Eilenberg-Maclane space

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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.

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