Eilenberg-MacLane space

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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).

1 Examples
2 Properties of Eilenberg–MacLane spaces
3 See also
4 References

[edit] Examples

The unit circle $S 1$ is a $K(\mathbb{Z},1)$ .
The infinite dimensional complex projective space $\mathbb{C}P^{\infty}$ is a $K(\mathbb{Z},2)$ .
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 $K(G,n)\times K(G',n)$ is a $K(G\times G',n)$ .

A K(G,n) can be constructed stage-by-stage, as a CW complex, starting with a wedge of 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.

[edit] Properties of Eilenberg–MacLane spaces

An important property of K(G,n) is that, for abelian 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

Hⁿ(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 Čech 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 Čech 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 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.

[edit] See also

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

[edit] References

Peter J. Huber (1961), Homotopical cohomology and Čech cohomology, Math. Ann. 144 , 73–76.
Kiiti Morita (1975), Čech cohomology and covering dimension for topological spaces, Fund. Math. 87, 31–52.

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Categories: Algebraic topology | Homotopy theory