Eilenberg-MacLane space

Let $\\pi$ be a discrete group. A based topological space $X$ is called an Eilenberg-MacLane space of type $K(\\pi,n)$ , where $n\\geq 1,$ if all the homotopy groups $\\pi_{k}(X)$ are trivial except for $\\pi_{n}(X),$ which is isomorphic to $\\pi.$ Clearly, for such a space to exist when $n\\geq 2,$ $\\pi$ must be abelian.

Given any group $\\pi,$ with $\\pi$ abelian if $n\\geq 2,$ there exists an Eilenberg-MacLane space of type $K(\\pi,n).$ Moreover, this space can be constructed as a CW complex. It turns out that any two Eilenberg-MacLane spaces of type $K(\\pi,n)$ are weakly homotopy equivalent. The Whitehead theorem then implies that there is a unique $K(\\pi,n)$ space up to homotopy equivalence in the category of topological spaces of the homotopy type of a CW complex. We will henceforth restrict ourselves to this category. With a slight abuse of notation, we refer to any such space as $K(\\pi,n).$

An important property of $K(\\pi,n)$ is that, for $\\pi$ abelian,there is a natural isomorphism

H^{n}(X;\\pi)\\cong[X,K(\\pi,n)]

of contravariant set-valued functors, where $[X,K(\\pi,n)]$ is the set of homotopy classes of based maps from $X$ to $K(\\pi,n).$ Thus one says that the $K(\\pi,n)$ are representing spaces for cohomology with coefficients in $\\pi.$

Remark 1.

Even when the group $\\pi$ is nonabelian, it can be seen that the set $[X,K(\\pi,1)]$ is naturally isomorphic to $\\mathop{\\mathrm{Hom}}(\\pi_{1}(X),\\pi)/\\pi;$ that is,to conjugacy classes of homomorphisms from $\\pi_{1}(X)$ to $\\pi.$ In fact, thisis a way to define $H^{1}(X;\\pi)$ when $\\pi$ is nonabelian.

Remark 2.

Though the above description does not include the case $n=0,$ it is natural to define a $K(\\pi,0)$ to be any space homotopy equivalent to $\\pi.$ The above statement about cohomology then becomes true for the reduced zeroth cohomology functor.