Eilenberg–MacLane space

• The unit circle ${\displaystyle S^{1}}$ is a ${\displaystyle K(\mathbb {Z} ,1)}$.
• The infinite-dimensional complex projective space ${\displaystyle \mathbb {CP} ^{\infty }}$ is a model of ${\displaystyle K(\mathbb {Z} ,2)}$. Its cohomology ring is ${\displaystyle \mathbb {Z} [x]}$, namely the free polynomial ring on a single 2-dimensional generator ${\displaystyle x}$ in degree 2. The generator can be represented in de Rham cohomology by the Fubini–Study 2-form. An application of ${\displaystyle K(\mathbb {Z} ,2)}$ is described at Abstract nonsense.
• The infinite-dimensional real projective space ${\displaystyle \mathbb {RP} ^{\infty }}$ is a ${\displaystyle K(\mathbb {Z} _{2},1)}$.
• The wedge sum of k unit circles ${\displaystyle \textstyle \bigvee _{i=1}^{k}S^{1}}$ is a ${\displaystyle K(G,1)}$ for G the free group on k generators.
• The complement to any knot in a 3-dimensional sphere ${\displaystyle S^{3}}$ is of type ${\displaystyle K(G,1)}$; this is called the "asphericity of knots", and is a 1957 theorem of Christos Papakyriakopoulos.[1]
• Any compact, connected, non-positively curved manifold M is a ${\displaystyle K(\Gamma ,1)}$, where ${\displaystyle \Gamma =\pi _{1}(M)}$ is the fundamental group of M.

Some further elementary examples can be constructed from these by using the fact that the product ${\displaystyle K(G,n)\times K(H,n)}$ is ${\displaystyle K(G\times H,n)}$.

A ${\displaystyle K(G,n)}$ 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. The corresponding chain complex is given by the Dold–Kan correspondence.

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

${\displaystyle H^{n}(K(G,n);G)=\operatorname {Hom} (H_{n}(K(G,n);\mathbf {Z} ),G)=\operatorname {Hom} (\pi _{n}(K(G,n)),G)=\operatorname {Hom} (G,G),}$

there is a distinguished element ${\displaystyle u\in H^{n}(K(G,n);G)}$ corresponding to the identity. The above bijection is given by pullback of that element — ${\displaystyle f\mapsto f^{*}u}$. This is similar to the Yoneda lemma of category theory.

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.

Another construction, in terms of classifying spaces and universal bundles, is given in May.[2]

The loop space of an Eilenberg–MacLane space is also an Eilenberg–MacLane space: ΩK(G, n) = K(G, n − 1). This property implies that Eilenberg–MacLane spaces with various n form an omega-spectrum, called Eilenberg–MacLane spectrum. This spectrum corresponds to the standard homology and cohomology theory.

It follows from the universal coefficient theorem for cohomology that the Eilenberg MacLane space is a quasi-functor of the group; that is, for each positive integer ${\displaystyle n}$ if ${\displaystyle a:G\to G'}$ is any homomorphism of Abelian groups, then there is a non-empty set

${\displaystyle K(a,n)=\{[f]:f:K(G,n)\to K(G',n),H_{n}(f)=a\},}$

satisfying ${\displaystyle K(a\circ b,n)\supset K(a,n)\circ K(b,n){\text{ and }}1\in K(1,n),}$ where ${\displaystyle [f]}$ denotes the homotopy class of a continuous map ${\displaystyle f}$ and ${\displaystyle S\circ T:=\{s\circ t:s\in S,t\in T\}.}$

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.

The loop space construction described above is used in string theory to obtain, for example, the string group, the fivebrane group and so on, as the Whitehead tower arising from the short exact sequence

${\displaystyle 0\rightarrow K(\mathbb {Z} ,2)\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow 0}$

with String(n) the string group, and Spin(n) the spin group. The construction generalizes: any given space ${\displaystyle K(\mathbb {Z} ,n)}$ can be used to start a short exact sequence that kills the homotopy group ${\displaystyle \pi _{n+1}}$ in a topological group.

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

• Eilenberg, Samuel; MacLane, Saunders (1945), "Relations between homology and homotopy groups of spaces", Annals of Mathematics, (Second Series), 46 (3): 480–509, doi:10.2307/1969165, MR 0013312
• Eilenberg, Samuel; MacLane, Saunders (1950), "Relations between homology and homotopy groups of spaces. II", Annals of Mathematics, (Second Series), 51 (3): 514–533, doi:10.2307/1969365, MR 0035435
• Peter J. Huber (1961), Homotopical cohomology and Čech cohomology, Mathematische Annalen 144 , 73–76.
• Morita, Kiiti (1975). "Čech cohomology and covering dimension for topological spaces". Fundamenta Mathematicae. 87: 31–52. doi:10.4064/fm-87-1-31-52.
• Papakyriakopoulos, C. D. (1957). "On Dehn's Lemma and the Asphericity of Knots". Proc. Natl. Acad. Sci. USA. 43 (1): 169–172. doi:10.1073/pnas.43.1.169. PMC 528404. PMID 16589993.
• Papakyriakopoulos, C. D. (1957). "On Dehn's Lemma and the Asphericity of Knots". Ann. Math. 66 (1): 1–26. doi:10.2307/1970113. JSTOR 1970113. PMC 528404.
• Rudyak, Yu.B. (2001) [1994], "Eilenberg−MacLane space", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Eilenberg-Mac Lane space in nLab