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| {{About|fibrations in algebraic topology|fibrations in category theory, as used in descent theory and categorical logic|Fibred category}}
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| In [[topology]], a branch of mathematics, a '''fibration''' is a generalization of the notion of a [[fiber bundle]]. A fiber bundle makes precise the idea of one [[topological space]] (called a fiber) being "parameterized" by another topological space (called a base). A fibration is like a fiber bundle, except that the fibers need not be the same space, rather they are just [[homotopy equivalent]]. Fibrations do not necessarily have the local [[Cartesian product]] structure that defines the more restricted fiber bundle case, but something weaker that still allows "sideways" movement from fiber to fiber. Fiber bundles have a particularly simple [[homotopy theory]] that allows topological information about the bundle to be inferred from information about one or both of these constituent spaces. A fibration satisfies an additional condition (the [[homotopy lifting property]]) guaranteeing that it will behave like a fiber bundle from the point of view of homotopy theory.
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| == Formal definition ==
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| A '''fibration''' (or '''Hurewicz fibration''') is a [[continuous function (topology)|continuous mapping]] {{math|''p'' : ''E'' → ''B''}} satisfying the [[homotopy lifting property]] with respect to any space. [[Fiber bundle]]s (over [[paracompact]] bases) constitute important examples. In [[homotopy theory]] any mapping is 'as good as' a fibration—i.e. any map can be decomposed as a homotopy equivalence into a "mapping path space" followed by a fibration. (See [[homotopy fiber]].)
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| The ''fibers'' are by definition the subspaces of {{mvar|E}} that are the inverse images of points {{mvar|b}} of {{mvar|B}}. If the base space {{mvar|B}} is path connected, it is a consequence of the definition that the fibers of two different points {{math|''b''<sub>1</sub>}} and {{math|''b''<sub>2</sub>}} in {{mvar|B}} are [[homotopy equivalence|homotopy equivalent]]. Therefore one usually speaks of "the fiber" {{mvar|F}}.
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| == Serre fibrations ==
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| A continuous mapping with the homotopy lifting property for [[CW complex]]es (or equivalently, just cubes {{math|''I''<sup>''n''</sup>}}) is called a ''Serre fibration'', in honor of the part played by the concept in the thesis of [[Jean-Pierre Serre]]. This thesis firmly established in [[algebraic topology]] the use of [[spectral sequence]]s, and clearly separated the notions of fiber bundles and fibrations from the notion of [[sheaf (mathematics)|sheaf]] (both concepts together having been implicit in the pioneer treatment of [[Jean Leray]]). Because a sheaf (thought of as an [[étalé space]]) can be considered a [[local homeomorphism]], the notions were closely interlinked at the time. One of the main desirable properties of the [[Serre spectral sequence]] is to account for the action of the [[fundamental group]] of the base {{mvar|B}} on the homology of the "total space" {{mvar|E}}.
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| == Examples ==
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| In the following examples a fibration is denoted
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| :{{bigmath|''F'' → ''E'' → ''B''}},
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| where the first map is the inclusion of "the" fiber {{mvar|F}} into the total space {{mvar|E}} and the second map is the fibration onto the basis {{mvar|B}}. This is also referred to as a fibration sequence.
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| *The projection map from a product space is very easily seen to be a fibration.
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| *[[Fiber bundle]]s have ''local trivializations,'' i.e. Cartesian product structures exist [[locally]] on {{mvar|B}}, and this is usually enough to show that a fiber bundle is a fibration. More precisely, if there are local trivializations over a "numerable open cover" of {{mvar|B}}, the bundle is a fibration. Any open cover of a [[paracompact]] space is numerable. For example, any open cover of a metric space has a locally finite refinement, so any bundle over such a space is a fibration. The local triviality also implies the existence of a [[well-defined]] ''fiber'' ([[up to]] [[homeomorphism]]), at least on each [[connected space|connected component]] of {{mvar|B}}.
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| * The [[Hopf fibration]] {{math|''S''<sup>1</sup> → ''S''<sup>3</sup> → ''S''<sup>2</sup>}} was historically one of the earliest non-trivial examples of a fibration.
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| * Over [[complex projective space]], there is a fibration {{math|''S''<sup>1</sup> → ''S''<sup>2''n''+1</sup> → [[complex projective space|'''CP'''<sup>''n''</sup>]]}}.(Notice that the first example ; that of the Hopf fibration, is a special case of this fibration for n=1, since [[ complex projective space| '''CP'''<sup>''1''</sup>]] is homeomorphic to {{math| ''S'' <sup> 2</sup>}} )
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| * The Serre fibration {{math|SO(2) → SO(3) → ''S''<sup>2</sup>}} comes from the action of the [[rotation group SO(3)|rotation group {{math|SO(3)}}]] on the [[sphere|2-sphere]] {{math|''S''<sup>2</sup>}}.
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| * The previous example can also be generalized to a fibration {{math|SO(''n'') → SO(''n''+1) → ''S''<sup>''n''</sup>}} for any non-negative integer {{mvar|n}} (though they only have a fiber that isn't just a point when {{math|''n'' > 1}}) that comes from the action of the [[orthogonal group|special orthogonal group {{math|SO(''n''+1)}}]] on the {{mvar|n}}-sphere.
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| == Properties == | |
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| ===Long exact sequence in homotopy groups===
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| Choose a base point {{math|''b''<sub>0</sub> ∈ ''B''}}. Let {{mvar|F}} refer to the fiber over {{math|''b''<sub>0</sub>}}, i.e. {{math|F {{=}} ''p''<sup>-1</sup>({''b''<sub>0</sub>})}}; and let {{mvar|i}} be the inclusion {{math|''F'' → ''E''}}. Choose a base point {{math|''f''<sub>0</sub> ∈ ''F''}} and let {{math|''e''<sub>0</sub> {{=}} ''i''(''f''<sub>0</sub>)}}. In terms of these base points, we have a [[long exact sequence]]
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| :<math>\cdots\to\pi_n(F)\to\pi_n(E)\to\pi_n(B)\to\pi_{n-1}(F)\to\cdots
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| </math>
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| constructed from the [[homotopy group]]s of the fiber {{mvar|F}}, total space {{mvar|E}}, and base space {{mvar|B}}. The homomorphisms {{math|''π''<sub>''n''</sub>(''F'') → ''π''<sub>''n''</sub>(''E'')}} and {{math|''π''<sub>''n''</sub>(''E'') → ''π''<sub>''n''</sub>(''B'')}} are just the induced homomorphisms from {{mvar|i}} and {{mvar|p}}, respectively.
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| The third set of homomorphisms {{math|''β''<sub>''n''</sub> : ''π''<sub>''n''</sub>(''B'') → ''π''<sub>''n''−1</sub>(''F'')}} (called the "connecting homomorphisms" (in reference to the [[snake lemma]]) or the "boundary maps") can be defined with the following steps.
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| # First, a little terminology: let {{math|''δ''<sub>''n''</sub> : ''S''<sup>''n''</sup> → ''D''<sup>''n''+1</sup>}} be the inclusion of the boundary [[n-sphere|{{mvar|n}}-sphere]] into the [[n-sphere#n-ball|{{math|(''n''+1)}}-ball]]. Let {{math|''γ''<sub>''n''</sub> : ''D''<sup>''n''</sup> → ''S''<sup>''n''</sup>}} be the map that collapses the image of {{math|''δ''<sub>''n''−1</sub>}} in {{math|''D''<sup>''n''</sup>}} to a point.
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| # Let {{math|''φ'' : ''S''<sup>''n''</sup> → ''B''}} be a representing map for an element of {{math|''π''<sub>''n''</sub>(''B'')}}.
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| # Because {{math|''D''<sup>''n''</sup>}} is homeomorphic to the {{mvar|n}}-dimensional cube, we can iteratively apply the homotopy lifting property to construct a lift {{math|''λ'' : ''D''<sup>''n''</sup> → ''E''}} of {{math|''φ'' ∘ ''γ''<sub>''n''</sub>}} (i.e., a map {{mvar|λ}} such that {{math|''p'' ∘ ''λ'' {{=}} ''φ'' ∘ ''γ''<sub>''n''</sub>}}).
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| # Because {{math|''γ''<sub>''n''</sub> ∘ ''δ''<sub>''n''−1</sub>}} is a point map (hereafter referred to as "{{math|{{=}} pt}}"), {{math|pt {{=}} ''φ'' ∘ ''γ''<sub>''n''</sub> ∘ ''δ''<sub>''n''−1</sub> {{=}} ''p'' ∘ ''λ'' ∘ ''δ''<sub>''n''−1</sub>}}, which implies that the image of {{math|''λ'' ∘ ''δ''<sub>''n''−1</sub>}} is in {{mvar|F}}. Therefore, there exists a map {{math|''ψ'' : ''S''<sup>''n''−1</sup> → ''F''}} such that {{math|''i'' ∘ ''ψ'' {{=}} ''λ'' ∘ ''δ''<sub>''n''−1</sub>}}.
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| # We define {{math|''β''<sub>''n''</sub> [''φ''] {{=}} [''ψ'']}}.
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| The above is summarized in the following [[commutative diagram]]:
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| :[[File:Fibration homotopy groups LES connecting morphism diagram.svg|300px]]
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| Repeated application of the homotopy lifting property is used to prove that {{math|''β''<sub>''n''</sub>}} is a well-defined homomorphism and that this sequence is exact.
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| === Euler characteristic ===
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| {{Main|Euler characteristic}}
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| The [[Euler characteristic]] {{mvar|χ}} is multiplicative for [[fibrations]] with certain conditions.
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| If {{math|''p'' : ''E'' → ''B''}} is a fibration with fiber {{mvar|F}}, with the base {{mvar|B}} [[path-connected]], and the fibration is orientable over a field {{mvar|K}}, then the Euler characteristic with coefficients in the field {{mvar|K}} satisfies the product property:<ref>{{citation
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| |title=Algebraic Topology
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| |first=Edwin Henry
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| |last=Spanier
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| |authorlink=Edwin Spanier
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| |publisher=Springer
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| |year=1982
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| |isbn=978-0-387-94426-5
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| |url=http://books.google.com/?id=h-wc3TnZMCcC
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| }}, [http://books.google.com/books?id=h-wc3TnZMCcC&pg=PA481 Applications of the homology spectral sequence, p. 481]</ref>
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| :{{bigmath|''χ''(''E'') {{=}} ''χ''(''F'') · ''χ''(''B'')}}.
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| This includes product spaces and covering spaces as special cases,
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| and can be proven by the [[Serre spectral sequence]] on homology of a fibration. | |
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| For fiber bundles, this can also be understood in terms of a [[transfer map]] {{math|''τ'' : ''H''<sub>∗</sub>(''B'') → ''H''<sub>∗</sub>(''E'')}}—note that this is a lifting and goes "the wrong way"—whose composition with the projection map {{math|''p''<sub>∗</sub> : ''H''<sub>∗</sub>(''E'') → ''H''<sub>∗</sub>(''B'')}} is multiplication by the Euler characteristic of the fiber:<ref>{{citation
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| |title=Fibre bundles and the Euler characteristic
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| |first=Daniel Henry
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| |last=Gottlieb
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| |journal=Journal of Differential Geometry
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| |volume=10
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| |year=1975
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| |pages=39–48
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| |url=http://www.math.purdue.edu/~gottlieb/Bibliography/17FibreBundlesAndtheEulerCharacteristic.pdf
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| |issue=1
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| }}</ref>
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| {{math|''p''<sub>∗</sub> ∘ ''τ'' {{=}} ''χ''(''F'') · 1}}.
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| ==Fibrations in closed model categories==
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| Fibrations of topological spaces fit into a more general framework, the so-called [[closed model category|closed model categories]]. In such categories, there are distinguished classes of morphisms, the so-called ''fibrations'', ''[[cofibration]]s'' and ''[[weak equivalence (homotopy theory)|weak equivalence]]s''. Certain [[axiom]]s, such as stability of fibrations under composition and [[pullback (category theory)|pullbacks]], factorization of every morphism into the composition of an acyclic cofibration followed by a fibration or a cofibration followed by an acyclic fibration, where the word "acyclic" indicates that the corresponding arrow is also a weak equivalence, and other requirements are set up to allow the abstract treatment of homotopy theory. (In the original treatment, due to [[Daniel Quillen]], the word "trivial" was used instead of "acyclic.")
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| It can be shown that the category of topological spaces is in fact a model category, where (abstract) fibrations are just the Serre fibrations introduced above and weak equivalences are weak [[homotopy equivalence]]s.<ref>{{citation
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| |title=Handbook of algebraic topology
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| |first1=William G.
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| |last1=Dwyer
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| |first2=J.
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| |last2=Spaliński
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| |authorlink=William Dwyer
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| |publisher=North-Holland
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| |location=Amsterdam
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| |year=1995
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| |chapter=Homotopy theories and model categories
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| |pages=73–126
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| |mr=1361887
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| |url=http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Spalinski/theories
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| }}</ref>
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| ==See also==
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| * [[Homotopy fiber]]
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| == References ==
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| <references/>
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| [[Category:Algebraic topology]]
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| [[Category:Homotopy theory]]
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| [[Category:Differential topology]]
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| [[Category:Category theory]]
| |
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