Fibration

The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.

Fibrations are used, for example, in Postnikov systems or obstruction theory.

In this article, all mappings are continuous mappings between topological spaces.

Homotopy lifting property
A mapping $$p \colon E \to B$$ satisfies the homotopy lifting property for a space $$X$$ if:


 * for every homotopy $$h \colon X \times [0, 1] \to B$$ and
 * for every mapping (also called lift) $$\tilde h_0 \colon X \to E$$ lifting $$h|_{X \times 0} = h_0$$ (i.e. $$h_0 = p \circ \tilde h_0$$)

there exists a (not necessarily unique) homotopy $$\tilde h \colon X \times [0, 1] \to E$$ lifting $$h$$ (i.e. $$h = p \circ \tilde h$$) with $$\tilde h_0 = \tilde h|_{X \times 0}.$$

The following commutative diagram shows the situation:

Fibration
A fibration (also called Hurewicz fibration) is a mapping $$p \colon E \to B$$ satisfying the homotopy lifting property for all spaces $$X.$$ The space $$B$$ is called base space and the space $$E$$ is called total space. The fiber over $$b \in B$$ is the subspace $$F_b = p^{-1}(b) \subseteq E.$$

Serre fibration
A Serre fibration (also called weak fibration) is a mapping $$p \colon E \to B$$ satisfying the homotopy lifting property for all CW-complexes.

Every Hurewicz fibration is a Serre fibration.

Quasifibration
A mapping $$p \colon E \to B$$ is called quasifibration, if for every $$b \in B,$$ $$e \in p^{-1}(b)$$ and $$i \geq 0$$ holds that the induced mapping $$p_* \colon \pi_i(E, p^{-1}(b), e) \to \pi_i(B, b)$$ is an isomorphism.

Every Serre fibration is a quasifibration.

Examples

 * The projection onto the first factor $$p \colon B \times F \to B$$ is a fibration. That is, trivial bundles are fibrations.
 * Every covering $$p \colon E \to B$$ is a fibration. Specifically, for every homotopy $$h \colon X \times [0, 1] \to B$$ and every lift $$\tilde h_0 \colon X \to E$$ there exists a uniquely defined lift $$\tilde h \colon X \times [0,1] \to E$$ with $$p \circ \tilde h = h.$$
 * Every fiber bundle $$p \colon E \to B$$ satisfies the homotopy lifting property for every CW-complex.
 * A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces.
 * An example of a fibration which is not a fiber bundle is given by the mapping $$i^* \colon X^{I^k} \to X^{\partial I^k}$$ induced by the inclusion $$i \colon \partial I^k \to I^k$$ where $$k \in \N,$$ $$X$$ a topological space and $$X^{A} = \{f \colon A \to X\}$$ is the space of all continuous mappings with the compact-open topology.
 * The Hopf fibration $$S^1 \to S^3 \to S^2$$ is a non-trivial fiber bundle and, specifically, a Serre fibration.

Fiber homotopy equivalence
A mapping $$f \colon E_1 \to E_2$$ between total spaces of two fibrations $$p_1 \colon E_1 \to B$$ and $$p_2 \colon E_2 \to B$$ with the same base space is a fibration homomorphism if the following diagram commutes: The mapping $$f$$ is a fiber homotopy equivalence if in addition a fibration homomorphism $$g \colon E_2 \to E_1$$ exists, such that the mappings $$f \circ g$$ and $$g \circ f$$ are homotopic, by fibration homomorphisms, to the identities $$\operatorname{Id}_{E_2}$$ and $$\operatorname{Id}_{E_1}.$$

Pullback fibration
Given a fibration $$p \colon E \to B$$ and a mapping $$f \colon A \to B$$, the mapping $$p_f \colon f^*(E) \to A$$ is a fibration, where $$f^*(E) = \{(a, e) \in A \times E | f(a) = p(e)\}$$ is the pullback and the projections of $$f^*(E)$$ onto $$A$$ and $$E$$ yield the following commutative diagram: The fibration $$p_f$$ 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 $$E_f$$ of the pathspace fibration for a continuous mapping $$f \colon A \to B$$ between topological spaces consists of pairs $$(a, \gamma)$$ with $$a \in A$$ and paths $$\gamma \colon I \to B$$ with starting point $$\gamma (0) = f(a),$$ where $$I = [0, 1]$$ is the unit interval. The space $$E_f = \{ (a, \gamma) \in A \times B^I | \gamma (0) = f(a) \}$$ carries the subspace topology of $$A \times B^I,$$ where $$B^I$$ describes the space of all mappings $$I \to B$$ and carries the compact-open topology.

The pathspace fibration is given by the mapping $$p \colon E_f \to B$$ with $$p(a, \gamma) = \gamma (1).$$ The fiber $$F_f$$ is also called the homotopy fiber of $$f$$ and consists of the pairs $$(a, \gamma)$$ with $$a \in A$$ and paths $$\gamma \colon [0, 1] \to B,$$ where $$\gamma(0) = f(a)$$ and $$\gamma(1) = b_0 \in B$$ holds.

For the special case of the inclusion of the base point $$i \colon b_0 \to B$$, an important example of the pathspace fibration emerges. The total space $$E_i$$ consists of all paths in $$B$$ which starts at $$b_0.$$ This space is denoted by $$PB$$ and is called path space. The pathspace fibration $$p \colon PB \to B$$ maps each path to its endpoint, hence the fiber $$p^{-1}(b_0)$$ consists of all closed paths. The fiber is denoted by $$\Omega B$$ and is called loop space.

Properties

 * The fibers $$p^{-1}(b)$$ over $$b \in B$$ are homotopy equivalent for each path component of $$B.$$
 * For a homotopy $$f \colon [0, 1] \times A \to B$$ the pullback fibrations $$f^*_0(E) \to A$$ and $$f^*_1(E) \to A$$ are fiber homotopy equivalent.
 * If the base space $$B$$ is contractible, then the fibration $$p \colon E \to B$$ is fiber homotopy equivalent to the product fibration $$B \times F \to B.$$
 * The pathspace fibration of a fibration $$p \colon E \to B$$ is very similar to itself. More precisely, the inclusion $$E \hookrightarrow E_p$$ is a fiber homotopy equivalence.
 * For a fibration $$p \colon E \to B$$ with fiber $$F$$ and contractible total space, there is a weak homotopy equivalence $$F \to \Omega B.$$

Puppe sequence
For a fibration $$p \colon E \to B$$ with fiber $$F$$ and base point $$b_0 \in B$$ the inclusion $$F \hookrightarrow F_p$$ of the fiber into the homotopy fiber is a homotopy equivalence. The mapping $$i \colon F_p \to E$$ with $$i (e, \gamma) = e$$, where $$e \in E$$ and $$\gamma \colon I \to B$$ is a path from $$p(e)$$ to $$b_0$$ in the base space, is a fibration. Specifically it is the pullback fibration of the pathspace fibration $$PB \to B$$. This procedure can now be applied again to the fibration $$i$$ and so on. This leads to a long sequence: "$ \cdots \to F_j \to F_i \xrightarrow {j} F_p \xrightarrow i E \xrightarrow p B.$" The fiber of $$i$$ over a point $$e_0 \in p^{-1}(b_0)$$ consists of the pairs $$(e_0, \gamma)$$ with closed paths $$\gamma$$ and starting point $$b_0$$, i.e. the loop space $$\Omega B$$. The inclusion $$\Omega B \hookrightarrow F_p(\simeq F)$$ is a homotopy equivalence and iteration yields the sequence:"$\cdots \Omega^2B \to \Omega F \to \Omega E \to \Omega B \to F \to E \to B.$"Due to the duality of fibration and cofibration, there also exists a sequence of cofibrations. These two sequences are known as the Puppe sequences or the sequences of fibrations and cofibrations.

Principal fibration
A fibration $$p \colon E \to B$$ with fiber $$F$$ is called principal, if there exists a commutative diagram: The bottom row is a sequence of fibrations and the vertical mappings are weak homotopy equivalences. Principal fibrations play an important role in Postnikov towers.

Long exact sequence of homotopy groups
For a Serre fibration $$p \colon E \to B$$ there exists a long exact sequence of homotopy groups. For base points $$b_0 \in B$$ and $$x_0 \in F = p^{-1}(b_0)$$ this is given by: $$\cdots \rightarrow \pi_n(F,x_0) \rightarrow \pi_n(E, x_0) \rightarrow \pi_n(B, b_0) \rightarrow \pi_{n - 1}(F, x_0) \rightarrow $$

$$\cdots \rightarrow \pi_0(F, x_0) \rightarrow \pi_0(E, x_0).$$ The homomorphisms $$\pi_n(F, x_0) \rightarrow \pi_n(E, x_0)$$ and $$\pi_n(E, x_0) \rightarrow \pi_n(B, b_0)$$ are the induced homomorphisms of the inclusion $$i \colon F \hookrightarrow E$$ and the projection $$p \colon E \rightarrow B.$$

Hopf fibration
Hopf fibrations are a family of fiber bundles whose fiber, total space and base space are spheres: $$S^0 \hookrightarrow S^1 \rightarrow S^1,$$

$$S^1 \hookrightarrow S^3 \rightarrow S^2,$$

$$S^3 \hookrightarrow S^7 \rightarrow S^4,$$

$$S^7 \hookrightarrow S^{15} \rightarrow S^8.$$ The long exact sequence of homotopy groups of the hopf fibration $$S^1 \hookrightarrow S^3 \rightarrow S^2$$ yields:"$\cdots \rightarrow \pi_n(S^1,x_0) \rightarrow \pi_n(S^3, x_0) \rightarrow \pi_n(S^2, b_0) \rightarrow \pi_{n - 1}(S^1, x_0) \rightarrow $ $\cdots \rightarrow \pi_1(S^1, x_0) \rightarrow \pi_1(S^3, x_0) \rightarrow \pi_1(S^2, b_0).$" This sequence splits into short exact sequences, as the fiber $$S^1$$ in $$S^3$$ is contractible to a point:"$0 \rightarrow \pi_i(S^3) \rightarrow \pi_i(S^2) \rightarrow \pi_{i-1}(S^1) \rightarrow 0.$"This short exact sequence splits because of the suspension homomorphism $$ \phi \colon \pi_{i - 1}(S^1) \to \pi_i(S^2)$$ and there are isomorphisms:"$\pi_i(S^2) \cong \pi_i(S^3) \oplus \pi_{i - 1}(S^1).$"The homotopy groups $$\pi_{i - 1}(S^1)$$ are trivial for $$i \geq 3,$$ so there exist isomorphisms between $$\pi_i(S^2)$$ and $$\pi_i(S^3)$$ for $$i \geq 3.$$

Analog the fibers $$S^3$$ in $$S^7$$ and $$S^7$$ in $$S^{15}$$ are contractible to a point. Further the short exact sequences split and there are families of isomorphisms: $$\pi_i(S^4) \cong \pi_i(S^7) \oplus \pi_{i - 1}(S^3)$$ and

$$\pi_i(S^8) \cong \pi_i(S^{15}) \oplus \pi_{i - 1}(S^7).$$

Spectral sequence
Spectral sequences are important tools in algebraic topology for computing (co-)homology groups.

The Leray-Serre spectral sequence connects the (co-)homology of the total space and the fiber with the (co-)homology of the base space of a fibration. For a fibration $$p \colon E \to B$$ with fiber $$F,$$ where the base space is a path connected CW-complex, and an additive homology theory $$G_*$$ there exists a spectral sequence:


 * $$H_k (B; G_q(F)) \cong E^2_{k, q} \implies G_{k + q}(E).$$

Fibrations do not yield long exact sequences in homology, as they do in homotopy. But under certain conditions, fibrations provide exact sequences in homology. For a fibration $$p \colon E \to B$$ with fiber $$F,$$ where base space and fiber are path connected, the fundamental group $$\pi_1(B)$$ acts trivially on $$H_*(F)$$ and in addition the conditions $$H_p(B) = 0$$ for $$0<p<m$$ and $$H_q(F) = 0$$ for $$0<q<n$$ hold, an exact sequence exists (also known under the name Serre exact sequence):"$H_{m+n-1}(F) \xrightarrow {i_*} H_{m+n-1}(E) \xrightarrow {f_*} H_{m+n-1} (B) \xrightarrow \tau H_{m+n-2} (F) \xrightarrow {i^*} \cdots \xrightarrow {f_*} H_1 (B) \to 0.$"This sequence can be used, for example, to prove Hurewicz's theorem or to compute the homology of loopspaces of the form $$\Omega S^n:$$ $$H_k (\Omega S^n) = \begin{cases} \Z & \exist q \in \Z \colon k = q (n-1)\\ 0 & \text{otherwise} \end{cases}.$$ For the special case of a fibration $$p \colon E \to S^n$$ where the base space is a $$n$$-sphere with fiber $$F,$$ there exist exact sequences (also called Wang sequences) for homology and cohomology: $$\cdots \to H_q(F) \xrightarrow{i_*} H_q(E) \to H_{q-n}(F) \to H_{q-1}(F) \to \cdots$$

$$\cdots \to H^q(E) \xrightarrow{i^*} H^q(F) \to H^{q-n+1}(F) \to H^{q+1}(E) \to \cdots$$

Orientability
For a fibration $$p \colon E \to B$$ with fiber $$F$$ and a fixed commutative ring $$R$$ with a unit, there exists a contravariant functor from the fundamental groupoid of $$B$$ to the category of graded $$R$$-modules, which assigns to $$b \in B$$ the module $$H_*(F_b, R)$$ and to the path class $$[\omega]$$ the homomorphism $$h[\omega]_* \colon H_*(F_{\omega (0)}, R) \to H_*(F_{\omega (1)}, R),$$ where $$h[\omega]$$ is a homotopy class in $$[F_{\omega(0)}, F_{\omega (1)}].$$

A fibration is called orientable over $$R$$ if for any closed path $$\omega$$ in $$B$$ the following holds: $$h[\omega]_* = 1.$$

Euler characteristic
For an orientable fibration $$p \colon E \to B$$ over the field $$\mathbb{K}$$ with fiber $$F$$ and path connected base space, the Euler characteristic of the total space is given by:"$\chi(E) = \chi(B)\chi(F).$"Here the Euler characteristics of the base space and the fiber are defined over the field $$\mathbb{K}$$.