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## Summary

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.

## 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:$^{p.66}$

### 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 $F_{b}=p^{-1}(b)\subseteq E.$ $^{p.66}$

### 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.$^{p.375-376}$

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.$^{p.241-242}$

## 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 $p\colon E\to B$  satisfies the homotopy lifting property for all spaces. 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 $p\circ {\tilde {h}}=h.$ $^{p.159}$ $^{p.50}$
• Every fiber bundle $p\colon E\to B$  satisfies the homotopy lifting property for every CW-complex.$^{p.379}$
• A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces.$^{p.379}$
• An example for 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.$^{p.198}$
• 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 $Id_{E_{2}}$  and $Id_{E_{1}}.$ $^{p.405-406}$

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

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

## Properties

• The fibers $p^{-1}(b)$  over $b\in B$  are homotopy equivalent for each path component of $B.$ $^{p.405}$
• 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.$^{p.406}$
• If the base space $B$  is contractible, then the fibration $p\colon E\to B$  is fiber homotopy equivalent to the product fibration $B\times F\to B.$ $^{p.406}$
• 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.$^{p.408}$
• For a fibration $p\colon E\to B$  with fiber $F$  and contractible total space, there is a weak homotopy equivalence $F\to \Omega B.$ $^{p.408}$

## 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 $PB\to B$ . 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 $\Omega B\to F$  is 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.$^{p.407-409}$

## 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.$^{p.412}$

## 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 $p\colon E\rightarrow B.$ $^{p.376}$

### 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:

$\pi _{i}(S^{4})\cong \pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3})$  and $\pi _{i}(S^{8})\cong \pi _{i}(S^{15})\oplus \pi _{i-1}(S^{7}).$ $^{p.111}$

## 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:

$H_{k}(B;G_{q}(F))\cong E_{k,q}^{2}\implies G_{k+q}(E).$ $^{p.242}$

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  and $H_{q}(F)=0$  for $0  hold, an exact sequence exists (also known under the name Serre exact sequence):

$H_{m+n-1}(F)\xrightarrow {i_{*}} H_{m+n-1}(E)\xrightarrow {f_{*}} H_{m+n-1}(B)\xrightarrow {\tau } H_{m+n-2}(F)\xrightarrow {i^{*}} \cdots \xrightarrow {f_{*}} H_{1}(B)\to 0.$ $^{p.250}$

This sequence can be used, for example, to prove Hurewicz`s theorem or to compute the homology of loopspaces of the form $\Omega S^{n}:$

$H_{k}(\Omega S^{n})={\begin{cases}\mathbb {Z} &\exists q\in \mathbb {Z} \colon k=q(n-1)\\0&else\end{cases}}.$ $^{p.162}$

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:

$\cdots \to H_{q}(F)\xrightarrow {i_{*}} H_{q}(E)\to H_{q-n}(F)\to H_{q-1}(F)\to \cdots$  $\cdots \to H^{q}(E)\xrightarrow {i^{*}} H^{q}(F)\to H^{q-n+1}(F)\to H^{q+1}(E)\to \cdots$ $^{p.456}$

## Orientability

For a fibration $p\colon E\to B$  with fiber $F$  and a fixed commuative 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: $h[\omega ]_{*}=1.$ $^{p.476}$

## 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 $\mathbb {K}$ .$^{p.481}$