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Fibration

## 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 ${\displaystyle p\colon E\to B}$  satisfies the homotopy lifting property for a space ${\displaystyle X}$  if:

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

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

The following commutative diagram shows the situation:${\displaystyle ^{[4]p.66}}$

### Fibration

A fibration (also called Hurewicz fibration) is a mapping ${\displaystyle p\colon E\to B}$  satisfying the homotopy lifting property for all spaces ${\displaystyle X.}$  The space ${\displaystyle B}$  is called base space and the space ${\displaystyle E}$  is called total space. The fiber over ${\displaystyle b\in B}$  is the subspace ${\displaystyle F_{b}=p^{-1}(b)\subseteq E.}$ ${\displaystyle ^{[4]p.66}}$

### Serre fibration

A Serre fibration (also called weak fibration) is a mapping ${\displaystyle p\colon E\to B}$  satisfying the homotopy lifting property for all CW-complexes.${\displaystyle ^{[1]p.375-376}}$

Every Hurewicz fibration is a Serre fibration.

### Quasifibration

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

Every Serre fibration is a quasifibration.${\displaystyle ^{[5]p.241-242}}$

## Examples

• The projection onto the first factor ${\displaystyle p\colon B\times F\to B}$  is a fibration. That is, trivial bundles are fibrations.
• Every covering ${\displaystyle p\colon E\to B}$  satisfies the homotopy lifting property for all spaces. Specifically, for every homotopy ${\displaystyle h\colon X\times [0,1]\to B}$  and every lift ${\displaystyle {\tilde {h}}_{0}\colon X\to E}$  there exists a uniquely defined lift ${\displaystyle {\tilde {h}}\colon X\times [0,1]\to E}$  with ${\displaystyle p\circ {\tilde {h}}=h.}$ ${\displaystyle ^{[2]p.159}}$ ${\displaystyle ^{[3]p.50}}$
• Every fiber bundle ${\displaystyle p\colon E\to B}$  satisfies the homotopy lifting property for every CW-complex.${\displaystyle ^{[1]p.379}}$
• A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces.${\displaystyle ^{[1]p.379}}$
• An example for a fibration, which is not a fiber bundle, is given by the mapping ${\displaystyle i^{*}\colon X^{I^{k}}\to X^{\partial I^{k}}}$  induced by the inclusion ${\displaystyle i\colon \partial I^{k}\to I^{k}}$  where ${\displaystyle k\in \mathbb {N} ,}$  ${\displaystyle X}$  a topological space and ${\displaystyle X^{A}=\{f\colon A\to X\}}$  is the space of all continuous mappings with the compact-open topology.${\displaystyle ^{[2]p.198}}$
• The Hopf fibration ${\displaystyle S^{1}\to S^{3}\to S^{2}}$  is a non trivial fiber bundle and specifically a Serre fibration.

## Basic concepts

### Fiber homotopy equivalence

A mapping ${\displaystyle f\colon E_{1}\to E_{2}}$  between total spaces of two fibrations ${\displaystyle p_{1}\colon E_{1}\to B}$  and ${\displaystyle p_{2}\colon E_{2}\to B}$  with the same base space is a fibration homomorphism if the following diagram commutes:

The mapping ${\displaystyle f}$  is a fiber homotopy equivalence if in addition a fibration homomorphism ${\displaystyle g\colon E_{2}\to E_{1}}$  exists, such that the mappings ${\displaystyle f\circ g}$  and ${\displaystyle g\circ f}$  are homotopic, by fibration homomorphisms, to the identities ${\displaystyle Id_{E_{2}}}$  and ${\displaystyle Id_{E_{1}}.}$ ${\displaystyle ^{[1]p.405-406}}$

### Pullback fibration

Given a fibration ${\displaystyle p\colon E\to B}$  and a mapping ${\displaystyle f\colon A\to B}$ , the mapping ${\displaystyle p_{f}\colon f^{*}(E)\to A}$  is a fibration, where ${\displaystyle f^{*}(E)=\{(a,e)\in A\times E|f(a)=p(e)\}}$  is the pullback and the projections of ${\displaystyle f^{*}(E)}$  onto ${\displaystyle A}$  and ${\displaystyle E}$  yield the following commutative diagram:

The fibration ${\displaystyle p_{f}}$  is called the pullback fibration or induced fibration.${\displaystyle ^{[1]p.405-406}}$

### 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 ${\displaystyle E_{f}}$  of the pathspace fibration for a continuous mapping ${\displaystyle f\colon A\to B}$  between topological spaces consists of pairs ${\displaystyle (a,\gamma )}$  with ${\displaystyle a\in A}$  and paths ${\displaystyle \gamma \colon I\to B}$  with starting point ${\displaystyle \gamma (0)=f(a),}$  where ${\displaystyle I=[0,1]}$  is the unit interval. The space ${\displaystyle E_{f}=\{(a,\gamma )\in A\times B^{I}|\gamma (0)=f(a)\}}$  carries the subspace topology of ${\displaystyle A\times B^{I},}$  where ${\displaystyle B^{I}}$  describes the space of all mappings ${\displaystyle I\to B}$  and carries the compact-open topology.

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

For the special case of the inclusion of the base point ${\displaystyle i\colon b_{0}\to B}$ , an important example of the pathspace fibration emerges. The total space ${\displaystyle E_{i}}$  consists of all paths in ${\displaystyle B}$  which starts at ${\displaystyle b_{0}.}$  This space is denoted by ${\displaystyle PB}$  and is called path space. The pathspace fibration ${\displaystyle p\colon PB\to B}$  maps each path to its endpoint, hence the fiber ${\displaystyle p^{-1}(b_{0})}$  consists of all closed paths. The fiber is denoted by ${\displaystyle \Omega B}$  and is called loop space.${\displaystyle ^{[1]p.407-408}}$

## Properties

• The fibers ${\displaystyle p^{-1}(b)}$  over ${\displaystyle b\in B}$  are homotopy equivalent for each path component of ${\displaystyle B.}$ ${\displaystyle ^{[1]p.405}}$
• For a homotopy ${\displaystyle f\colon [0,1]\times A\to B}$  the pullback fibrations ${\displaystyle f_{0}^{*}(E)\to A}$  and ${\displaystyle f_{1}^{*}(E)\to A}$  are fiber homotopy equivalent.${\displaystyle ^{[1]p.406}}$
• If the base space ${\displaystyle B}$  is contractible, then the fibration ${\displaystyle p\colon E\to B}$  is fiber homotopy equivalent to the product fibration ${\displaystyle B\times F\to B.}$ ${\displaystyle ^{[1]p.406}}$
• The pathspace fibration of a fibration ${\displaystyle p\colon E\to B}$  is very similar to itself. More precisely, the inclusion ${\displaystyle E\hookrightarrow E_{p}}$  is a fiber homotopy equivalence.${\displaystyle ^{[1]p.408}}$
• For a fibration ${\displaystyle p\colon E\to B}$  with fiber ${\displaystyle F}$  and contractible total space, there is a weak homotopy equivalence ${\displaystyle F\to \Omega B.}$ ${\displaystyle ^{[1]p.408}}$

## Puppe sequence

For a fibration ${\displaystyle p\colon E\to B}$  with fiber ${\displaystyle F}$  and base point ${\displaystyle b_{0}\in B}$  the inclusion ${\displaystyle F\hookrightarrow F_{p}}$  of the fiber into the homotopy fiber is a homotopy equivalence. The mapping ${\displaystyle i\colon F_{p}\to E}$  with ${\displaystyle i(e,\gamma )=e}$ , where ${\displaystyle e\in E}$  and ${\displaystyle \gamma \colon I\to B}$  is a path from ${\displaystyle p(e)}$  to ${\displaystyle b_{0}}$  in the base space, is a fibration. Specifically it is the pullback fibration of the pathspace fibration ${\displaystyle PB\to B}$ . This procedure can now be applied again to the fibration ${\displaystyle i}$  and so on. This leads to a long sequence:

${\displaystyle \cdots \to F_{j}\to F_{i}\xrightarrow {j} F_{p}\xrightarrow {i} E\xrightarrow {p} B.}$

The fiber of ${\displaystyle i}$  over a point ${\displaystyle e_{0}\in p^{-1}(b_{0})}$  consists of the pairs ${\displaystyle (e_{0},\gamma )}$  with closed paths ${\displaystyle \gamma }$  and starting point ${\displaystyle b_{0}}$ , i.e. the loop space ${\displaystyle \Omega B}$ . The inclusion ${\displaystyle \Omega B\to F}$  is a homotopy equivalence and iteration yields the sequence:

${\displaystyle \cdots \Omega ^{2}B\to \Omega F\to \Omega E\to \Omega B\to F\to E\to B.}$

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.${\displaystyle ^{[1]p.407-409}}$

## Principal fibration

A fibration ${\displaystyle p\colon E\to B}$  with fiber ${\displaystyle F}$  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.${\displaystyle ^{[1]p.412}}$

## Long exact sequence of homotopy groups

For a Serre fibration ${\displaystyle p\colon E\to B}$  there exists a long exact sequence of homotopy groups. For base points ${\displaystyle b_{0}\in B}$  and ${\displaystyle x_{0}\in F=p^{-1}(b_{0})}$  this is given by:

${\displaystyle \cdots \rightarrow \pi _{n}(F,x_{0})\rightarrow \pi _{n}(E,x_{0})\rightarrow \pi _{n}(B,b_{0})\rightarrow \pi _{n-1}(F,x_{0})\rightarrow }$  ${\displaystyle \cdots \rightarrow \pi _{0}(F,x_{0})\rightarrow \pi _{0}(E,x_{0}).}$

The homomorphisms ${\displaystyle \pi _{n}(F,x_{0})\rightarrow \pi _{n}(E,x_{0})}$  and ${\displaystyle \pi _{n}(E,x_{0})\rightarrow \pi _{n}(B,b_{0})}$  are the induced homomorphisms of the inclusion ${\displaystyle i\colon F\hookrightarrow E}$  and the projection ${\displaystyle p\colon E\rightarrow B.}$ ${\displaystyle ^{[1]p.376}}$

### Hopf fibration

Hopf fibrations are a family of fiber bundles whose fiber, total space and base space are spheres:

${\displaystyle S^{0}\hookrightarrow S^{1}\rightarrow S^{1},}$

${\displaystyle S^{1}\hookrightarrow S^{3}\rightarrow S^{2},}$

${\displaystyle S^{3}\hookrightarrow S^{7}\rightarrow S^{4},}$

${\displaystyle S^{7}\hookrightarrow S^{15}\rightarrow S^{8}.}$

The long exact sequence of homotopy groups of the hopf fibration ${\displaystyle S^{1}\hookrightarrow S^{3}\rightarrow S^{2}}$  yields:

${\displaystyle \cdots \rightarrow \pi _{n}(S^{1},x_{0})\rightarrow \pi _{n}(S^{3},x_{0})\rightarrow \pi _{n}(S^{2},b_{0})\rightarrow \pi _{n-1}(S^{1},x_{0})\rightarrow }$  ${\displaystyle \cdots \rightarrow \pi _{1}(S^{1},x_{0})\rightarrow \pi _{1}(S^{3},x_{0})\rightarrow \pi _{1}(S^{2},b_{0}).}$

This sequence splits into short exact sequences, as the fiber ${\displaystyle S^{1}}$  in ${\displaystyle S^{3}}$  is contractible to a point:

${\displaystyle 0\rightarrow \pi _{i}(S^{3})\rightarrow \pi _{i}(S^{2})\rightarrow \pi _{i-1}(S^{1})\rightarrow 0.}$

This short exact sequence splits because of the suspension homomorphism ${\displaystyle \phi \colon \pi _{i-1}(S^{1})\to \pi _{i}(S^{2})}$  and there are isomorphisms:

${\displaystyle \pi _{i}(S^{2})\cong \pi _{i}(S^{3})\oplus \pi _{i-1}(S^{1}).}$

The homotopy groups ${\displaystyle \pi _{i-1}(S^{1})}$  are trivial for ${\displaystyle i\geq 3,}$  so there exist isomorphisms between ${\displaystyle \pi _{i}(S^{2})}$  and ${\displaystyle \pi _{i}(S^{3})}$  for ${\displaystyle i\geq 3.}$  Analog the fibers ${\displaystyle S^{3}}$  in ${\displaystyle S^{7}}$  and ${\displaystyle S^{7}}$  in ${\displaystyle S^{15}}$  are contractible to a point. Further the short exact sequences split and there are families of isomorphisms:

${\displaystyle \pi _{i}(S^{4})\cong \pi _{i}(S^{7})\oplus \pi _{i-1}(S^{3})}$  and ${\displaystyle \pi _{i}(S^{8})\cong \pi _{i}(S^{15})\oplus \pi _{i-1}(S^{7}).}$ ${\displaystyle ^{[6]p.111}}$

## 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 ${\displaystyle p\colon E\to B}$  with fiber ${\displaystyle F,}$  where the base space is a path connected CW-complex, and an additive homology theory ${\displaystyle G_{*}}$  there exists a spectral sequence:

${\displaystyle H_{k}(B;G_{q}(F))\cong E_{k,q}^{2}\implies G_{k+q}(E).}$ ${\displaystyle ^{[7]p.242}}$

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 ${\displaystyle p\colon E\to B}$  with fiber ${\displaystyle F,}$  where base space and fiber are path connected, the fundamental group ${\displaystyle \pi _{1}(B)}$  acts trivially on ${\displaystyle H_{*}(F)}$  and in addition the conditions ${\displaystyle H_{p}(B)=0}$  for ${\displaystyle 0  and ${\displaystyle H_{q}(F)=0}$  for ${\displaystyle 0  hold, an exact sequence exists (also known under the name Serre exact sequence):

${\displaystyle H_{m+n-1}(F)\xrightarrow {i_{*}} H_{m+n-1}(E)\xrightarrow {f_{*}} H_{m+n-1}(B)\xrightarrow {\tau } H_{m+n-2}(F)\xrightarrow {i^{*}} \cdots \xrightarrow {f_{*}} H_{1}(B)\to 0.}$ ${\displaystyle ^{[7]p.250}}$

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

${\displaystyle H_{k}(\Omega S^{n})={\begin{cases}\mathbb {Z} &\exists q\in \mathbb {Z} \colon k=q(n-1)\\0&else\end{cases}}.}$ ${\displaystyle ^{[8]p.162}}$

For the special case of a fibration ${\displaystyle p\colon E\to S^{n}}$  where the base space is a ${\displaystyle n}$ -sphere with fiber ${\displaystyle F,}$  there exist exact sequences (also called Wang sequences) for homology and cohomology:

${\displaystyle \cdots \to H_{q}(F)\xrightarrow {i_{*}} H_{q}(E)\to H_{q-n}(F)\to H_{q-1}(F)\to \cdots }$  ${\displaystyle \cdots \to H^{q}(E)\xrightarrow {i^{*}} H^{q}(F)\to H^{q-n+1}(F)\to H^{q+1}(E)\to \cdots }$ ${\displaystyle ^{[4]p.456}}$

## Orientability

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

A fibration is called orientable over ${\displaystyle R}$  if for any closed path ${\displaystyle \omega }$  in ${\displaystyle B}$  the following holds: ${\displaystyle h[\omega ]_{*}=1.}$ ${\displaystyle ^{[4]p.476}}$

## Euler characteristic

For an orientable fibration ${\displaystyle p\colon E\to B}$  over the field ${\displaystyle \mathbb {K} }$  with fiber ${\displaystyle F}$  and path connected base space, the Euler characteristic of the total space is given by:

${\displaystyle \chi (E)=\chi (B)\chi (F).}$

Here the Euler characteristics of the base space and the fiber are defined over the field ${\displaystyle \mathbb {K} }$ .${\displaystyle ^{[4]p.481}}$