Fibration

This article is about fibrations in algebraic topology. For fibrations in category theory, as used in descent theory and categorical logic, see Fibred category.

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 $p : E \to B$ 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 $E$ that are the inverse images of points $b$ of $B$ . If the base space $B$ is path connected, it is a consequence of the definition that the fibers of two different points $b 1$ and $b 2$ in $B$ are homotopy equivalent. Therefore one usually speaks of "the fiber" $F$ .

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 $B$ on the homology of the "total space" $E$ .

Examples

In the following examples a fibration is denoted

F \to E \to B

where the first map is the inclusion of "the" fiber $F$ into the total space $E$ and the second map is the fibration onto the basis $B$ . 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 $B$ , 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 $B$ , the bundle is a fibration. Any open cover of a paracompact space has a numerable refinement. 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 $B$ .
The Hopf fibration $S 1 \to S 3 \to S 2$ was historically one of the earliest non-trivial examples of a fibration.
Over complex projective space, there is a fibration $S 1 \to S 2 n +1 \to CP n$ .(Note that the Hopf fibration is a special case of this fibration for n=1, since CP¹ is homeomorphic to $S 2$ )
The Serre fibration $SO(2) \to SO(3) \to S 2$ 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 $SO(n) \to SO(n +1) \to S n$ for any non-negative integer $n$ (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 $n$ -sphere.

Properties

Long exact sequence in homotopy groups

Choose a base point $b 0 \in B$ . Let $F$ refer to the fiber over $b 0$ , i.e. $F = p -1 ({b 0})$ ; and let $i$ be the inclusion $F \to E$ . Choose a base point $f 0 \in F$ and let $e 0 = i (f 0)$ . In terms of these base points, we have a long exact sequence

\cdots\to\pi_n(F)\to\pi_n(E)\to\pi_n(B)\to\pi_{n-1}(F)\to\cdots \to \pi_{0}(F) \to \pi_{0}(E).

constructed from the homotopy groups of the fiber $F$ , total space $E$ , and base space $B$ . The homomorphisms $π n (F) \to π n (E)$ and $π n (E) \to π n (B)$ are just the induced homomorphisms from $i$ and $p$ , respectively. The maps involving π₀ are not group homomorphisms because the π₀ 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 $β n : π n (B) \to π n - 1 (F)$ (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 $δ n : S n \to D n +1$ be the inclusion of the boundary $n$ -sphere into the $(n +1)$ -ball. Let $γ n : D n \to S n$ be the map that collapses the image of $δ n - 1$ in $D n$ to a point.
Let $φ : S n \to B$ be a representing map for an element of $π n (B)$ .
Because $D n$ is homeomorphic to the $n$ -dimensional cube, we can iteratively apply the homotopy lifting property to construct a lift $λ : D n \to E$ of $φ \circ γ n$ (i.e., a map $λ$ such that $p \circ λ = φ \circ γ n$ ).
Because $γ n \circ δ n - 1$ is a point map (hereafter referred to as " $= pt$ "), $pt = φ \circ γ n \circ δ n - 1 = p \circ λ \circ δ n - 1$ , which implies that the image of $λ \circ δ n - 1$ is in $F$ . Therefore, there exists a map $ψ : S n - 1 \to F$ such that $i \circ ψ = λ \circ δ n - 1$ .
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

Main article: Euler characteristic

The Euler characteristic $χ$ is multiplicative for fibrations with certain conditions.

If $p : E \to B$ is a fibration with fiber $F$ , with the base $B$ path-connected, and the fibration is orientable over a field $K$ , then the Euler characteristic with coefficients in the field $K$ satisfies the product property:^[1]

χ (E) = χ (F) \cdot χ (B)

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 $τ : H * (B) \to H * (E)$ —note that this is a lifting and goes "the wrong way"—whose composition with the projection map $p * : H * (E) \to H * (B)$ is multiplication by the Euler characteristic of the fiber:^[2] $p * \circ τ = χ (F) \cdot 1$ .

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]

References

↑ Spanier, Edwin Henry (1982), Algebraic Topology, Springer, ISBN 978-0-387-94426-5 , Applications of the homology spectral sequence, p. 481
↑ Gottlieb, Daniel Henry (1975), "Fibre bundles and the Euler characteristic", Journal of Differential Geometry 10 (1): 39–48
↑ Dwyer, William G.; Spaliński, J. (1995), "Homotopy theories and model categories", Handbook of algebraic topology, Amsterdam: North-Holland, pp. 73–126, MR 1361887