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 $n\ge 1,$ 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 $n\ge 2,$ $\pi$ must be abelian.

Given any group $\pi ,$ with $\pi$ abelian if $n\ge 2,$ 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 $\mathrm{[}X\mathrm{,}K\mathit{}\mathrm{(}\pi \mathrm{,}\mathrm{1}\mathrm{)}\mathrm{]}$ is naturally isomorphic to $\mathrm{Hom}\mathrm{(}{\pi }_{\mathrm{1}}\mathit{}\mathrm{(}X\mathrm{)}\mathrm{,}\pi \mathrm{)}\mathrm{/}\pi \mathrm{;}$ that is, to conjugacy classes of homomorphisms from ${\pi }_{\mathrm{1}}\mathit{}\mathrm{(}X\mathrm{)}$ to $\pi \mathrm{.}$ In fact, this is a way to define $H^{\mathrm{1}}\mathit{}\mathrm{(}X\mathrm{;}\pi \mathrm{)}$ when $\pi$ is nonabelian.

Remark 2.

Though the above description does not include the case $n\mathrm{=}\mathrm{0}\mathrm{,}$ it is natural to define a $K\mathit{}\mathrm{(}\pi \mathrm{,}\mathrm{0}\mathrm{)}$ to be any space homotopy equivalent to $\pi \mathrm{.}$ 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