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

• for every homotopy $h:X\times [0,1]\to B$ and
• for every mapping (also called lift) ${\tilde{h}}_0:X\to E$ lifting $h{|}_{X\times 0}=h_0$ (i.e. $h_0=p\circ {\tilde{h}}_0$)

there exists a (not necessarily unique) homotopy $\tilde{h}:X\times [0,1]\to E$ 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 $p:E\to B$ 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 $p:E\to B$ satisfying the homotopy lifting property for all CW-complexes.

Every Hurewicz fibration is a Serre fibration.

Quasifibration

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

Every Serre fibration is a quasifibration.

Examples

• The projection onto the first factor $p:B\times F\to B$ is a fibration. That is, trivial bundles are fibrations.
• Every covering $p:E\to B$ is a fibration. Specifically, for every homotopy $h:X\times [0,1]\to B$ and every lift ${\tilde{h}}_0:X\to E$ there exists a uniquely defined lift $\tilde{h}:X\times [0,1]\to E$ with $p\circ \tilde{h}=h.$
• Every fiber bundle $p:E\to B$ 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 $i^{*}:X^{I^k}\to X^{\partial I^k}$ induced by the inclusion $i:\partial I^k\to I^k$ where $k\in \mathbb{\mathbb{N}},$ $X$ a topological space and $X^A=\{f:A\to X\}$ 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.

Basic concepts

Fiber homotopy equivalence

A mapping $f:E_1\to E_2$ between total spaces of two fibrations $p_1:E_1\to B$ and $p_2:E_2\to B$ 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 $g:E_2\to E_1$ exists, such that the mappings $f\circ g$ and $g\circ f$ are homotopic, by fibration homomorphisms, to the identities ${\mathrm{Id}}_{E_2}$ and ${\mathrm{Id}}_{E_1}.$

Pullback fibration

Given a fibration $p:E\to B$ and a mapping $f:A\to B$, the mapping $p_f:f^{*}(E)\to A$ 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 $f:A\to B$ between topological spaces consists of pairs $(a,\gamma )$ with $a\in A$ and paths $\gamma :I\to B$ 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 $p:E_f\to B$ 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 $\gamma :[0,1]\to B,$ where $\gamma (0)=f(a)$ and $\gamma (1)=b_0\in B$ holds.

For the special case of the inclusion of the base point $i:b_0\to B$, 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 $p:PB\to B$ maps each path to its endpoint, hence the fiber $p^{-1}(b_0)$ consists of all closed paths. The fiber is denoted by $\mathrm{\Omega }B$ 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 $f:[0,1]\times A\to B$ 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 $p:E\to B$ is fiber homotopy equivalent to the product fibration $B\times F\to B.$
• The pathspace fibration of a fibration $p:E\to B$ is very similar to itself. More precisely, the inclusion $E\hookrightarrow E_p$ is a fiber homotopy equivalence.
• For a fibration $p:E\to B$ with fiber $F$ and contractible total space, there is a weak homotopy equivalence $F\to \mathrm{\Omega }B.$

Puppe sequence

For a fibration $p:E\to B$ 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 $i:F_p\to E$ with $i(e,\gamma )=e$, where $e\in E$ and $\gamma :I\to B$ 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:

$\cdots \to F_j\to F_i\stackrel{j}{\to }{F}_p\stackrel{i}{\to }E\stackrel{p}{\to }B.$

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

$\mathrm{\Omega }B$

. The inclusion

$\mathrm{\Omega }B\hookrightarrow F_i$

of the fiber of

$i$

into the homotopy fiber of

$i$

is again a homotopy equivalence and iteration yields the sequence:

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

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 $p:E\to B$ 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

$p:E\to B$

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:

$\cdots \to {\pi }_n(F,x_0)\to {\pi }_n(E,x_0)\to {\pi }_n(B,b_0)\to {\pi }_{n-1}(F,x_0)\to$ $\cdots \to {\pi }_0(F,x_0)\to {\pi }_0(E,x_0).$

The homomorphisms

${\pi }_n(F,x_0)\to {\pi }_n(E,x_0)$

and

${\pi }_n(E,x_0)\to {\pi }_n(B,b_0)$

are the induced homomorphisms of the inclusion

$i:F\hookrightarrow E$

and the projection

$p:E\to B.$

Hopf fibration

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

$S^0\hookrightarrow S^1\to S^1,$

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

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

$S^7\hookrightarrow S^{15}\to S^8.$

The long exact sequence of homotopy groups of the hopf fibration

$S^1\hookrightarrow S^3\to S^2$

yields:

$\cdots \to {\pi }_n(S^1,x_0)\to {\pi }_n(S^3,x_0)\to {\pi }_n(S^2,b_0)\to {\pi }_{n-1}(S^1,x_0)\to$ $\cdots \to {\pi }_1(S^1,x_0)\to {\pi }_1(S^3,x_0)\to {\pi }_1(S^2,b_0).$

This sequence splits into short exact sequences, as the fiber

$S^1$

in

$S^3$

is contractible to a point:

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

This short exact sequence splits because of the suspension homomorphism

$\varphi :{\pi }_{i-1}(S^1)\to {\pi }_i(S^2)$

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

$i\ge 3,$

so there exist isomorphisms between

${\pi }_i(S^2)$

and

${\pi }_i(S^3)$

for

$i\ge 3.$

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

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

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

$p:E\to B$

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):

$H_{m+n-1}(F)\stackrel{i_{*}}{\to }{H}_{m+n-1}(E)\stackrel{f_{*}}{\to }{H}_{m+n-1}(B)\stackrel{\tau }{\to }{H}_{m+n-2}(F)\stackrel{i^{*}}{\to }\cdots \stackrel{f_{*}}{\to }{H}_1(B)\to 0.$

This sequence can be used, for example, to prove Hurewicz's theorem or to compute the homology of loopspaces of the form

$\mathrm{\Omega }{S}^n:$

$H_k(\mathrm{\Omega }{S}^n)=\{\begin{array}{ll}\mathbb{\mathbb{Z}}& \mathrm{\exists }q\in \mathbb{\mathbb{Z}}:k=q(n-1)\\ 0& \text{otherwise}\end{array}.$

For the special case of a fibration

$p:E\to S^n$

where the base space is a

$n$

-sphere with fiber

$F,$

there exist exact sequences (also called Wang sequences) for homology and cohomology:

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

Orientability

For a fibration $p:E\to B$ 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 $h[\omega {]}_{*}:H_{*}(F_{\omega (0)},R)\to H_{*}(F_{\omega (1)},R),$ 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

$p:E\to B$

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

$\mathbb{𝕂}$

.

See also

• Approximate fibration