Eilenberg–MacLane space

In mathematics, and algebraic topology in particular, an Eilenberg–MacLane space^{[note 1]} is a topological space with a single nontrivial homotopy group. As such, an Eilenberg–MacLane space is a special kind of topological space that can be regarded as a building block for homotopy theory; general topological spaces can be constructed from these via the Postnikov system. 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 $\pi _{n}(X)$ isomorphic to G and all other homotopy groups trivial. If $n>1$ then G must be abelian. Such a space exists, is a CW-complex, and is unique up to a weak homotopy equivalence. By abuse of language, any such space is often called just $K(G,n)$ .

A generalised Eilenberg-Maclane space is a space which has the homotopy type of a product of Eilenberg-Maclane spaces $\prod _{n}K(G_{n},n)$ .

Examples

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

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

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. The corresponding chain complex is given by the Dold–Kan correspondence.

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)=\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 $u\in H^{n}(K(G,n);G)$ corresponding to the identity. The above bijection is given by pullback of that element — $f\mapsto f^{*}u$ . 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 $n$ if $a:G\to G'$ is any homomorphism of Abelian groups, then there is a non-empty set

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

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

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.

Applications

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

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 $K(\mathbb {Z} ,n)$ can be used to start a short exact sequence that kills the homotopy group $\pi _{n+1}$ in a topological group.

Notes

^ 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. MR 13312) In this context it is therefore conventional to write the name without a space.

^ (Papakyriakopoulos 1957) harv error: multiple targets (2×): CITEREFPapakyriakopoulos1957 (help)
^ J.P. May, A Concise Course in Algebraic Topology University of Chicago Press (See chapter 16, section 5.)

References

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", Encyclopedia of Mathematics, EMS Press
Eilenberg-Mac Lane space at the nLab

[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. MR 13312) In this context it is therefore conventional to write the name without a space.

[2] (Papakyriakopoulos 1957) harv error: multiple targets (2×): CITEREFPapakyriakopoulos1957 (help)

[3] J.P. May, A Concise Course in Algebraic Topology University of Chicago Press (See chapter 16, section 5.)

[note 1]

[1]

[2]

Examples

Properties of Eilenberg–MacLane spaces

Applications

See also

Notes

References