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Eilenberg-Mac Lane Space

For any Abelian Group $G$ and any Natural Number $n$, there is a unique Space (up to Homotopy type) such that all Homotopy Groups except for the $n$th are trivial (including the 0th Homotopy Groups, meaning the Space is path-connected), and the $n$th Homotopy Group is Isomorphic to the Group $G$. In the case where $n=1$, the Group $G$ 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.

© 1996-9 Eric W. Weisstein