Eilenberg-Mac Lane Space

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 pathwise-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.