In mathematics, an Eilenberg-MacLane space is a special kind of topological space that is important in many contexts in algebraic topology, including stage-by-stage constructions of spaces, computations of homotopy groups of spheres, and definition of cohomological operations . The name is for Samuel Eilenberg and Saunders MacLane, who introduced such spaces in the late 1940s.

Let π be a group and n a positive integer number. A connected topological space X is called an Eilenberg—Mac Lane space of type K(π,n), if it has n-th homotopy group π_n(X) isomorphic to π and all other homotopy groups trivial. If n > 1 then π must be abelian. Then an Eilenberg—MacLane space exists, as a CW complex, and is unique up to a weak homotopy equivalence. By abuse of language, any such space is often called just K(pi;,n).

A K(π,n) can be constructed stage-by-stage, as a CW complex, starting with a wedge of spheres, one for each generator of the group π, and adding cells in (possibly infinite number of) higher dimensions so as to kill all extra homotopy.

An important property of K(π,n) is that, for abelian π and any topological space X, a set [X, K(π,n)] of homotopy classes of based maps from X to K(π,n) is in natural bijection with n-th cohomology group

Hⁿ(X; π)

of the space X. Thus one says that the K(π,n) are representing spaces for cohomology with coefficients in π.

Every CW complex possesses a Postnikov tower, that is, it is homotopy equivalent to an iterated fibration with fibers the Eilenberg—Mac Lane spaces.

There is a method due to Serre which allows one, at least theoretically, to compute homotopy groups of spaces using spectral sequence for special fibrations with Eilenberg—Mac Lane spaces for fibers.