For any Abelian Group and any Natural Number , there is a unique Space (up to Homotopy type) such that all Homotopy Groups except for the th are trivial (including the 0th Homotopy Groups, meaning the Space is path-connected), and the th Homotopy Group is Isomorphic to the Group . In the case where , the Group can be non-Abelian as well.
Eilenberg-Mac Lane spaces have many important applications. One of them is that every Topological Space has the Homotopy type of an iterated Fibration of Eilenberg-Mac Lane spaces (called a Postnikov System). In addition, there is a spectral sequence relating the Cohomology of Eilenberg-Mac Lane spaces to the Homotopy Groups of Spheres.