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Difference between revisions of "Killing space"

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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k0554101.png" />''
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A [[Fibre space|fibre space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k0554102.png" /> for which the homotopy groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k0554103.png" /> vanish if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k0554104.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k0554105.png" /> is an isomorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k0554106.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k0554107.png" /> is constructed by induction with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k0554108.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k0554109.png" /> has already been constructed, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k05541010.png" /> is taken to be the homotopy fibre of the canonical mapping
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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/k/k055/k055410/k05541011.png" /></td> </tr></table>
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'' $  ( X , n ) $''
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k05541012.png" /> denoting an [[Eilenberg–MacLane space|Eilenberg–MacLane space]]. The sequence of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k05541013.png" /> and mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k05541014.png" /> is a Moore–Postnikov system of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055410/k05541015.png" />.
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A [[Fibre space|fibre space]] $  p _ {n} :  ( X , n ) \rightarrow X $
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for which the homotopy groups  $  \pi _ {i} ( X , n ) $
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vanish if  $  i < n $,
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and $  p _ {n*} : \pi _ {i} ( X , n ) \rightarrow \pi _ {i} ( X) $
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is an isomorphism if  $  i \geq  n $.  
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The space  $  ( X , n ) $
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is constructed by induction with respect to  $  n $.  
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If  $  ( X , n - 1 ) $
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has already been constructed, then  $  ( X , n ) $
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is taken to be the homotopy fibre of the canonical mapping
  
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$$
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( X , n - 1 )  \rightarrow  K ( \pi _ {n-} 1 ( X) , n - 1 ) ,
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$$
  
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$  K ( \pi _ {n-} 1 ( X) , n - 1 ) $
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denoting an [[Eilenberg–MacLane space|Eilenberg–MacLane space]]. The sequence of spaces  $  ( X , n ) $
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and mappings  $  p _ {n} $
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is a Moore–Postnikov system of the mapping  $  * \rightarrow X $.
  
 
====Comments====
 
====Comments====

Latest revision as of 22:14, 5 June 2020


$ ( X , n ) $

A fibre space $ p _ {n} : ( X , n ) \rightarrow X $ for which the homotopy groups $ \pi _ {i} ( X , n ) $ vanish if $ i < n $, and $ p _ {n*} : \pi _ {i} ( X , n ) \rightarrow \pi _ {i} ( X) $ is an isomorphism if $ i \geq n $. The space $ ( X , n ) $ is constructed by induction with respect to $ n $. If $ ( X , n - 1 ) $ has already been constructed, then $ ( X , n ) $ is taken to be the homotopy fibre of the canonical mapping

$$ ( X , n - 1 ) \rightarrow K ( \pi _ {n-} 1 ( X) , n - 1 ) , $$

$ K ( \pi _ {n-} 1 ( X) , n - 1 ) $ denoting an Eilenberg–MacLane space. The sequence of spaces $ ( X , n ) $ and mappings $ p _ {n} $ is a Moore–Postnikov system of the mapping $ * \rightarrow X $.

Comments

See also [a1], Chapt. 8, Sect. 3.

References

[a1] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Chapt. 2
How to Cite This Entry:
Killing space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Killing_space&oldid=47498
This article was adapted from an original article by A.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article