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 rank 1.[1]

One-dimensional real line bundles (as just described) and one-dimensional complex line bundles differ. The topology of the 1×1 invertible real matrices is a space homotopy equivalent to a discrete two-point space (positive and negative reals contracted down), while 1×1 invertible complex matrices have the homotopy type of a circle.

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 is like the orientable 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 strip corresponds to a double cover of the circle (the θ → 2θ mapping) and can be viewed if 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 fibrations of spheres to spheres.

Contents

The tautological bundle on projective space

Main article: Tautological line bundle

The most important line bundle is the tautological line bundle on projective space. The projectivization P(V) of a vector space V over a field k is defined to be the quotient of V \setminus \{0\} by the action of the multiplicative group k×. Each point of P(V) therefore corresponds to a copy of k×, and these copies of k× can be assembled into a k×-bundle over P(V). k× differs from k only by a single point, and by adjoining that point to each fiber, we get a line bundle on P(V). This line bundle is called the tautological line bundle. This line bundle is sometimes denoted \mathcal{O}(-1) since it corresponds to the dual of the Serre twisting sheaf \mathcal{O}(1).

Maps to projective space

Suppose that X is a space and that L is a line bundle on X. A global section of L is a function s : X → L such that if p : L → X is the natural projection, then ps = idX. In a small neighborhood U in X in which L is trivial, the total space of the line bundle is the product of U and the underlying field k, and the section s restricts to a function U → k. However, the values of s depend on the choice of trivialization, and so they are determined only up to multiplication by a nowhere-vanishing function.

Global sections determine maps to projective spaces in the following way: Choosing r + 1 not all zero points in a fiber of L chooses a fiber of the tautological line bundle on Pr, so choosing r + 1 non-simultaneously vanishing global sections of L determines a map from X into projective space Pr. This map sends the fibers of L to the fibers of the dual of the tautological bundle. More specifically, suppose that s0, ..., sr are global sections of L. In a small neighborhood U in X, these sections determine k-valued functions on U whose values depend on the choice of trivialization. However, they are determined up to simultaneous multiplication by a non-zero function, so their ratios are well-defined. That is, over a point x, the values s0(x), ..., sr(x) are not well-defined because a change in trivialization will multiply them each by a non-zero constant λ. But it will multiply them by the same constant λ, so the homogeneous coordinates [s0(x) : ... : sr(x)] are well-defined as long as the sections s0, ..., sr do not simultaneously vanish at x. Therefore, if the sections never simultaneously vanish, they determine a form [s0 : ... : sr] which gives a map from X to Pr, and the pullback of the dual of the tautological bundle under this map is L. In this way, projective space acquires a universal property.

The universal way to determine a map to projective space is to map to the projectivization of the vector space of all sections of L. In the topological case, there is a non-vanishing section at every point which can be constructed using a bump function which vanishes outside a small neighborhood of the point. Because of this, the resulting map is defined everywhere. However, the codomain is usually far, far too big to be useful. The opposite is true in the algebraic and holomorphic settings. Here the space of global sections is often finite dimensional, but there may not be any non-vanishing global sections at a given point. In fact, it is possible for a bundle to have no non-zero global sections at all; this is the case for the tautological line bundle.

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 cotangent 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.

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 Z/2Z 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 structures (and thus the same first Chern class) but different holomorphic structures. The Chern class statements are easily proven using the exponential sequence of 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 spaces, the heuristic is to 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.

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 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 quaternionic (real dimension four) line bundles. This gives rise to one of the Pontryagin classes, in real four-dimensional cohomology.

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).

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.

Notes

  1. ^ Hartshorne (1975), p. 7

References

See also