View source for Eilenberg-MacLane space

{{TEX|done}}
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|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|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====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  S. Eilenberg,   S. MacLane,   "Relations between homology and homotopy groups of spaces"  ''Ann. of Math.'' , '''46'''  (1945)  pp. 480–509</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  S. Eilenberg,   S. MacLane,   "Relations between homology and homotopy groups of spaces. II"  ''Ann. of Math.'' , '''51'''  (1950)  pp. 514–533</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Mosher,   M.C. Tangora,   "Cohomology operations and applications in homotopy theory" , Harper &amp; Row  (1968)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , McGraw-Hill  (1966)</TD></TR></table>