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''section surface, of a [[Fibre space|fibre space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083730/s0837301.png" />''
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{{MSC|14}}
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{{TEX|done}}
  
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A continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083730/s0837302.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083730/s0837303.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083730/s0837304.png" /> is a [[Serre fibration|Serre fibration]], then
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A ''section'' or
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''section surface'' of a
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surjective (continuous) map or of a
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[[Fibre space|fibre space]] $p:X\to Y$'' is
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a (continuous) mapping $s:Y\to X$ such that $p\circ s={\rm id}_Y$.  
  
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<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/s083/s083730/s0837305.png" /></td> </tr></table>
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If $(X,p,Y)$ is a
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[[Serre fibration|Serre fibration]], then
  
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For a [[Principal fibre bundle|principal fibre bundle]] the existence of a section implies its triviality. A [[Vector bundle|vector bundle]] always possesses the so-called zero section.
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$$\pi_n(X) = \pi_n(p^{-1}(pt))\oplus \pi_n(Y).$$
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For a
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[[Principal fibre bundle|principal fibre bundle]] the existence of a section implies its triviality. A
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[[Vector bundle|vector bundle]] always possesses the so-called zero section.
  
  
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====References====
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{|
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|-
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|valign="top"|{{Ref|Sp}}||valign="top"|  E.H. Spanier,  "Algebraic topology", McGraw-Hill  (1966)  pp. 77  {{MR|0210112}} {{MR|1325242}}  {{ZBL|0145.43303}}
  
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====Comments====
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|}
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====References====
 
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)  pp. 77</TD></TR></table>
 

Latest revision as of 22:27, 24 November 2013

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

A section or section surface of a surjective (continuous) map or of a fibre space $p:X\to Y$ is a (continuous) mapping $s:Y\to X$ such that $p\circ s={\rm id}_Y$.

If $(X,p,Y)$ is a Serre fibration, then

$$\pi_n(X) = \pi_n(p^{-1}(pt))\oplus \pi_n(Y).$$ For a principal fibre bundle the existence of a section implies its triviality. A vector bundle always possesses the so-called zero section.


References

[Sp] E.H. Spanier, "Algebraic topology", McGraw-Hill (1966) pp. 77 MR0210112 MR1325242 Zbl 0145.43303
How to Cite This Entry:
Section. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Section&oldid=18525
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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