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 nth are trivial (including the 0th homotopy groups, meaning the space is pathwise-connected), and the nth 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.

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Cite this as:

Weisstein, Eric W. "Eilenberg-Mac Lane Space." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Eilenberg-MacLaneSpace.html

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