Fibration Explained

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 \colon E \to B satisfies the homotopy lifting property for a space X if:h \colon X x [ 0 , 1 ] \to B and

• for every mapping (also called lift)

\tilde h _{0} \colon X \to E lifting h| _{X} = h _{0} (i.e. h _{0} = p \circ \tilde h _{0} )

there exists a (not necessarily unique) homotopy

\tilde h \colon X x [ 0 , 1 ] \to E lifting h (i.e. h = p \circ \tilde h ) with \tilde h _{0} = \tilde h| _{X} .

The following commutative diagram shows the situation:

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.

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.

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.

Examples

• The projection onto the first factor

p \colon B x F \to B is a fibration. That is, trivial bundles are fibrations.p \colon E \to B is a fibration. Specifically, for every homotopy h \colon X x [ 0 , 1 ] \to B and every lift \tilde h _{0} \colon X \to E there exists a uniquely defined lift \tilde h \colon X x [ 0 , 1 ] \to E with p \circ \tilde h = h. p \colon 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 ^{*} \colon X ^{I ^{k}} \to X ^{\partial I ^{k}} induced by the inclusion i \colon \partial I ^{k} \to I ^{k} where k \in \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.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 \operatorname{Id} _{E _{2}} and \operatorname{Id} _{E _{1}} .

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 x 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 \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 x B ^{I} | \gamma ( 0 ) = f(a) \} carries the subspace topology of A x 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.

Properties

• The fibers

p ^{-1} (b) over b \in B are homotopy equivalent for each path component of B.

• For a homotopy

f \colon [ 0 , 1 ] x 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 \colon E \to B is fiber homotopy equivalent to the product fibration B x F \to B.

• 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.

• 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.

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 along p . This procedure can now be applied again to the fibration i and so on. This leads to a long sequence:

… \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) where \gamma is a path from p(e _{0)} = b _{0} to b _{0} , i.e. the loop space \Omega B . The inclusion \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:

… \Omega ^{2B} \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.

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.

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:

… → \pi _{n(F,x} _{0)} → \pi _{n(E,} x _{0)} → \pi _{n(B,} b _{0)} → \pi _{n} (F, x _{0)} → … → \pi _{0(F,} x _{0)} → \pi _{0(E,} x _{0).}

The homomorphisms \pi _{n(F,} x _{0)} → \pi _{n(E,} x _{0)} and \pi _{n(E,} x _{0)} → \pi _{n(B,} b _{0)} are the induced homomorphisms of the inclusion i \colon F \hookrightarrow E and the projection p \colon E → 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} → S ^{1,} S ^{1} \hookrightarrow S ^{3} → S ^{2,} S ^{3} \hookrightarrow S ^{7} → S ^{4,} S ^{7} \hookrightarrow S ^{15} → S ^{8.}

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

… → \pi ^{1 ,x} _{0 )} → \pi ^{3 ,} _{n(S} x _{0)} → \pi ^{2 ,} _{n(S} b _{0)} → \pi _{n} (S ^{1,} x _{0)} → … → \pi ^{1 ,} _{1 (S} x _{0)} → \pi ^{3 ,} _{1 (S} x _{0)} → \pi ^{2 ,} _{1 (S} b _{0).}

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

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

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

\pi ^{2 )} _{i(S} \cong \pi ^{3 )} _{i(S} ⊕ \pi _{i} (S ^{1).}

The homotopy groups \pi _{i} (S ^{1)} are trivial for i \geq 3 , so there exist isomorphisms between \pi ^{2 )} _{i(S} and \pi ^{3 )} _{i(S} 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 ^{4 )} _{i(S} \cong \pi ^{7 )} _{i(S} ⊕ \pi _{i} (S ^{3)} and\pi ^{8 )} _{i(S} \cong \pi ^{15} _{i(S} ) ⊕ \pi _{i} (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 \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 ^{2} _{k, q} \implies G _{k} (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 \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 <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) \xrightarrow {i _{*}} H _{m+n-1} (E) \xrightarrow {f _{*}} H _{m+n-1} (B) \xrightarrow \tau H _{m+n-2} (F) \xrightarrow {i ^{*}} … \xrightarrow {f _{*}} 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\Omega S ^{n:}

H _{k} (\Omega S ^{n)} = \begin{cases} \Z & \exist q \in \Z \colon k = q (n -1 )\ 0 & otherwise \end{cases}.

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:

… \to H _{q(F)} \xrightarrow{i _{*}} H _{q(E)} \to H _{q-n} (F) \to H _{q-1} (F) \to … … \to H ^{q(E)} \xrightarrow{i ^{*}} H ^{q(F)} \to H ^{q-n+1} (F) \to H ^{q+1} (E) \to …

Orientability

For a fibration

p \colon 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] _{*} \colon H _{*(F} _{\omega} , R) \to H _{*(F} _{\omega} , R), where h[\omega] is a homotopy class in [F _{\omega(0)} , F _{\omega} ].

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 \colon E \to B over the field 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 K .

See also

• Approximate fibration