# Killing space

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$( X , n )$

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