Eilenberg–MacLane space

In mathematics, an Eilenberg–MacLane space[note 1] 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 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 πn(X) isomorphic to G and all other homotopy groups trivial. If n > 1 then G must be abelian. Then an Eilenberg–MacLane 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

Examples

Some further elementary examples can be constructed from these by using the obvious fact that the product K(G,n) × K(H,n) is K(G × H,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

Hn(X; G)

of the space X. Thus one says that the K(G,n) are representing spaces for cohomology with coefficients in G. Since H^n(K(G,n);G) = \mathrm{Hom}(H_n(K(G,n);\mathbb{Z}), G) = \mathrm{Hom}(\pi_n(K(G,n)), G) = \mathrm{Hom}(G,G), there is a distinguished element u \in H^n(K(G,n);G) corresponding to the identity. The above bijection is given by pullback of that element —  f \mapsto f^*u .

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–MacLane 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–MacLane spaces for fibers.

The cohomology groups of Eilenberg–MacLane spaces can be used to classify all cohomology operations.

See also

Notes

  1. ^ Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. MR13312.) In this context it is therefore conventional to write the name without a space.

References