Let be a discrete group. A based topological space is called an Eilenberg-MacLane space of type , where if all the homotopy groups are trivial except for which is isomorphic to Clearly, for such a space to exist when must be abelian.
Given any group with abelian if there exists an Eilenberg-MacLane space of type Moreover, this space can be constructed as a CW complex. It turns out that any two Eilenberg-MacLane spaces of type are weakly homotopy equivalent. The Whitehead theorem then implies that there is a unique 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
An important property of is that, for abelian, there is a natural isomorphism
Though the above description does not include the case it is natural to define a to be any space homotopy equivalent to The above statement about cohomology then becomes true for the reduced zeroth cohomology functor.
|Date of creation||2013-03-22 13:25:42|
|Last modified on||2013-03-22 13:25:42|
|Last modified by||antonio (1116)|
|Synonym||Eilenberg-Mac Lane space|