Eilenberg-Maclane space

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 $n^{th}$ 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