# 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, this is 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.

 Title Eilenberg-MacLane space Canonical name EilenbergMacLaneSpace Date of creation 2013-03-22 13:25:42 Last modified on 2013-03-22 13:25:42 Owner antonio (1116) Last modified by antonio (1116) Numerical id 6 Author antonio (1116) Entry type Definition Classification msc 55P20 Synonym Eilenberg-Mac Lane space Related topic NaturalTransformation Related topic LoopSpace Related topic HomotopyGroups Related topic RepresentableFunctor Related topic FundamentalGroupoid2 Related topic CohomologyGroupTheorem Related topic ProofOfCohomologyGroupTheorem Related topic OmegaSpectrum