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).

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

[edit] Examples

• The unit circle S¹ is a .
• The infinite-dimensional complex projective space is a .
• The infinite-dimensional real projective space is a .
• The wedge sum of k unit circles 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 is a .

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.

[edit] 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

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 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.

[edit] See also

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

[edit] 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 Čech cohomology, Mathematische Annalen 144 , 73–76.
• Kiiti Morita (1975), Čech cohomology and covering dimension for topological spaces, Fundamenta Mathematicae 87, 31–52.
• Rudyak, Yu.B. (2001), “Eilenberg-MacLane space”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104