Eilenberg-Maclane space

Definition
Let $$G$$ be a group and $$n \ge 1$$ 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 $$n^{th}$$ 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