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 ${\displaystyle K(G,n)}$, if it has n-th homotopy group ${\displaystyle \pi _{n}(X)}$ isomorphic to G and all other homotopy groups trivial. If ${\displaystyle n>1}$ 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 ${\displaystyle K(G,n)}$.

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

Examples

• 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 as 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 ${\displaystyle K(F_{k},1)}$, where ${\displaystyle F_{k}}$ is 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.
• An infinite lens space given by the quotient ${\displaystyle S^{\infty }/(\mathbb {Z} /q)=L(\infty ,q)}$ is a ${\displaystyle K(\mathbb {Z} /q,1)}$. This can be shown using the long exact sequence on homotopy groups for the fibration ${\displaystyle \mathbb {Z} /q\to S^{\infty }\to L(\infty ,q)}$ since ${\displaystyle \pi _{1}(S^{\infty })=0}$ because the infinite sphere is contractible.[2] Note this includes ${\displaystyle \mathbb {RP} ^{\infty }}$ as a ${\displaystyle K(\mathbb {Z} /2,1)}$.
• The configuration space of ${\displaystyle n}$ points in the plane is a ${\displaystyle K(P_{n},1)}$, where ${\displaystyle P_{n}}$ is the pure braid group on ${\displaystyle n}$ strands.

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.

Remark on constructing higher Eilenberg-Maclane spaces

There are multiple techniques for constructing higher Eilenberg-Maclane spaces. One of which is to construct a Moore space ${\displaystyle M(A,n)}$ for an abelian group ${\displaystyle A}$ and iteratively kill the higher homotopy groups of ${\displaystyle M(A,n)}$ since the lower homotopy groups ${\displaystyle \pi _{i<n}=0}$ are all trivial. This follows from the Hurewicz theorem.

Another useful technique is to first construct ${\displaystyle K(G,1)}$ for every group ${\displaystyle G}$ using simplicial techniques,[3] and then construct the higher Eilenberg-Maclane spaces using homotopy cofibers. Note that for non-abelian ${\displaystyle G}$,

${\displaystyle K(G,2)\simeq K(G/[G,G],2)}$

since all higher homotopy groups are abelian. The higher groups can be constructed using the ${\displaystyle K(G,1)}$ because we can recursively use the homotopy cofiber of the fibration

${\displaystyle K(G,n)\to *}$

to construct ${\displaystyle K(G,n+1)}$, giving a fibration sequence

${\displaystyle K(G,n)\to *\to K(G,n+1)}$

which can be used to study the cohomology of ${\displaystyle K(G,n+1)}$ from ${\displaystyle K(G,n)}$ using the Leray spectral sequence. This was exploited by Jean-Pierre Serre while he studied the homotopy groups of spheres using the Postnikov system and spectral sequences.

One other technique is to use the geometric realization of simplicial abelian groups.[4] This gives an explicit presentations of simplicial abelian groups which represent Eilenberg-Maclane spaces. Another simplicial construction, in terms of classifying spaces and universal bundles, is given in J. Peter May's book.[5]

Properties of Eilenberg–MacLane spaces

Bijection between homotopy classes of maps and cohomology

An important property of ${\displaystyle K(G,n)}$ is that, for any abelian group G, and any CW-complex X, the set

${\displaystyle [X,K(G,n)]}$

of homotopy classes of maps from X to ${\displaystyle K(G,n)}$ is in natural bijection with the n-th singular cohomology group

${\displaystyle H^{n}(X;G)}$

of the space X. Thus one says that the ${\displaystyle [X,K(G,n)]}$ are representing spaces for cohomology with coefficients in G. Since

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

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 Kiiti Morita establishes a bijection with the n-th numerable Čech cohomology group for an arbitrary topological space X and G an arbitrary abelian group.

Loop spaces

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

Functoriality

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 ${\displaystyle a\colon G\to G'}$ is any homomorphism of abelian groups, then there is a non-empty set

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

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 ${\displaystyle S\circ T:=\{s\circ t:s\in S,t\in T\}.}$

Relation with Postnikov tower

Every CW-complex possesses a Postnikov tower, that is, it is homotopy equivalent to an iterated fibration whose fibers are Eilenberg–MacLane spaces.

Cohomology operations

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

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

with ${\displaystyle {\text{String}}(n)}$ the string group, and ${\displaystyle {\text{Spin}}(n)}$ the spin group. The relevance of ${\displaystyle K(\mathbb {Z} ,2)}$ lies in the fact that there are the homotopy equivalences

${\displaystyle K(\mathbb {Z} ,1)\simeq U(1)\simeq B\mathbb {Z} }$

for the classifying space ${\displaystyle B\mathbb {Z} }$, and the fact ${\displaystyle K(\mathbb {Z} ,2)\simeq BU(1)}$. Notice that because the complex spin group is a group extension

${\displaystyle 0\to K(\mathbb {Z} ,1)\to {\text{Spin}}^{\mathbb {C} }(n)\to {\text{Spin}}(n)\to 0}$

the String group can be thought of as a "higher" complex spin group extension, in the sense of higher group theory since the space ${\displaystyle K(\mathbb {Z} ,2)}$ is an example of a higher group. It can be thought of the topological realization of the groupoid ${\displaystyle \mathbf {B} U(1)}$ whose object is a single point and whose morphisms are the group ${\displaystyle U(1)}$. Because of these homotopical properties, 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.

See also

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

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.

1. (Papakyriakopoulos 1957) harv error: multiple targets (2×): CITEREFPapakyriakopoulos1957 (help)
2. "general topology - Unit sphere in $\mathbb{R}^\infty$ is contractible?". Mathematics Stack Exchange. Retrieved 2020-09-01.
3. Yin, Xi. "On Eilenberg-Maclane spaces" (PDF). Archived (PDF) from the original on 21 Aug 2018.
4. "gt.geometric topology - Explicit constructions of K(G,2)?". MathOverflow. Retrieved 2020-10-28.
5. May, J. Peter. A Concise Course in Algebraic Topology (PDF). Chapter 16, section 5: University of Chicago Press.CS1 maint: location (link)

References

Foundational articles

• 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.
• Eilenberg, Samuel; MacLane, Saunders (1954). "On the groups ${\displaystyle H(\Pi ,n)}$. III. Operations and obstructions". Annals of Mathematics. 60 (3): 513–557. doi:10.2307/1969849. MR 0065163.

Cartan seminar and applications

The Cartan seminar contains many fundamental results about Eilenberg-Maclane spaces including their homology and cohomology, and applications for calculating the homotopy groups of spheres.

• http://www.numdam.org/volume/SHC_1954-1955__7/

Computing integral cohomology rings

• Derived functors of the divided power functors
• (Co)homology of the Eilenberg-MacLane spaces K(G,n)

Applications

• Huber, Peter J. (1961). "Homotopical cohomology and Čech cohomology". Mathematische Annalen. 144 (1): 73–76. doi:10.1007/BF01396544. MR 0133821.
• 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, Christos D. (1957). "On Dehn's Lemma and the Asphericity of Knots". Proceedings of the National Academy of Sciences of the United States of America. 43 (1): 169–172. doi:10.1073/pnas.43.1.169. PMC 528404. PMID 16589993.
• Papakyriakopoulos, Christos D. (1957). "On Dehn's Lemma and the Asphericity of Knots". Annals of Mathematics. 66 (1): 1–26. doi:10.2307/1970113. JSTOR 1970113. PMC 528404.

Other encyclopedic references

• Rudyak, Yu.B. (2001) [1994], "Eilenberg−MacLane space", Encyclopedia of Mathematics, EMS Press
• Eilenberg-Mac Lane space in nLab