# Fibration

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In topology, a branch of mathematics, a **fibration** is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space (called a fiber) being "parameterized" by another topological space (called a base). A fibration is like a fiber bundle, except that the fibers need not be the same space, rather they are just homotopy equivalent. Fibrations do not necessarily have the local Cartesian product structure that defines the more restricted fiber bundle case, but something weaker that still allows "sideways" movement from fiber to fiber. Fiber bundles have a particularly simple homotopy theory that allows topological information about the bundle to be inferred from information about one or both of these constituent spaces. A fibration satisfies an additional condition (the homotopy lifting property) guaranteeing that it will behave like a fiber bundle from the point of view of homotopy theory.

## Formal definition

A **fibration** (or **Hurewicz fibration**) is a continuous mapping *p* : *E* → *B* satisfying the homotopy lifting property with respect to any space. Fiber bundles (over paracompact bases) constitute important examples. In homotopy theory any mapping is 'as good as' a fibration—i.e. any map can be decomposed as a homotopy equivalence into a "mapping path space" followed by a fibration. (See homotopy fiber.)

The *fibers* are by definition the subspaces of Template:Mvar that are the inverse images of points Template:Mvar of Template:Mvar. If the base space Template:Mvar is path connected, it is a consequence of the definition that the fibers of two different points *b*_{1} and *b*_{2} in Template:Mvar are homotopy equivalent. Therefore one usually speaks of "the fiber" Template:Mvar.

## Serre fibrations

A continuous mapping with the homotopy lifting property for CW complexes (or equivalently, just cubes *I*^{n}) is called a *Serre fibration*, in honor of the part played by the concept in the thesis of Jean-Pierre Serre. This thesis firmly established in algebraic topology the use of spectral sequences, and clearly separated the notions of fiber bundles and fibrations from the notion of sheaf (both concepts together having been implicit in the pioneer treatment of Jean Leray). Because a sheaf (thought of as an étalé space) can be considered a local homeomorphism, the notions were closely interlinked at the time. One of the main desirable properties of the Serre spectral sequence is to account for the action of the fundamental group of the base Template:Mvar on the homology of the "total space" Template:Mvar.

## Examples

In the following examples a fibration is denoted

where the first map is the inclusion of "the" fiber Template:Mvar into the total space Template:Mvar and the second map is the fibration onto the basis Template:Mvar. This is also referred to as a fibration sequence.

- The projection map from a product space is very easily seen to be a fibration.
- Fiber bundles have
*local trivializations,*i.e. Cartesian product structures exist locally on Template:Mvar, and this is usually enough to show that a fiber bundle is a fibration. More precisely, if there are local trivializations over a "numerable open cover" of Template:Mvar, the bundle is a fibration. Any open cover of a paracompact space is numerable. For example, any open cover of a metric space has a locally finite refinement, so any bundle over such a space is a fibration. The local triviality also implies the existence of a well-defined*fiber*(up to homeomorphism), at least on each connected component of Template:Mvar. - The Hopf fibration
*S*^{1}→*S*^{3}→*S*^{2}was historically one of the earliest non-trivial examples of a fibration. - Over complex projective space, there is a fibration
*S*^{1}→*S*^{2n+1}→**CP**^{n}.(Note that the Hopf fibration is a special case of this fibration for n=1, since**CP**^{1}is homeomorphic to*S*^{ 2}) - The Serre fibration SO(2) → SO(3) →
*S*^{2}comes from the action of the rotation group SO(3) on the 2-sphere*S*^{2}. - The previous example can also be generalized to a fibration SO(
*n*) → SO(*n*+1) →*S*^{n}for any non-negative integer Template:Mvar (though they only have a fiber that isn't just a point when*n*> 1) that comes from the action of the special orthogonal group SO(*n*+1) on the Template:Mvar-sphere.

## Properties

### Long exact sequence in homotopy groups

Choose a base point *b*_{0} ∈ *B*. Let Template:Mvar refer to the fiber over *b*_{0}, i.e. F = *p*^{-1}({*b*_{0}}); and let Template:Mvar be the inclusion *F* → *E*. Choose a base point *f*_{0} ∈ *F* and let *e*_{0} = *i*(*f*_{0}). In terms of these base points, we have a long exact sequence

constructed from the homotopy groups of the fiber Template:Mvar, total space Template:Mvar, and base space Template:Mvar. The homomorphisms *π*_{n}(*F*) → *π*_{n}(*E*) and *π*_{n}(*E*) → *π*_{n}(*B*) are just the induced homomorphisms from Template:Mvar and Template:Mvar, respectively. The maps involving π_{0} are not group homomorphisms because the π_{0} are not groups, but they are exact in the sense that the image equals the kernel (here the "neutral element" is the connected component containing the base point).

The third set of homomorphisms *β*_{n} : *π*_{n}(*B*) → *π*_{n−1}(*F*) (called the "connecting homomorphisms" (in reference to the snake lemma) or the "boundary maps") can be defined with the following steps.

- First, a little terminology: let
*δ*_{n}:*S*^{n}→*D*^{n+1}be the inclusion of the boundary [[n-sphere|Template:Mvar-sphere]] into the (*n*+1)-ball. Let*γ*_{n}:*D*^{n}→*S*^{n}be the map that collapses the image of*δ*_{n−1}in*D*^{n}to a point. - Let
*φ*:*S*^{n}→*B*be a representing map for an element of*π*_{n}(*B*). - Because
*D*^{n}is homeomorphic to the Template:Mvar-dimensional cube, we can iteratively apply the homotopy lifting property to construct a lift*λ*:*D*^{n}→*E*of*φ*∘*γ*_{n}(i.e., a map Template:Mvar such that*p*∘*λ*=*φ*∘*γ*_{n}). - Because
*γ*_{n}∘*δ*_{n−1}is a point map (hereafter referred to as "= pt"), pt =*φ*∘*γ*_{n}∘*δ*_{n−1}=*p*∘*λ*∘*δ*_{n−1}, which implies that the image of*λ*∘*δ*_{n−1}is in Template:Mvar. Therefore, there exists a map*ψ*:*S*^{n−1}→*F*such that*i*∘*ψ*=*λ*∘*δ*_{n−1}. - We define
*β*_{n}[*φ*] = [*ψ*].

The above is summarized in the following commutative diagram:

Repeated application of the homotopy lifting property is used to prove that *β*_{n} is a well-defined homomorphism and that this sequence is exact.

### Euler characteristic

{{#invoke:main|main}} The Euler characteristic Template:Mvar is multiplicative for fibrations with certain conditions.

If *p* : *E* → *B* is a fibration with fiber Template:Mvar, with the base Template:Mvar path-connected, and the fibration is orientable over a field Template:Mvar, then the Euler characteristic with coefficients in the field Template:Mvar satisfies the product property:^{[1]}

This includes product spaces and covering spaces as special cases, and can be proven by the Serre spectral sequence on homology of a fibration.

For fiber bundles, this can also be understood in terms of a transfer map *τ* : *H*_{∗}(*B*) → *H*_{∗}(*E*)—note that this is a lifting and goes "the wrong way"—whose composition with the projection map *p*_{∗} : *H*_{∗}(*E*) → *H*_{∗}(*B*) is multiplication by the Euler characteristic of the fiber:^{[2]}
*p*_{∗} ∘ *τ* = *χ*(*F*) · 1.

## Fibrations in closed model categories

Fibrations of topological spaces fit into a more general framework, the so-called closed model categories. In such categories, there are distinguished classes of morphisms, the so-called *fibrations*, *cofibrations* and *weak equivalences*. Certain axioms, such as stability of fibrations under composition and pullbacks, factorization of every morphism into the composition of an acyclic cofibration followed by a fibration or a cofibration followed by an acyclic fibration, where the word "acyclic" indicates that the corresponding arrow is also a weak equivalence, and other requirements are set up to allow the abstract treatment of homotopy theory. (In the original treatment, due to Daniel Quillen, the word "trivial" was used instead of "acyclic.")

It can be shown that the category of topological spaces is in fact a model category, where (abstract) fibrations are just the Serre fibrations introduced above and weak equivalences are weak homotopy equivalences.^{[3]}

## See also

## References

- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}, Applications of the homology spectral sequence, p. 481
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