Eilenberg-MacLane space

A space, denoted by $K(\pi,n)$, representing the functor $X\to H^n(X;\pi)$, where $n$ is a non-negative number, $\pi$ is a group which is commutative for $n>1$ and $H^n(X;\pi)$ is the $n$-dimensional cohomology group of a cellular space $X$ with coefficients in $\pi$. It exists for any such $n$ and $\pi$.

The Eilenberg–MacLane space $K(\pi,n)$ can also be characterized by the condition: $\pi_i(K(\pi,n))=\pi$ for $i=n$ and $\pi_i(K(\pi,n))=0$ for $i\neq n$, where $\pi_i$ is the $i$-th homotopy group. Thus, $K(\pi,n)$ is uniquely defined up to a weak homotopy equivalence. An arbitrary topological space can, up to a weak homotopy equivalence, be decomposed into a twisted product of Eilenberg–MacLane spaces (see Postnikov system). The cohomology groups of $K(\pi,1)$ coincide with those of $\pi$. Eilenberg–MacLane spaces were introduced by S. Eilenberg and S. MacLane .

References

[1a]	S. Eilenberg, S. MacLane, "Relations between homology and homotopy groups of spaces" Ann. of Math. , 46 (1945) pp. 480–509
[1b]	S. Eilenberg, S. MacLane, "Relations between homology and homotopy groups of spaces. II" Ann. of Math. , 51 (1950) pp. 514–533
[2]	R.E. Mosher, M.C. Tangora, "Cohomology operations and applications in homotopy theory" , Harper & Row (1968)
[4]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)