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 <math>p \colon E \to B</math> satisfies the homotopy lifting property for a space <math>X</math> if:

* for every homotopy <math>h \colon X \times [0, 1] \to B</math> and * for every mapping (also called lift) <math>\tilde h_0 \colon X \to E</math> lifting <math>h|_{X \times 0} = h_0</math> (i.e. <math>h_0 = p \circ \tilde h_0</math>)

there exists a (not necessarily unique) homotopy <math>\tilde h \colon X \times [0, 1] \to E</math> lifting <math>h</math> (i.e. <math>h = p \circ \tilde h</math>) with <math>\tilde h_0 = \tilde h|_{X \times 0}.</math>

The following commutative diagram shows the situation: {{r|Spanier1966|p=66}} center|frameless|220x220px

=== Fibration === A ''fibration'' (also called ''Hurewicz fibration'') is a mapping <math>p \colon E \to B</math> satisfying the homotopy lifting property for all spaces <math>X.</math> The space <math>B</math> is called the ''base space'' and the space <math>E</math> is called the ''total space''. The ''fiber over'' <math>b \in B</math> is the subspace <math>F_b = p^{-1}(b) \subseteq E.</math>{{r|Spanier1966|p=66}} === Serre fibration === A ''Serre fibration'' (also called ''weak fibration'') is a mapping <math>p \colon E \to B</math> satisfying the homotopy lifting property for all CW-complexes.{{r|Hatcher2001|p=375-376}}

Every Hurewicz fibration is a Serre fibration.

=== Quasifibration === A mapping <math>p \colon E \to B</math> is called ''quasifibration'', if for every <math>b \in B,</math> <math>e \in p^{-1}(b)</math> and <math>i \geq 0</math> holds that the induced mapping <math>p_* \colon \pi_i(E, p^{-1}(b), e) \to \pi_i(B, b)</math> is an isomorphism.

Every Serre fibration is a quasifibration.{{r|Dold1958|p=241-242}}

== Examples ==

* The projection onto the first factor <math>p \colon B \times F \to B</math> is a fibration. That is, trivial bundles are fibrations. * Every covering <math>p \colon E \to B</math> is a fibration. Specifically, for every homotopy <math>h \colon X \times [0, 1] \to B</math> and every lift <math>\tilde h_0 \colon X \to E</math> there exists a uniquely defined lift <math>\tilde h \colon X \times [0,1] \to E</math> with <math>p \circ \tilde h = h.</math>{{r|Laures2014|p=159}} {{r|May1999|p=50}} * Every fiber bundle <math>p \colon E \to B</math> satisfies the homotopy lifting property for every CW-complex.{{r|Hatcher2001|p=379}} * A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces.{{r|Hatcher2001|p=379}} * An example of a fibration which is not a fiber bundle is given by the mapping <math>i^* \colon X^{I^k} \to X^{\partial I^k}</math> induced by the inclusion <math>i \colon \partial I^k \to I^k</math> where <math>k \in \N,</math> <math>X</math> a topological space and <math>X^{A} = \{f \colon A \to X\}</math> is the space of all continuous mappings with the compact-open topology.{{r|Laures2014|p=198}} * The Hopf fibration <math>S^1 \to S^3 \to S^2</math> is a non-trivial fiber bundle and, specifically, a Serre fibration.

== Basic concepts ==

=== Fiber homotopy equivalence === A mapping <math>f \colon E_1 \to E_2</math> between total spaces of two fibrations <math>p_1 \colon E_1 \to B</math> and <math>p_2 \colon E_2 \to B</math> with the same base space is a ''fibration homomorphism'' if the following diagram commutes: center|frameless The mapping <math>f</math> is a ''fiber homotopy equivalence'' if in addition a fibration homomorphism <math>g \colon E_2 \to E_1</math> exists, such that the mappings <math>f \circ g</math> and <math>g \circ f</math> are homotopic, by fibration homomorphisms, to the identities <math>\operatorname{Id}_{E_2}</math> and <math>\operatorname{Id}_{E_1}.</math> {{r|Hatcher2001|p=405-406}}

=== Pullback fibration === Given a fibration <math>p \colon E \to B</math> and a mapping <math>f \colon A \to B</math>, the mapping <math>p_f \colon f^*(E) \to A</math> is a fibration, where <math>f^*(E) = \{(a, e) \in A \times E\ |\ f(a) = p(e)\}</math> is the pullback and the projections of <math>f^*(E)</math> onto <math>A</math> and <math>E</math> yield the following commutative diagram: center|frameless|140x140px The fibration <math>p_f</math> is called the ''pullback fibration'' or induced fibration.{{r|Hatcher2001|p=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 <math>E_f</math> of the pathspace fibration for a continuous mapping <math>f \colon A \to B</math> between topological spaces consists of pairs <math>(a, \gamma)</math> with <math>a \in A</math> and paths <math>\gamma \colon I \to B</math> with starting point <math>\gamma (0) = f(a),</math> where <math>I = [0, 1]</math> is the unit interval. The space <math>E_f = \{ (a, \gamma) \in A \times B^I | \gamma (0) = f(a) \}</math> carries the subspace topology of <math>A \times B^I,</math> where <math>B^I</math> describes the space of all mappings <math>I \to B</math> and carries the compact-open topology.

The pathspace fibration is given by the mapping <math>p \colon E_f \to B</math> with <math>p(a, \gamma) = \gamma (1).</math> The fiber <math>F_f</math> is also called the homotopy fiber of <math>f</math> and consists of the pairs <math>(a, \gamma)</math> with <math>a \in A</math> and paths <math>\gamma \colon [0, 1] \to B,</math> where <math>\gamma(0) = f(a)</math> and <math>\gamma(1) = b_0 \in B</math> holds.

For the special case of the inclusion of the base point <math>i \colon b_0 \to B</math>, an important example of the pathspace fibration emerges. The total space <math>E_i</math> consists of all paths in <math>B</math> which starts at <math>b_0.</math> This space is denoted by <math>PB</math> and is called path space. The pathspace fibration <math>p \colon PB \to B</math> maps each path to its endpoint, hence the fiber <math>p^{-1}(b_0)</math> consists of all closed paths. The fiber is denoted by <math>\Omega B</math> and is called loop space.{{r|Hatcher2001|p=407-408}}

== Properties ==

* The fibers <math>p^{-1}(b)</math> over <math>b \in B</math> are homotopy equivalent for each path component of <math>B.</math>{{r|Hatcher2001|p=405}} * For a homotopy <math>f \colon [0, 1] \times A \to B</math> the pullback fibrations <math>f^*_0(E) \to A</math> and <math>f^*_1(E) \to A</math> are fiber homotopy equivalent.{{r|Hatcher2001|p=406}} * If the base space <math>B</math> is contractible, then the fibration <math>p \colon E \to B</math> is fiber homotopy equivalent to the product fibration <math>B \times F \to B.</math>{{r|Hatcher2001|p=406}} * The pathspace fibration of a fibration <math>p \colon E \to B</math> is very similar to itself. More precisely, the inclusion <math>E \hookrightarrow E_p</math> is a fiber homotopy equivalence.{{r|Hatcher2001|p=408}} * For a fibration <math>p \colon E \to B</math> with fiber <math>F</math> and contractible total space, there is a weak homotopy equivalence <math>F \to \Omega B.</math>{{r|Hatcher2001|p=408}}

== Puppe sequence ==

For a fibration <math>p \colon E \to B</math> with fiber <math>F</math> and base point <math>b_0 \in B</math> the inclusion <math>F \hookrightarrow F_p</math> of the fiber into the homotopy fiber is a homotopy equivalence. The mapping <math>i \colon F_p \to E</math> with <math>i (e, \gamma) = e</math>, where <math>e \in E</math> and <math>\gamma \colon I \to B</math> is a path from <math>p(e)</math> to <math>b_0</math> in the base space, is a fibration. Specifically it is the pullback fibration of the pathspace fibration <math>PB \to B</math> along <math>p</math>. This procedure can now be applied again to the fibration <math>i</math> and so on. This leads to a long sequence: <blockquote><math> \cdots \to F_j \to F_i \xrightarrow {j} F_p \xrightarrow i E \xrightarrow p B.</math></blockquote> The fiber of <math>i</math> over a point <math>e_0 \in p^{-1}(b_0)</math> consists of the pairs <math>(e_0, \gamma)</math> where <math>\gamma</math> is a path from <math>p(e_0) = b_0</math> to <math>b_0</math>, i.e. the loop space <math>\Omega B</math>. The inclusion <math>\Omega B \hookrightarrow F_i</math> of the fiber of <math>i</math> into the homotopy fiber of <math>i</math> is again a homotopy equivalence and iteration yields the sequence:<blockquote><math>\cdots \Omega^2B \to \Omega F \to \Omega E \to \Omega B \to F \to E \to B.</math></blockquote>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.{{r|Hatcher2001|p=407-409}}

== Principal fibration == A fibration <math>p \colon E \to B</math> with fiber <math>F</math> is called ''principal'', if there exists a commutative diagram: center|frameless 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.{{r|Hatcher2001|p=412}}

== Long exact sequence of homotopy groups == For a Serre fibration <math>p \colon E \to B</math> there exists a long exact sequence of homotopy groups. For base points <math>b_0 \in B</math> and <math>x_0 \in F = p^{-1}(b_0)</math> this is given by:<blockquote><math>\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 </math>

<math>\cdots \rightarrow \pi_0(F, x_0) \rightarrow \pi_0(E, x_0).</math></blockquote>The homomorphisms <math>\pi_n(F, x_0) \rightarrow \pi_n(E, x_0)</math> and <math>\pi_n(E, x_0) \rightarrow \pi_n(B, b_0)</math> are the induced homomorphisms of the inclusion <math>i \colon F \hookrightarrow E</math> and the projection <math>p \colon E \rightarrow B.</math>{{r|Hatcher2001|p=376}}

=== Hopf fibration === Hopf fibrations are a family of fiber bundles whose fiber, total space and base space are spheres:<blockquote><math>S^0 \hookrightarrow S^1 \rightarrow S^1,</math>

<math>S^1 \hookrightarrow S^3 \rightarrow S^2,</math>

<math>S^3 \hookrightarrow S^7 \rightarrow S^4,</math>

<math>S^7 \hookrightarrow S^{15} \rightarrow S^8.</math></blockquote>The long exact sequence of homotopy groups of the hopf fibration <math>S^1 \hookrightarrow S^3 \rightarrow S^2</math> yields:<blockquote><math>\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 </math> <math>\cdots \rightarrow \pi_1(S^1, x_0) \rightarrow \pi_1(S^3, x_0) \rightarrow \pi_1(S^2, b_0).</math></blockquote> This sequence splits into short exact sequences, as the fiber <math>S^1</math> in <math>S^3</math> is contractible to a point:<blockquote><math>0 \rightarrow \pi_i(S^3) \rightarrow \pi_i(S^2) \rightarrow \pi_{i-1}(S^1) \rightarrow 0.</math></blockquote>This short exact sequence splits because of the suspension homomorphism <math> \phi \colon \pi_{i - 1}(S^1) \to \pi_i(S^2)</math> and there are isomorphisms:<blockquote><math>\pi_i(S^2) \cong \pi_i(S^3) \oplus \pi_{i - 1}(S^1).</math></blockquote>The homotopy groups <math>\pi_{i - 1}(S^1)</math> are trivial for <math>i \geq 3,</math> so there exist isomorphisms between <math>\pi_i(S^2)</math> and <math>\pi_i(S^3)</math> for <math>i \geq 3.</math>

Analog the fibers <math>S^3</math> in <math>S^7</math> and <math>S^7</math> in <math>S^{15}</math> are contractible to a point. Further the short exact sequences split and there are families of isomorphisms:{{r|Steenrod1951|p=111}} <blockquote><math>\pi_i(S^4) \cong \pi_i(S^7) \oplus \pi_{i - 1}(S^3)</math> and

<math>\pi_i(S^8) \cong \pi_i(S^{15}) \oplus \pi_{i - 1}(S^7).</math></blockquote>

== 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 <math>p \colon E \to B</math> with fiber <math>F,</math> where the base space is a path connected CW-complex, and an additive homology theory <math>G_*</math> there exists a spectral sequence:{{r|Davis1991|p=242}}

:<math>H_k (B; G_q(F)) \cong E^2_{k, q} \implies G_{k + q}(E).</math>

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 <math>p \colon E \to B</math> with fiber <math>F,</math> where base space and fiber are path connected, the fundamental group <math>\pi_1(B)</math> acts trivially on <math>H_*(F)</math> and in addition the conditions <math>H_p(B) = 0</math> for <math>0<p<m</math> and <math>H_q(F) = 0</math> for <math>0<q<n</math> hold, an exact sequence exists (also known under the name Serre exact sequence):<blockquote><math>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.</math>{{r|Davis1991|p=250}}</blockquote>This sequence can be used, for example, to prove Hurewicz's theorem or to compute the homology of loopspaces of the form <math>\Omega S^n:</math> {{r|Cohen1998|p=162}} <blockquote><math>H_k (\Omega S^n) = \begin{cases} \Z & \exist q \in \Z \colon k = q (n-1)\\ 0 & \text{otherwise} \end{cases}.</math></blockquote>For the special case of a fibration <math>p \colon E \to S^n</math> where the base space is a <math>n</math>-sphere with fiber <math>F,</math> there exist exact sequences (also called Wang sequences) for homology and cohomology:{{r|Spanier1966|p=456}} <blockquote><math>\cdots \to H_q(F) \xrightarrow{i_*} H_q(E) \to H_{q-n}(F) \to H_{q-1}(F) \to \cdots</math>

<math>\cdots \to H^q(E) \xrightarrow{i^*} H^q(F) \to H^{q-n+1}(F) \to H^{q+1}(E) \to \cdots</math></blockquote>

== Orientability == For a fibration <math>p \colon E \to B</math> with fiber <math>F</math> and a fixed commutative ring <math>R</math> with a unit, there exists a contravariant functor from the fundamental groupoid of <math>B</math> to the category of graded <math>R</math>-modules, which assigns to <math>b \in B</math> the module <math>H_*(F_b, R)</math> and to the path class <math>[\omega]</math> the homomorphism <math>h[\omega]_* \colon H_*(F_{\omega (0)}, R) \to H_*(F_{\omega (1)}, R),</math> where <math>h[\omega]</math> is a homotopy class in <math>[F_{\omega(0)}, F_{\omega (1)}].</math>

A fibration is called ''orientable'' over <math>R</math> if for any closed path <math>\omega</math> in <math>B</math> the following holds: <math>h[\omega]_* = 1.</math>{{r|Spanier1966|p=476}}

== Euler characteristic == For an orientable fibration <math>p \colon E \to B</math> over the field <math>\mathbb{K}</math> with fiber <math>F</math> and path connected base space, the Euler characteristic of the total space is given by:<blockquote><math>\chi(E) = \chi(B)\chi(F).</math></blockquote>Here the Euler characteristics of the base space and the fiber are defined over the field <math>\mathbb{K}</math>.{{r|Spanier1966|p=481}}

== See also == *Approximate fibration *Cofibration

== References == <references> # <ref name=Hatcher2001>{{Cite book |last=Hatcher |first=Allen|author-link=Allen Hatcher |title=Algebraic Topology |publisher=Cambridge University Press |year=2001 |isbn=0-521-79160-X |location=NY}}</ref> # <ref name=Laures2014>{{Cite book |last1=Laures |first1=Gerd |title=Grundkurs Topologie |last2=Szymik |first2=Markus |publisher=Springer Spektrum |year=2014 |isbn=978-3-662-45952-2 |edition=2nd |language=German |doi=10.1007/978-3-662-45953-9}}</ref> # <ref name=May1999>{{Cite book |last=May |first=J.P. |author-link=J. Peter May|title=A Concise Course in Algebraic Topology |isbn=0-226-51182-0 |oclc=41266205 |url=https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |year=1999 |publisher=University of Chicago Press}}</ref> # <ref name=Spanier1966>{{Cite book|last=Spanier|first=Edwin H.|author-link=Edwin Spanier|title=Algebraic Topology|publisher=McGraw-Hill Book Company|year=1966|isbn=978-0-387-90646-1}}</ref> # <ref name=Dold1958>{{Cite journal |last1=Dold |first1=Albrecht|author-link= Albrecht Dold|title=Quasifaserungen und Unendliche Symmetrische Produkte |last2=Thom |first2=René |journal=Annals of Mathematics |year=1958 |volume=67 |issue=2 |pages=239–281 |doi=10.2307/1970005|jstor=1970005 }}</ref> # <ref name=Steenrod1951>{{Cite book |last=Steenrod |first=Norman|author-link=Norman Steenrod |title=The Topology of Fibre Bundles |publisher=Princeton University Press |year=1951 |isbn=0-691-08055-0 }}</ref> # <ref name=Davis1991>{{Cite book |last1=Davis |first1=James F. |title=Lecture Notes in Algebraic Topology |last2=Kirk |first2=Paul |year=1991 |publisher=Department of Mathematics, Indiana University |url=https://jfdmath.sitehost.iu.edu/teaching/m623/book.pdf}}</ref> # <ref name=Cohen1998>{{Cite book |last=Cohen |first=Ralph L. |author-link=Ralph Louis Cohen|title=The Topology of Fiber Bundles Lecture Notes |year=1998 |publisher=Stanford University |url=https://math.stanford.edu/~ralph/fiber.pdf}}</ref> </references>

Category:Algebraic topology Category:Topological spaces