Trivial bundle Let E = B \times F and let \pi : E \to B be the projection onto the first factor. Then \pi is a fiber bundle (of F) over B. Here E is not just locally a product but
globally one. Any such fiber bundle is called a ''''''. Any fiber bundle over a
contractible CW-complex is trivial.
Nontrivial bundles Möbius strip Perhaps the simplest example of a nontrivial bundle E is the
Möbius strip. It has the
circle that runs lengthwise along the center of the strip as a base B and a
line segment for the fiber F, so the Möbius strip is a bundle of the line segment over the circle. A
neighborhood U of \pi(x) \in B (where x \in E) is an
arc; in the picture, this is the
length of one of the squares. The
preimage \pi^{-1}(U) in the picture is a (somewhat twisted) slice of the strip four squares wide and one long (i.e. all the points that project to U). A homeomorphism (\varphi in ) exists that maps the preimage of U (the trivializing neighborhood) to a slice of a cylinder: curved, but not twisted. This pair locally trivializes the strip. The corresponding trivial bundle B\times F would be a
cylinder, but the Möbius strip has an overall "twist". This twist is visible only globally; locally the Möbius strip and the cylinder are identical (making a single vertical cut in either gives the same space).
Klein bottle A similar nontrivial bundle is the
Klein bottle, which can be viewed as a "twisted" circle bundle over another circle. The corresponding non-twisted (trivial) bundle is the 2-
torus, S^1 \times S^1.
Covering map A
covering space is a fiber bundle such that the bundle projection is a
local homeomorphism. It follows that the fiber is a
discrete space.
Vector and principal bundles A special class of fiber bundles, called
vector bundles, are those whose fibers are
vector spaces (to qualify as a vector bundle the structure group of the bundle—see below—must be a
linear group). Important examples of vector bundles include the
tangent bundle and
cotangent bundle of a smooth manifold. From any vector bundle, one can construct the
frame bundle of
bases, which is a principal bundle (see below). Another special class of fiber bundles, called
principal bundles, are bundles on whose fibers a
free and
transitive action by a group G is given, so that each fiber is a
principal homogeneous space. The bundle is often specified along with the group by referring to it as a principal G-bundle. The group G is also the structure group of the bundle. Given a
representation \rho of G on a vector space V, a vector bundle with \rho(G) \subseteq \text{Aut}(V) as a structure group may be constructed, known as the
associated bundle.
Sphere bundles A
sphere bundle is a fiber bundle whose fiber is an
n-sphere. Given a vector bundle E with a
metric (such as the tangent bundle to a
Riemannian manifold) one can construct the associated
unit sphere bundle, for which the fiber over a point x is the set of all
unit vectors in E_x. When the vector bundle in question is the tangent bundle TM, the unit sphere bundle is known as the
unit tangent bundle. A sphere bundle is partially characterized by its
Euler class, which is a degree n + 1
cohomology class in the total space of the bundle. In the case n = 1 the sphere bundle is called a
circle bundle and the Euler class is equal to the first
Chern class, which characterizes the topology of the bundle completely. For any n, given the Euler class of a bundle, one can calculate its cohomology using a
long exact sequence called the
Gysin sequence.
Mapping tori If X is a
topological space and f : X \to X is a
homeomorphism then the
mapping torus M_f has a natural structure of a fiber bundle over the
circle with fiber X. Mapping tori of homeomorphisms of
surfaces are of particular importance in
3-manifold topology.
Quotient spaces If G is a
topological group and H is a
closed subgroup, then under some circumstances, the
quotient space G/H together with the quotient map \pi : G \to G/H is a fiber bundle, whose fiber is the topological space H. A
necessary and sufficient condition for (G,\, G/H,\, \pi,\, H) to form a fiber bundle is that the mapping \pi admits
local cross-sections . The most general conditions under which the
quotient map will admit local cross-sections are not known, although if G is a
Lie group and H a closed subgroup (and thus a
Lie subgroup by
Cartan's theorem), then the quotient map is a fiber bundle. One example of this is the
Hopf fibration, S^3 \to S^2, which is a fiber bundle over the sphere S^2 whose total space is S^3. From the perspective of Lie groups, S^3 can be identified with the
special unitary group SU(2). The
abelian subgroup of
diagonal matrices is
isomorphic to the
circle group U(1), and the quotient SU(2)/U(1) is
diffeomorphic to the sphere. More generally, if G is any topological group and H a closed subgroup that also happens to be a Lie group, then G \to G/H is a fiber bundle. == Sections ==