# 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 |