The notion of a fibration generalizes the notion of a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
and plays an important role in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, a branch of mathematics.
Fibrations are used, for example, in
postnikov-systems or
obstruction theory In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.
In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the ex ...
.
In this article, all mappings are
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
mappings between
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s.
Formal definitions
Homotopy lifting property
A mapping
satisfies the
homotopy lifting property In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from ...
for a space
if:
* for every
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
and
* for every mapping (also called lift)
lifting
(i.e.
)
there exists a (not necessarily unique) homotopy
lifting
(i.e.
) with
The following
commutative diagram shows the situation:
Fibration
A fibration (also called Hurewicz fibration) is a mapping
satisfying the homotopy lifting property for all spaces
The space
is called base space and the space
is called total space. The fiber over
is the subspace
Serre fibration
A Serre fibration (also called weak fibration) is a mapping
satisfying the homotopy lifting property for all
CW-complexes
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
.
Every Hurewicz fibration is a Serre fibration.
Quasifibration
A mapping
is called quasifibration, if for every
and
holds that the induced mapping
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
.
Every Serre fibration is a quasifibration.
Examples
* The
projection
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
onto the first factor
is a fibration. That is, trivial bundles are fibrations.
* Every
covering satisfies the homotopy lifting property for all spaces. Specifically, for every homotopy
and every lift
there exists a uniquely defined lift
with
* Every
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
satisfies the homotopy lifting property for every CW-complex.
* A fiber bundle with a
paracompact
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
and Hausdorff base space satisfies the homotopy lifting property for all spaces.
* An example for a fibration, which is not a fiber bundle, is given by the mapping
induced by the inclusion
where
a topological space and
is the space of all continuous mappings with the
compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and ...
.
* The
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz H ...
is a non trivial fiber bundle and specifically a Serre fibration.
Basic concepts
Fiber homotopy equivalence
A mapping
between total spaces of two fibrations
and
with the same base space is a fibration homomorphism if the following diagram commutes:
The mapping
is a fiber homotopy equivalence if in addition a fibration homomorphism
exists, such that the mappings
and
are homotopic, by fibration homomorphisms, to the identities
and
Pullback fibration
Given a fibration
and a mapping
, the mapping
is a fibration, where
is the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
and the projections of
onto
and
yield the following commutative diagram:
The fibration
is called the pullback fibration or induced fibration.
Pathspace fibration
With the pathspace construction, any continuous mapping can be extended to a fibration by enlarging its domain to a homotopy equivalent space. This fibration is called pathspace fibration.
The total space
of the
pathspace fibration for a continuous mapping
between topological spaces consists of pairs
with
and
paths with starting point
where