Difference between revisions of "Killing space"
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| − | + | '' $ ( X , n ) $'' | |
| − | + | A [[Fibre space|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|Eilenberg–MacLane space]]. The sequence of spaces $ ( X , n ) $ | ||
| + | and mappings $ p _ {n} $ | ||
| + | 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
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