as the classifying spaces <math>BG</math>. In these cases we can find those explicitly, in the infinite-dimensional analogues of real and complex [[projective space]].

Therefore the classifying space <math>BC_2</math> is of the homotopy type of <math>\mathbb{R}\mathbf{P}^{\infty}</math>, 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 <math>L</math> on a [[CW complex]] <math>X</math> determines a ''classifying map'' from <math>X</math> to <math>\mathbb{R}\mathbf{P}^{\infty}</math>, making <math>L</math> a bundle isomorphic to the pullback of the universal bundle. This classifying map can be used to define the [[Stiefel-Whitney class]] of <math>L</math>, in the first cohomology of <math>X</math> with <math>\mathbb{Z}/2\mathbb{Z}</math> coefficients, from a standard class on <math>\mathbb{R}\mathbf{P}^{\infty}</math>.

In an analogous way, the complex projective space <math>\mathbb{C}\mathbf{P}^{\infty}</math> carries a universal complex line bundle. In this case classifying maps give rise to the first [[Chern class]] of <math>X</math>, in <math>H^2(X)</math> (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 sheaf|invertible sheaves]] in [[algebraic geometry]], that work out a line bundle theory in those areas.