Eilenberg–MacLane space
In mathematics, and algebraic topology in particular, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. As such, an Eilenberg–MacLane space is a special kind of topological space that can be regarded as a building block for homotopy theory; general topological spaces can be constructed from these via the Postnikov system. These spaces are important in many contexts in algebraic topology, including constructions of spaces, computations of homotopy groups of spheres, and definition of cohomology operations. The name is for Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.
Let G be a group and n a positive integer. A connected topological space X is called an Eilenberg–MacLane space of type K(G, n), if it has n-th homotopy group πn(X) isomorphic to G and all other homotopy groups trivial. If n > 1 then G must be abelian. Such a space exists, is a CW-complex, and is unique up to a weak homotopy equivalence. By abuse of language, any such space is often called just K(G, n).
Some further elementary examples can be constructed from these by using the fact that the product K(G, n) × K(H, n) is K(G × H, n).
A K(G, n) can be constructed stage-by-stage, as a CW complex, starting with a wedge of n-spheres, one for each generator of the group G, and adding cells in (possibly infinite number of) higher dimensions so as to kill all extra homotopy. The corresponding chain complex is given by the Dold–Kan correspondence.
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