Fibration

Examples

• The projection onto the first factor ${\displaystyle p:B\times F\to B}$ is a fibration. That is, trivial bundles are fibrations.
• Every covering ${\displaystyle p:E\to B}$ is a fibration. Specifically, for every homotopy ${\displaystyle h:X\times [0,1]\to B}$ and every lift ${\displaystyle {\tilde{h}}_0:X\to E}$ there exists a uniquely defined lift ${\displaystyle \tilde{h}:X\times [0,1]\to E}$ with ${\displaystyle p\circ \tilde{h}=h.}$
• Every fiber bundle ${\displaystyle p:E\to B}$ 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 ${\displaystyle i^{\ast }:X^{I^k}\to X^{\mathrm{\partial }{I}^k}}$ induced by the inclusion ${\displaystyle i:\mathrm{\partial }{I}^k\to I^k}$ where ${\displaystyle k\in \mathbb{N},}$ ${\displaystyle X}$ a topological space and ${\displaystyle X^A=\{f:A\to X\}}$ is the space of all continuous mappings with the compact-open topology.
• 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 ${\displaystyle f:E_1\to E_2}$ between total spaces of two fibrations ${\displaystyle p_1:E_1\to B}$ and ${\displaystyle p_2:E_2\to B}$ 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 ${\displaystyle g:E_2\to E_1}$ exists, such that the mappings ${\displaystyle f\circ g}$ and ${\displaystyle g\circ f}$ are homotopic, by fibration homomorphisms, to the identities ${\displaystyle {\mathrm{Id}}_{E_2}}$ and ${\displaystyle {\mathrm{Id}}_{E_1}.}$

Pullback fibration

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

The fibration ${\displaystyle p_f}$ is called the pullback fibration or induced fibration.

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 ${\displaystyle f:A\to B}$ between topological spaces consists of pairs ${\displaystyle (a,\gamma )}$ with ${\displaystyle a\in A}$ and paths ${\displaystyle \gamma :I\to B}$ 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 ${\displaystyle p:E_f\to B}$ 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 ${\displaystyle \gamma :[0,1]\to B,}$ 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 ${\displaystyle i:b_0\to B}$, 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 ${\displaystyle p:PB\to B}$ maps each path to its endpoint, hence the fiber ${\displaystyle p^{-1}(b_0)}$ consists of all closed paths. The fiber is denoted by ${\displaystyle \mathrm{\Omega }B}$ and is called loop space.

Properties

• The fibers ${\displaystyle p^{-1}(b)}$ over ${\displaystyle b\in B}$ are homotopy equivalent for each path component of ${\displaystyle B.}$
• For a homotopy ${\displaystyle f:[0,1]\times A\to B}$ the pullback fibrations ${\displaystyle f_0^{\ast }(E)\to A}$ and ${\displaystyle f_1^{\ast }(E)\to A}$ are fiber homotopy equivalent.
• If the base space ${\displaystyle B}$ is contractible, then the fibration ${\displaystyle p:E\to B}$ is fiber homotopy equivalent to the product fibration ${\displaystyle B\times F\to B.}$
• The pathspace fibration of a fibration ${\displaystyle p:E\to B}$ is very similar to itself. More precisely, the inclusion ${\displaystyle E\hookrightarrow E_p}$ is a fiber homotopy equivalence.
• For a fibration ${\displaystyle p:E\to B}$ with fiber ${\displaystyle F}$ and contractible total space, there is a weak homotopy equivalence ${\displaystyle F\to \mathrm{\Omega }B.}$

Puppe sequence

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

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

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 ${\displaystyle \mathrm{\Omega }B}$. The inclusion ${\displaystyle \mathrm{\Omega }B\hookrightarrow F_i}$ of the fiber of ${\displaystyle i}$ into the homotopy fiber of ${\displaystyle i}$ is again a homotopy equivalence and iteration yields the sequence:

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

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 ${\displaystyle p:E\to B}$ 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.

Long exact sequence of homotopy groups

For a Serre fibration ${\displaystyle p:E\to B}$ 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:

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

The homomorphisms ${\displaystyle {\pi }_n(F,x_0)\to {\pi }_n(E,x_0)}$ and ${\displaystyle {\pi }_n(E,x_0)\to {\pi }_n(B,b_0)}$ are the induced homomorphisms of the inclusion ${\displaystyle i:F\hookrightarrow E}$ and the projection ${\displaystyle p:E\to B.}$

Hopf fibration

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

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

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

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

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

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

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

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

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

This short exact sequence splits because of the suspension homomorphism ${\displaystyle \varphi :{\pi }_{i-1}(S^1)\to {\pi }_i(S^2)}$ 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 ${\displaystyle i\ge 3,}$ so there exist isomorphisms between ${\displaystyle {\pi }_i(S^2)}$ and ${\displaystyle {\pi }_i(S^3)}$ for ${\displaystyle i\ge 3.}$

Analogously, 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 ${\displaystyle {\pi }_i(S^8)\cong {\pi }_i(S^{15})\oplus {\pi }_{i-1}(S^7).}$

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

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

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 ${\displaystyle p:E\to B}$ with fiber ${\displaystyle F,}$ where base space and fiber are path connected, the fundamental group ${\displaystyle {\pi }_1(B)}$ acts trivially on ${\displaystyle H_{\ast }(F)}$ 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):

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

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

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

For the special case of a fibration ${\displaystyle p:E\to S^n}$ 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:

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

Orientability

For a fibration ${\displaystyle p:E\to B}$ 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 ${\displaystyle H_{\ast }(F_b,R)}$ and to the path class ${\displaystyle [\omega ]}$ the homomorphism ${\displaystyle h[\omega {]}_{\ast }:H_{\ast }(F_{\omega (0)},R)\to H_{\ast }(F_{\omega (1)},R),}$ 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: ${\displaystyle h[\omega {]}_{\ast }=1.}$

Euler characteristic

For an orientable fibration ${\displaystyle p:E\to B}$ 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 ${\displaystyle \mathbb{K}}$.