# Killing space

* $ ( 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