In topological applications, this vector space is usually real or complex.
The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane yields the 1×1 invertible complex matrices, which have the homotopy type of a circle.
From the perspective of homotopy theory, a real line bundle therefore behaves much the same as a fiber bundle with a two-point fiber, that is, like a double cover.
since it corresponds to the dual of the Serre twisting sheaf
is trivial, the total space of the line bundle is the product of
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
However, they are determined up to simultaneous multiplication by a non-zero function, so their ratios are well-defined.
are not well-defined because a change in trivialization will multiply them each by a non-zero constant λ.
Therefore, if the sections never simultaneously vanish, they determine a form
, and the pullback of the dual of the tautological bundle under this map is
In this way, projective space acquires a universal property.
The opposite is true in the algebraic and holomorphic settings.
When the line bundle is sufficiently ample this construction verifies the Kodaira embedding theorem.
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 nonvanishing global section, and its tensor powers with any real exponent may be defined and used to 'twist' any vector bundle by tensor product.
The same construction (taking the top exterior power) applies to a finitely generated projective module
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
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.
The Chern class statements are easily proven using the exponential sequence of sheaves on the manifold.
In these cases we can find those explicitly, in the infinite-dimensional analogues of real and complex projective space.
, 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
In this case classifying maps give rise to the first Chern class of
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.