Eilenberg-MacLane space

Let π be a discrete group. A based topological spacePlanetmathPlanetmath X is called an Eilenberg-MacLane space of type K(π,n), where n1, if all the homotopy groupsMathworldPlanetmath πk(X) are trivial except for πn(X), which is isomorphicPlanetmathPlanetmathPlanetmath to π. Clearly, for such a space to exist when n2, π must be abelianMathworldPlanetmath.

Given any group π, with π abelian if n2, there exists an Eilenberg-MacLane space of type K(π,n). Moreover, this space can be constructed as a CW complex. It turns out that any two Eilenberg-MacLane spaces of type K(π,n) are weakly homotopy equivalent. The Whitehead theoremMathworldPlanetmath then implies that there is a unique K(π,n) space up to homotopy equivalenceMathworldPlanetmathPlanetmath in the category of topological spaces of the homotopy type of a CW complex. We will henceforth restrict ourselves to this categoryMathworldPlanetmath. With a slight abuse of notation, we refer to any such space as K(π,n).

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


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

Remark 1.

Even when the group π is nonabelianPlanetmathPlanetmathPlanetmath, it can be seen that the set [X,K(π,1)] is naturally isomorphic to Hom(π1(X),π)/π; that is, to conjugacy classesMathworldPlanetmathPlanetmath of homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from π1(X) to π. In fact, this is a way to define H1(X;π) when π is nonabelian.

Remark 2.

Though the above description does not include the case n=0, it is natural to define a K(π,0) to be any space homotopy equivalent to π. 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