Eilenberg–MacLane space ($K(A,n)$)

A connected Eilenberg–MacLane space $K(G,n)$, $n\ge 1$, has ${\pi }_n\cong G$ and all other positive-degree homotopy groups trivial. The group $G$ must be abelian when $n\ge 2$, but may be nonabelian when $n=1$. For an abelian group, the Dold–Kan correspondence constructs a simplicial model from the chain complex concentrated in degree $n$. A classifying space of a discrete group gives the general $K(G,1)$ case.

For $n\ge 4$, the first four possible degrees have integral groups $\mathbb{Z}/2,0,\mathbb{Z}/2,\mathbb{Z}/2$ in degrees $n,n+1,n+2,n+3$. Mod-two Betti numbers count cyclic summands, while the nonzero first Bockstein actions on $u,{\mathrm{Sq}}^{2}u,{\mathrm{Sq}}^2{\mathrm{Sq}}^{1}u$ show that their orders are two. This prevents mistaking a nonzero mod-two group for a higher-order integral cyclic group.

For $n\ge 1$, the mod-two cohomology of $K(\mathbb{Z}/2,n)$ is polynomial on the admissible operations in the displayed strict-excess range, including the empty operation. It is obtained from path-space transgressions and compatibility with Steenrod squares. Products must still be included when listing low-degree vector-space bases.

For $n\ge 2$, the positive-degree dimensions in degrees $n,n+1,n+2,n+3$ are $1,1,1,2$. The first three classes are $u,{\mathrm{Sq}}^{1}u,{\mathrm{Sq}}^{2}u$. In degree $n+3$, use ${\mathrm{Sq}}^{3}u$ and ${\mathrm{Sq}}^2{\mathrm{Sq}}^{1}u$ for $n\ge 3$; at $n=2$ replace the unstable first class by $u{\mathrm{Sq}}^{1}u$. At $n=2$ the degree-four class is $u^2$, and at $n=3$ the degree-six class ${\mathrm{Sq}}^{3}u$ is $u^2$.

The ring is exterior on its degree-$n$ universal class for odd $n$, and polynomial for even $n$. The path-loop Serre spectral sequence alternates these two types by transgression. In the polynomial-fiber case its derivation has nonzero coefficients $k$ on powers, which become invertible over the rationals.

For a CW complex $X$, abelian $G$ and $n\ge 1$, homotopy classes of maps to the Eilenberg–MacLane space correspond to cohomology classes by pullback of the universal cohomology class of an Eilenberg–MacLane space. Pointed maps give the reduced version. This converts a cohomology class into an actual map realizing its effect on homotopy.

For abelian $G$ and $n\ge 1$, the identity of $G$ determines this class under $H^n(K(G,n);G)\cong \mathrm{Hom}(G,G)$. Pulling it back realizes representability of cohomology by Eilenberg–MacLane spaces. A map inducing this class on a fiber induces the identity of its unique positive homotopy group.

A path-connected space is aspherical when its higher homotopy groups vanish, equivalently when its universal cover is contractible. An aspherical space with fundamental group $G$ is a $K(G,1)$, so based homotopy classes of maps into it are controlled by homomorphisms of fundamental groups.