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.

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: ^[1]: 66

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 ${\displaystyle F_{b}=p^{-1}(b)\subseteq E.}$^[1]: 66

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.^{[2]: 375-376}

Every Hurewicz fibration is a Serre fibration.

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.^{[3]: 241-242}

• 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 ${\displaystyle p\colon E\to B}$ is a fibration. 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 ${\displaystyle p\circ {\tilde {h}}=h.}$^{[4]: 159  [5]: 50}
• Every fiber bundle ${\displaystyle p\colon E\to B}$ satisfies the homotopy lifting property for every CW-complex.^[2]: 379
• A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces.^[2]: 379
• An example of 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.^[4]: 198
• The Hopf fibration ${\displaystyle S^{1}\to S^{3}\to S^{2}}$ is a non-trivial fiber bundle and, specifically, a Serre fibration.

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 ${\displaystyle \operatorname {Id} _{E_{2}}}$ and ${\displaystyle \operatorname {Id} _{E_{1}}.}$ ^{[2]: 405-406}

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.^{[2]: 405-406}

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.^{[2]: 407-408}

• The fibers ${\displaystyle p^{-1}(b)}$ over ${\displaystyle b\in B}$ are homotopy equivalent for each path component of ${\displaystyle B.}$^[2]: 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.^[2]: 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 ${\displaystyle B\times F\to B.}$^[2]: 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.^[2]: 408
• For a fibration ${\displaystyle p\colon E\to B}$ with fiber ${\displaystyle F}$ and contractible total space, there is a weak homotopy equivalence ${\displaystyle F\to \Omega B.}$^[2]: 408

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 ${\displaystyle PB\to B}$ along ${\displaystyle p}$. 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 ${\displaystyle (e_{0},\gamma )}$ where ${\displaystyle \gamma }$ is a path from ${\displaystyle p(e_{0})=b_{0}}$ to ${\displaystyle b_{0}}$, i.e. the loop space ${\displaystyle \Omega B}$. The inclusion ${\displaystyle \Omega B\hookrightarrow F_{i}}$ of the fiber of ${\displaystyle i}$ into the homotopy fiber of ${\displaystyle i}$ is again 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.^{[2]: 407-409}

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.^[2]: 412

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 ${\displaystyle p\colon E\rightarrow B.}$^[2]: 376

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:^[6]: 111

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

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:^[7]: 242

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

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 ${\displaystyle p\colon E\to B}$ 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 ${\displaystyle 0<p<m}$ and ${\displaystyle H_{q}(F)=0}$ for ${\displaystyle 0<q<n}$ hold, an exact sequence exists (also known under the name Serre exact sequence):

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

This sequence can be used, for example, to prove Hurewicz's theorem or to compute the homology of loopspaces of the form ${\displaystyle \Omega S^{n}:}$ ^[8]: 162

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

For the special case of a fibration ${\displaystyle p\colon E\to S^{n}}$ 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:^[1]: 456

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

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

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 ${\displaystyle \mathbb {K} }$.^[1]: 481

• Approximate fibration