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Difference between revisions of "Serre fibration"

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A triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084690/s0846901.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084690/s0846902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084690/s0846903.png" /> are topological spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084690/s0846904.png" /> is a continuous mapping, with the following property (called the property of the existence of a [[Covering homotopy|covering homotopy]] for polyhedra). For any finite polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084690/s0846905.png" /> and for any mappings
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{{MSC|15}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084690/s0846906.png" /></td> </tr></table>
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A triple $(X,p,Y)$, where $X$ and $Y$ are topological spaces and $p:X\to Y$ is a continuous mapping, with the following property (called the property of the existence of a
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[[Covering homotopy|covering homotopy]] for polyhedra). For any finite polyhedron $K$ and for any mappings
  
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$$f:K\times[0,1]\to Y, \qquad F_0:K=K\times\{0\}\to X$$
 
with
 
with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084690/s0846907.png" /></td> </tr></table>
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$$f\mid_{K\times\{0\}} = p\circ F_0$$
 
 
 
there is a mapping
 
there is a mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084690/s0846908.png" /></td> </tr></table>
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$$F : K\times[0,1]\to X$$
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such that $F\mid_{K\times\{0\}} = F_0$, $p\circ F=f$. It was introduced by J.-P. Serre in 1951 (see
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{{Cite|Se}}).
  
such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084690/s0846909.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084690/s08469010.png" />. It was introduced by J.-P. Serre in 1951 (see [[#References|[1]]]).
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====Comments====
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A Serre fibration is also called a weak fibration. If the defining homotopy lifting property holds for every space (not just polyhedra), $p:X\to Y$ is called a fibration or Hurewicz fibre space.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.P. Serre,  "Homologie singulière des espaces fibrés. Applications"  ''Ann. of Math.'' , '''54'''  (1951)  pp. 425–505</TD></TR></table>
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{|
 
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|-
 
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|valign="top"|{{Ref|Se}}||valign="top"| J.P. Serre,  "Homologie singulière des espaces fibrés. Applications"  ''Ann. of Math.'', '''54'''  (1951)  pp. 425–505 {{MR|0045386}} {{MR|0039255}} {{MR|0039254}}  {{ZBL|0045.26003}}
  
====Comments====
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|-
A Serre fibration is also called a weak fibration. If the defining homotopy lifting property holds for every space (not just polyhedra), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084690/s08469011.png" /> is called a fibration or Hurewicz fibre space.
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|valign="top"|{{Ref|Sp}}||valign="top"|  E.H. Spanier,  "Algebraic topology", McGraw-Hill  (1966)  pp. Chapt. 2, §2; Chapt. 7, §2  {{MR|0210112}} {{MR|1325242}}  {{ZBL|0145.43303}}
  
====References====
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|-
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)  pp. Chapt. 2, §2; Chapt. 7, §2</TD></TR></table>
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|}

Revision as of 22:24, 24 November 2013

2020 Mathematics Subject Classification: Primary: 15-XX [MSN][ZBL]

A triple $(X,p,Y)$, where $X$ and $Y$ are topological spaces and $p:X\to Y$ is a continuous mapping, with the following property (called the property of the existence of a covering homotopy for polyhedra). For any finite polyhedron $K$ and for any mappings

$$f:K\times[0,1]\to Y, \qquad F_0:K=K\times\{0\}\to X$$ with

$$f\mid_{K\times\{0\}} = p\circ F_0$$ there is a mapping

$$F : K\times[0,1]\to X$$ such that $F\mid_{K\times\{0\}} = F_0$, $p\circ F=f$. It was introduced by J.-P. Serre in 1951 (see [Se]).

Comments

A Serre fibration is also called a weak fibration. If the defining homotopy lifting property holds for every space (not just polyhedra), $p:X\to Y$ is called a fibration or Hurewicz fibre space.

References

[Se] J.P. Serre, "Homologie singulière des espaces fibrés. Applications" Ann. of Math., 54 (1951) pp. 425–505 MR0045386 MR0039255 MR0039254 Zbl 0045.26003
[Sp] E.H. Spanier, "Algebraic topology", McGraw-Hill (1966) pp. Chapt. 2, §2; Chapt. 7, §2 MR0210112 MR1325242 Zbl 0145.43303
How to Cite This Entry:
Serre fibration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Serre_fibration&oldid=30773
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article