# 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.

## Formal definitions
### 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:$^{:66}$

### 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.$$^{:66}$

### 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.$^{:375-376}$
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.$^{:241-242}$

## 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.$$^{:159}$$^{:50}$
* Every fiber bundle $p\colon E\to B$ satisfies the homotopy lifting property for every CW-complex.$^{:379}$
* A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces.$^{:379}$
* 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 $k\in \mathbb {N} ,$ $X$ a topological space and $X^{A}=\{f\colon A\to X\}$ is the space of all continuous mappings with the compact-open topology.$^{:198}$
* The Hopf fibration $S^{1}\to S^{3}\to S^{2}$ is a non-trivial fiber bundle and, specifically, a Serre fibration.

## Basic concepts
### 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}}.$$^{:405-406}$

### 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.$^{:405-406}$

### 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.$^{:407-408}$

## Properties

* The fibers $p^{-1}(b)$ over $b\in B$ are homotopy equivalent for each path component of $B.$$^{:405}$
* For a homotopy $f\colon [0,1]\times A\to B$ the pullback fibrations $f_{0}^{*}(E)\to A$ and $f_{1}^{*}(E)\to A$ are fiber homotopy equivalent.$^{:406}$
* If the base space $B$ is contractible, then the fibration $p\colon E\to B$ is fiber homotopy equivalent to the product fibration $B\times F\to B.$$^{:406}$
* 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.$^{:408}$
* 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.$$^{:408}$

## 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 $PB\to B$ along $p$. 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 $(e_{0},\gamma )$ where $\gamma$ is a path from $p(e_{0})=b_{0}$ to $b_{0}$, i.e. the loop space $\Omega B$. The inclusion $\Omega B\hookrightarrow F_{i}$ of the fiber of $i$ into the homotopy fiber of $i$ is again a homotopy equivalence and iteration yields the sequence:
   $\cdots \Omega ^{2}B\to \Omega F\to \Omega E\to \Omega B\to F\to E\to B.$
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.$^{:407-409}$

## 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.$^{:412}$

## 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.$$^{:376}$

### 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:$^{:111}$
   $\pi _{i}(S^{4})\cong \pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3})$ 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:$^{:242}$
    $H_{k}(B;G_{q}(F))\cong E_{k,q}^{2}\implies G_{k+q}(E).$
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.$$^{:250}$
This sequence can be used, for example, to prove Hurewicz's theorem or to compute the homology of loopspaces of the form $\Omega S^{n}:$$^{:162}$
   $H_{k}(\Omega S^{n})={\begin{cases}\mathbb {Z} &\exists q\in \mathbb {Z} \colon k=q(n-1)\\0&{\text{otherwise}}\end{cases}}.$
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:$^{:456}$
   $\cdots \to H_{q}(F)\xrightarrow {i_{*}} H_{q}(E)\to H_{q-n}(F)\to H_{q-1}(F)\to \cdots$ $\cdots \to H^{q}(E)\xrightarrow {i^{*}} H^{q}(F)\to H^{q-n+1}(F)\to H^{q+1}(E)\to \cdots$

## 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.$$^{:476}$

## 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 $\mathbb {K}$.$^{:481}$