Eilenberg-Maclane space: Difference between revisions

Revision as of 19:06, 2 December 2007

Definition

Let $G$ be a group and $n \geq 1$ an integer. If $n > 1$ , we require that $G$ be Abelian. An Eilenberg-Maclane space for the pair $(G, n)$ denoted $K (G, n)$ , is defined as a path-connected space whose $n^{t h}$ homotopy group is $G$ , and for which all the other homotopy groups are trivial.

Eilenberg-Maclane spaces are unique upto weak homotopy-equivalence. In particular, among the class of CW-spaces, the Eilenberg-Maclane spaces are unique upto homotopy type.

In the particular case where $n = 1$ the Eilenberg-Maclane space coincides with the classifying space for $G$ , viewed as a discrete group.

Related notions

Moore space

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 Let <math>G</math> be a [[group]] and <math>n \ge 1</math> an integer. If <math>n > 1</math>, we require that <math>G</math> be [[Abelian group|Abelian]]. An '''Eilenberg-Maclane space''' for the pair <math>(G,n)</math> denoted <math>K(G,n)</math>, is defined as a [[path-connected space]] whose <math>n^{th}</math> [[homotopy group]] is <math>G</math>, and for which all the other homotopy groups are trivial.
-Eilenberg-Maclane spaces are unique upto weak homotopy-equivalence. In particular, among the class of [[CW-spaces]], the Eilenberg-Maclane spaces are unique upto homotopy type.
+Eilenberg-Maclane spaces are unique upto weak homotopy-equivalence. In particular, among the class of [[CW-space]]s, the Eilenberg-Maclane spaces are unique upto homotopy type.
 In the particular case where <math>n = 1</math> the Eilenberg-Maclane space coincides with the [[classifying space]] for <math>G</math>, viewed as a discrete group.
+==Related notions==
+* [[Moore space]]