Eilenberg-MacLane space

In mathematics, an Eilenberg-MacLane space is a special kind of topological space that can be regarded as a building block for homotopy theory. These spaces are important in many contexts in algebraic topology, including stage-by-stage constructions of spaces, computations of homotopy groups of spheres, and definition of cohomology operations. The name is for Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.

Let "G" be a group and "n" a positive integer. A connected topological space "X" is called an Eilenberg–MacLane space of type "K"("G","n"), if it has "n"-th homotopy group "π"_n(X) isomorphic to "G" and all other homotopy groups trivial. If "n" > 1 then "G" must be abelian. Then an Eilenberg–Mac Lane space exists, as a CW-complex, and is unique up to a weak homotopy equivalence. By abuse of language, any such space is often called just "K"("G","n").

Examples

* The unit circle $S^1$ is a $K(mathbb\{Z\},1)$.
* The infinite-dimensional complex projective space $mathbb\{C\}P^\{infty\}$ is a model of $K(mathbb\{Z\},2)$.
* The infinite-dimensional real projective space $mathbb\{R\}P^\{infty\}$is a $K(mathbb\{Z\}\_2,1)$.
* The wedge sum of "k" unit circles $vee\_\{i=1\}^kS^1$ is a "K"("G","1") for "G" the free group on "k" generators.Further elementary examples can be constructed from these by using the obvious fact that the product $K(G,n)\; imes\; K(G\text{'},n)$ is a $K(G\; imes\; G\text{'},n)$.

A "K"("G","n") can be constructed stage-by-stage, as a CW complex, starting with a wedge of "n"-spheres, one for each generator of the group "G", and adding cells in (possibly infinite number of) higher dimensions so as to kill all extra homotopy.

Properties of Eilenberg–MacLane spaces

An important property of "K"("G","n") is that, for any abelian group "G", and any CW-complex "X", the set

: ["X", "K"("G","n")]

of homotopy classes of maps from "X" to "K"("G","n") is in natural bijection with the "n"-th singular cohomology group

:"H"^"n"("X"; "G")

of the space "X". Thus one says that the "K"("G","n") are representing spaces for cohomology with coefficients in "G".

Another version of this result, due to Peter J. Huber, establishes a bijection with the "n"-th ech cohomology group when "X" is Hausdorff and paracompact and "G" is countable, or when "X" is Hausdorff, paracompact and compactly generated and "G" is arbitrary. A further result of Morita establishes a bijection with the "n"-th numerable ech cohomology group for an arbitrary topological space "X" and "G" an arbitrary abelian group.

Every CW-complex possesses a Postnikov tower, that is, it is homotopy equivalent to an iterated fibration with fibers the Eilenberg–Mac Lane spaces.

There is a method due to Jean-Pierre Serre which allows one, at least theoretically, to compute homotopy groups of spaces using a spectral sequence for special fibrations with Eilenberg–Mac Lane spaces for fibers.

ee also

*Brown representability theorem, regarding representation spaces
*Moore space, the homology analogue.

References

*S. Eilenberg, S. MacLane, Relations between homology and homotopy groups of spaces Ann. of Math. 46 (1945) pp. 480–509
*S. Eilenberg, S. MacLane, Relations between homology and homotopy groups of spaces. II Ann. of Math. 51 (1950) pp. 514–533
*Peter J. Huber (1961), Homotopical cohomology and ech cohomology, "Mathematische Annalen" 144 , 73–76.
*Kiiti Morita (1975), ech cohomology and covering dimension for topological spaces, "Fundamenta Mathematicae" 87, 31–52.
*springer|title=Eilenberg-MacLane space|id=E/e035200|first=Yu.B.|last= Rudyak