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: ^: 66

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 the base space and the space $E$ is called the total space. The fiber over $b \in B$ is the subspace $F_{b} = p^{- 1} (b) \subseteq E .$ ^: 66

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.^: 375-376

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 $i \geq 0$ holds that the induced mapping $p_{*} : π_{i} (E, p^{- 1} (b), e) \to π_{i} (B, b)$ is an isomorphism.

Every Serre fibration is a quasifibration.^: 241-242

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 $p \circ \tilde{h} = h .$ ^: 159 ^: 50
Every fiber bundle $p : E \to B$ satisfies the homotopy lifting property for every CW-complex.^: 379
A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces.^: 379
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 $k \in ℕ,$ $X$ a topological space and $X^{A} = {f : A \to X}$ is the space of all continuous mappings with the compact-open topology.^: 198
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 ${Id}_{E_{2}}$ and ${Id}_{E_{1}} .$ ^: 405-406

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.^: 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 $E_{f}$ of the pathspace fibration for a continuous mapping $f : A \to B$ between topological spaces consists of pairs $(a, γ)$ with $a \in A$ and paths $γ : I \to B$ with starting point $γ (0) = f (a),$ where $I = [0, 1]$ is the unit interval. The space $E_{f} = {(a, γ) \in A \times B^{I} | γ (0) = f (a)}$ 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 $p (a, γ) = γ (1) .$ The fiber $F_{f}$ is also called the homotopy fiber of $f$ and consists of the pairs $(a, γ)$ with $a \in A$ and paths $γ : [0, 1] \to B,$ where $γ (0) = f (a)$ and $γ (1) = b_{0} \in B$ 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 $P B$ and is called path space. The pathspace fibration $p : P B \to B$ maps each path to its endpoint, hence the fiber $p^{- 1} (b_{0})$ consists of all closed paths. The fiber is denoted by $Ω B$ and is called loop space.^: 407-408

Properties

The fibers $p^{- 1} (b)$ over $b \in B$ are homotopy equivalent for each path component of $B .$ ^: 405
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.^: 406
If the base space $B$ is contractible, then the fibration $p : E \to B$ is fiber homotopy equivalent to the product fibration $B \times F \to B .$ ^: 406
The pathspace fibration of a fibration $p : E \to B$ is very similar to itself. More precisely, the inclusion $E ↪ E_{p}$ is a fiber homotopy equivalence.^: 408
For a fibration $p : E \to B$ with fiber $F$ and contractible total space, there is a weak homotopy equivalence $F \to Ω B .$ ^: 408

Puppe sequence

For a fibration $p : E \to B$ with fiber $F$ and base point $b_{0} \in B$ the inclusion $F ↪ F_{p}$ of the fiber into the homotopy fiber is a homotopy equivalence. The mapping $i : F_{p} \to E$ with $i (e, γ) = e$ , where $e \in E$ and $γ : I \to B$ 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 $P B \to B$ along $p$ . This procedure can now be applied again to the fibration $i$ and so on. This leads to a long sequence:

$\dots \to F_{j} \to F_{i} \overset{j}{\to} F_{p} \overset{i}{\to} E \overset{p}{\to} B .$

The fiber of $i$ over a point $e_{0} \in p^{- 1} (b_{0})$ consists of the pairs $(e_{0}, γ)$ where $γ$ is a path from $p (e_{0}) = b_{0}$ to $b_{0}$ , i.e. the loop space $Ω B$ . The inclusion $Ω B ↪ F_{i}$ of the fiber of $i$ into the homotopy fiber of $i$ is again a homotopy equivalence and iteration yields the sequence:

$\dots Ω^{2} B \to Ω F \to Ω E \to Ω B \to F \to E \to B .$

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.^: 407-409

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.^: 412

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:

$\dots \to π_{n} (F, x_{0}) \to π_{n} (E, x_{0}) \to π_{n} (B, b_{0}) \to π_{n - 1} (F, x_{0}) \to$ $\dots \to π_{0} (F, x_{0}) \to π_{0} (E, x_{0}) .$

The homomorphisms $π_{n} (F, x_{0}) \to π_{n} (E, x_{0})$ and $π_{n} (E, x_{0}) \to π_{n} (B, b_{0})$ are the induced homomorphisms of the inclusion $i : F ↪ E$ and the projection $p : E \to B .$ ^: 376

Hopf fibration

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

$S^{0} ↪ S^{1} \to S^{1},$

$S^{1} ↪ S^{3} \to S^{2},$

$S^{3} ↪ S^{7} \to S^{4},$

$S^{7} ↪ S^{15} \to S^{8} .$

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

$\dots \to π_{n} (S^{1}, x_{0}) \to π_{n} (S^{3}, x_{0}) \to π_{n} (S^{2}, b_{0}) \to π_{n - 1} (S^{1}, x_{0}) \to$ $\dots \to π_{1} (S^{1}, x_{0}) \to π_{1} (S^{3}, x_{0}) \to π_{1} (S^{2}, b_{0}) .$

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

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

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

$π_{i} (S^{2}) ≅ π_{i} (S^{3}) \oplus π_{i - 1} (S^{1}) .$

The homotopy groups $π_{i - 1} (S^{1})$ are trivial for $i \geq 3,$ so there exist isomorphisms between $π_{i} (S^{2})$ and $π_{i} (S^{3})$ for $i \geq 3.$

Analogously, 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:^: 111

$π_{i} (S^{4}) ≅ π_{i} (S^{7}) \oplus π_{i - 1} (S^{3})$ and $π_{i} (S^{8}) ≅ π_{i} (S^{15}) \oplus π_{i - 1} (S^{7}) .$

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:^: 242

H_{k} (B; G_{q} (F)) ≅ 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 $π_{1} (B)$ 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):

$H_{m + n - 1} (F) \overset{i_{*}}{\to} H_{m + n - 1} (E) \overset{f_{*}}{\to} H_{m + n - 1} (B) \overset{τ}{\to} H_{m + n - 2} (F) \overset{i^{*}}{\to} \dots \overset{f_{*}}{\to} H_{1} (B) \to 0.$ ^: 250

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

$H_{k} (Ω S^{n}) = {\begin{matrix} ℤ & \exists q \in ℤ : k = q (n - 1) \\ 0 & otherwise \end{matrix} .$

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:^: 456

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

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 $[ω]$ the homomorphism $h [ω]_{*} : H_{*} (F_{ω (0)}, R) \to H_{*} (F_{ω (1)}, R),$ where $h [ω]$ is a homotopy class in $[F_{ω (0)}, F_{ω (1)}] .$

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

Euler characteristic

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

$χ (E) = χ (B) χ (F) .$

Here the Euler characteristics of the base space and the fiber are defined over the field $𝕂$ .^: 481

References

1 2 3 4 5 Spanier, Edwin H. (1966). Algebraic Topology. McGraw-Hill Book Company. ISBN 978-0-387-90646-1.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 Hatcher, Allen (2001). Algebraic Topology. NY: Cambridge University Press. ISBN 0-521-79160-X.
↑ Dold, Albrecht; Thom, René (1958). "Quasifaserungen und Unendliche Symmetrische Produkte". Annals of Mathematics. 67 (2): 239–281. doi:10.2307/1970005. JSTOR 1970005.
1 2 Laures, Gerd; Szymik, Markus (2014). Grundkurs Topologie (in German) (2nd ed.). Springer Spektrum. doi:10.1007/978-3-662-45953-9. ISBN 978-3-662-45952-2.
↑ May, J.P. (1999). A Concise Course in Algebraic Topology (PDF). University of Chicago Press. ISBN 0-226-51182-0. OCLC 41266205.
↑ Steenrod, Norman (1951). The Topology of Fibre Bundles. Princeton University Press. ISBN 0-691-08055-0. {{cite book}}: ISBN / Date incompatibility (help)
1 2 Davis, James F.; Kirk, Paul (1991). Lecture Notes in Algebraic Topology (PDF). Department of Mathematics, Indiana University.
↑ Cohen, Ralph L. (1998). The Topology of Fiber Bundles Lecture Notes (PDF). Stanford University.

[Spanier1966-1] 1 2 3 4 5 Spanier, Edwin H. (1966). Algebraic Topology. McGraw-Hill Book Company. ISBN 978-0-387-90646-1.

[Hatcher2001-2] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Hatcher, Allen (2001). Algebraic Topology. NY: Cambridge University Press. ISBN 0-521-79160-X.

[Dold1958-3] Dold, Albrecht; Thom, René (1958). "Quasifaserungen und Unendliche Symmetrische Produkte". Annals of Mathematics. 67 (2): 239–281. doi:10.2307/1970005. JSTOR 1970005.

[Laures2014-4] 1 2 Laures, Gerd; Szymik, Markus (2014). Grundkurs Topologie (in German) (2nd ed.). Springer Spektrum. doi:10.1007/978-3-662-45953-9. ISBN 978-3-662-45952-2.

[May1999-5] May, J.P. (1999). A Concise Course in Algebraic Topology (PDF). University of Chicago Press. ISBN 0-226-51182-0. OCLC 41266205.

[Steenrod1951-6] Steenrod, Norman (1951). The Topology of Fibre Bundles. Princeton University Press. ISBN 0-691-08055-0. {{cite book}}: ISBN / Date incompatibility (help)

[Davis1991-7] 1 2 Davis, James F.; Kirk, Paul (1991). Lecture Notes in Algebraic Topology (PDF). Department of Mathematics, Indiana University.

[Cohen1998-8] Cohen, Ralph L. (1998). The Topology of Fiber Bundles Lecture Notes (PDF). Stanford University.

Fibration

Formal definitions

Homotopy lifting property

Fibration

Serre fibration

Quasifibration

Examples

Basic concepts

Fiber homotopy equivalence

Pullback fibration

Pathspace fibration

Properties

Puppe sequence

Principal fibration

Long exact sequence of homotopy groups

Hopf fibration

Spectral sequence

Orientability

Euler characteristic

See also

References