Editing Eilenberg–MacLane space

{{short description|Topological space with only one nontrivial homotopy group}}
In [[mathematics]], specifically [[algebraic topology]], an '''Eilenberg–MacLane space'''<ref group="note">[[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.</ref> is a [[topological space]] with a single nontrivial [[homotopy group]].

Let ''G'' be a group and ''n'' a positive [[integer]]. A [[connected space|connected]] topological space ''X'' is called an Eilenberg–MacLane space of type <math>K(G,n)</math>, if it has ''n''-th [[homotopy group]] <math>\pi_n(X)</math> [[group isomorphism|isomorphic]] to ''G'' and all other homotopy groups [[trivial group|trivial]]. Assuming that ''G'' is [[abelian group|abelian]] in the case that <math>n > 1</math>, Eilenberg–MacLane spaces of type <math>K(G,n)</math> always exist, and are all weak homotopy equivalent. Thus, one may consider <math>K(G,n)</math> as referring to a weak homotopy equivalence class of spaces. It is common to refer to any representative as "a <math>K(G,n)</math>" or as "a model of <math>K(G,n)</math>". Moreover, it is common to assume that this space is a CW-complex (which is always possible via [[CW approximation]]).

The name is derived from [[Samuel Eilenberg]] and [[Saunders Mac Lane]], who introduced such spaces in the late 1940s.

As such, an Eilenberg–MacLane space is a special kind of [[topological space]] that in [[homotopy theory]] can be regarded as a building block for CW-complexes via [[fibration | fibrations]] in a [[Postnikov system]]. These spaces are important in many contexts in [[algebraic topology]], including computations of [[homotopy groups of spheres]], definition of [[cohomology operation]]s, and for having a strong connection to [[ cohomology | singular cohomology]].

A generalised Eilenberg–MacLane space is a space which has the homotopy type of a [[product topology|product]] of Eilenberg–MacLane spaces
<math>\prod_{m}K(G_m,m)</math>.

==Examples==
* The [[unit circle]] <math>S^1</math> is a <math>K(\Z,1)</math>.
* The infinite-dimensional [[complex projective space]] <math>\mathbb{CP}^{\infty}</math> is a model of <math>K(\Z,2)</math>.
* The infinite-dimensional [[real projective space]] <math>\mathbb{RP}^{\infty}</math> is a <math>K(\Z/2,1)</math>.
* The [[wedge sum]] of ''k'' [[unit circle]]s <math>\textstyle\bigvee_{i=1}^k S^1</math> is a <math>K(F_k,1)</math>, where <math>F_k</math> is the [[free group]] on ''k'' generators.
* The complement to any connected [[Knot theory|knot]] or graph in a 3-dimensional sphere <math>S^3</math> is of type <math>K(G,1)</math>; this is called the "[[Aspherical space|asphericity]] of knots", and is a 1957 theorem of [[Christos Papakyriakopoulos]].<ref>{{cite journal |last1=Papakyriakopoulos |first1=C. D. |title=On Dehn's lemma and the asphericity of knots |journal=Proceedings of the National Academy of Sciences |date=15 January 1957 |volume=43 |issue=1 |pages=169–172 |doi=10.1073/pnas.43.1.169 |pmid=16589993 |pmc=528404 |bibcode=1957PNAS...43..169P |doi-access=free }}</ref>
* Any [[compact space|compact]], connected, [[non-positive curvature|non-positively curved]] [[manifold]] ''M'' is a <math>K(\Gamma,1)</math>, where <math>\Gamma=\pi_1(M)</math> is the [[fundamental group]] of ''M''. This is a consequence of the [[Cartan–Hadamard theorem]].
* An infinite [[lens space]] <math> L(\infty, q)</math> given by the quotient of <math>S^\infty</math> by the free action <math> (z \mapsto e^{2\pi i m/q}z) </math> for <math> m \in \Z/q </math> is a <math>K(\mathbb{Z}/q,1)</math>. This can be shown using [[Covering space#Deck transformation|covering space theory]] and the fact that the infinite dimensional sphere is [[Contractible space|contractible]].<ref>{{Cite web|title=general topology - Unit sphere in $\mathbb{R}^\infty$ is contractible?|url=https://math.stackexchange.com/q/282268 |access-date=2020-09-01|website=Mathematics Stack Exchange}}</ref> Note this includes <math>\mathbb{RP}^{\infty}</math> as a <math>K(\Z/2,1)</math>.
* The [[Configuration space (mathematics)|configuration space]] of <math>n</math> points in the plane is a <math>K(P_n,1)</math>, where <math>P_n</math> is the [[Braid_group#Relation_with_symmetric_group_and_the_pure_braid_group|pure braid group]] on <math>n</math> strands.
* Correspondingly, the [[Configuration space (mathematics)| {{var|n}}th unordered configuration space]] of <math> \mathbb{R}^2 </math> is a <math>K(B_n,1)</math>, where <math>B_n</math> denotes the [[Braid group |{{var|n}}-strand braid group]]. <ref>Lucas Williams [https://arxiv.org/pdf/1911.11186.pdf "Configuration spaces for the working undergraduate"], ''arXiv'', November 5, 2019. Retrieved 2021-06-14</ref>
* The [[Symmetric product (topology) |infinite symmetric product]] <math> SP(S^n)</math> of a [[Sphere|''n''-sphere]] is a <math>K(\mathbb{Z},n)</math>. More generally <math> SP(M(G,n)) </math> is a <math> K(G,n) </math> for all [[Moore space (algebraic topology) | Moore spaces]] <math> M(G,n) </math>.

Some further elementary examples can be constructed from these by using the fact that the product <math>K(G,n) \times K(H,n)</math> is <math>K(G\times H,n)</math>. For instance the [[Torus#n-dimensional_torus| {{var|n}}-dimensional Torus]] <math>\mathbb{T}^n</math> is a <math> K(\Z^n, 1)</math>.

==Constructing Eilenberg–MacLane spaces==
For <math> n = 1 </math> and <math> G </math> an arbitrary [[Group (mathematics)|group]] the construction of <math> K(G,1) </math> is identical to that of the [[classifying space]] of the group <math> G </math>. Note that if G has a torsion element, then every CW-complex of type K(G,1) has to be infinite-dimensional.

There are multiple techniques for constructing higher Eilenberg–MacLane spaces. One of which is to construct a [[Moore space (algebraic topology)|Moore space]] <math>M(A,n)</math> for an abelian group <math>A</math>: Take the [[wedge sum|wedge]] of ''n''-[[sphere]]s, one for each generator of the group ''A'' and realise the relations between these generators by attaching ''(n+1)''-cells via corresponding maps in <math> \pi_n(\bigvee S^n) </math> of said wedge sum. Note that the lower homotopy groups <math>\pi_{i < n} (M(A,n)) </math> are already trivial by construction. Now iteratively kill all higher homotopy groups <math>\pi_{i > n} (M(A,n)) </math> by successively attaching cells of dimension greater than <math> n + 1 </math>, and define <math> K(A,n) </math> as [[direct limit]] under inclusion of this iteration.

Another useful technique is to use the geometric realization of [[Simplicial abelian group|simplicial abelian groups]].<ref>{{Cite web|title=gt.geometric topology - Explicit constructions of K(G,2)?|url=https://mathoverflow.net/questions/61546/explicit-constructions-of-kg-2|access-date=2020-10-28|publisher=[[MathOverflow]]}}</ref> This gives an explicit presentation of simplicial abelian groups which represent Eilenberg–MacLane spaces.

Another simplicial construction, in terms of [[classifying space]]s and [[universal bundle]]s, is given in [[J. Peter May]]'s book.<ref>{{cite book|last=May|first=J. Peter|author-link=J. Peter May|url= http://www.maths.ed.ac.uk/~aar/papers/maybook.pdf|title=A Concise Course in Algebraic Topology|publisher=[[University of Chicago Press]]|place=Chapter 16, section 5}}</ref>

Since taking the loop space lowers the homotopy groups by one slot, we have a canonical homotopy equivalence <math>K(G,n)\simeq\Omega K(G,n+1)</math>, hence there is a fibration sequence
:<math>K(G,n) \to * \to K(G,n+1)</math>.
Note that this is not a cofibration sequence ― the space <math>K(G,n+1)</math> is not the homotopy cofiber of <math>K(G,n) \to *</math>.

This fibration sequence can be used to study the cohomology of <math>K(G,n+1)</math> from <math>K(G,n)</math> 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.

==Properties of Eilenberg–MacLane spaces==

===Bijection between homotopy classes of maps and cohomology===
An important property of <math>K(G, n)</math>'s is that for any abelian group ''G'', and any based CW-complex ''X'', the set <math>[X, K(G,n)]</math> of based homotopy classes of based maps from ''X'' to <math> K(G,n)</math> is in natural bijection with the ''n''-th [[singular cohomology]] group <math>H^n(X, G)</math> of the space ''X''. Thus one says that the <math>K(G,n)s</math> are [[representing space]]s for singular cohomology with coefficients in ''G''. Since
:<math>\begin{array}{rcl}
H^n(K(G,n),G) &=& \operatorname{Hom}(H_n(K(G,n);\Z), G) \\
&=& \operatorname{Hom}(\pi_n(K(G,n)), G) \\
&=& \operatorname{Hom}(G,G),
\end{array}</math>
there is a distinguished element <math>u \in H^n(K(G,n),G)</math> corresponding to the identity. The above bijection is given by the pullback of that element <math> f \mapsto f^*u </math>. This is similar to the [[Yoneda lemma]] of [[category theory]].

A constructive proof of this theorem can be found here,<ref>Xi Yin [http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/AT.pdf "On Eilenberg-MacLanes Spaces"] {{Webarchive|url=https://web.archive.org/web/20210929112440/http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/AT.pdf |date=2021-09-29 }}, Retrieved 2021-06-14</ref> another making use of the relation between [[spectrum (topology)|omega-spectra]] and [[Cohomology#Axioms and generalized cohomology theories |generalized reduced cohomology theories]] can be found here,<ref>[[Allen Hatcher]] [https://pi.math.cornell.edu/~hatcher/AT/AT.pdf "Algebraic Topology"], ''[[Cambridge University Press]]'', 2001. Retrieved 2021-06-14</ref> and the main idea is sketched later as well.

=== Loop spaces and Omega spectra ===
The [[loop space]] of an Eilenberg–MacLane space is again an Eilenberg–MacLane space: <math>\Omega K(G,n) \cong K(G,n-1)</math>. Further there is an adjoint relation between the loop-space and the reduced suspension: <math>[\Sigma X, Y] = [X,\Omega Y]</math>, which gives <math>[X,K(G,n)] \cong [X,\Omega^2K(G,n+2)]</math> the structure of an abelian group, where the operation is the concatenation of loops. This makes the bijection <math>[X, K(G,n)] \to H^n(X, G) </math> mentioned above a group isomorphism.

Also this property implies that Eilenberg–MacLane spaces with various ''n'' form an [[spectrum (topology)|omega-spectrum]], called an "Eilenberg–MacLane spectrum". This spectrum defines via <math> X \mapsto h^n(X):= [X, K(G,n)] </math> a reduced cohomology theory on based CW-complexes and for any reduced cohomology theory <math> h^* </math> on CW-complexes with <math> h^n(S^0) = 0 </math> for <math> n \neq 0</math> there is a natural isomorphism  <math> h^n(X) \cong \tilde{H}^n(X, h^0(S^0)) </math>, where <math> \tilde{H^*} </math>  denotes reduced singular cohomology. Therefore these two cohomology theories coincide.

In a more general context, [[Brown's representability theorem|Brown representability]] says that every reduced cohomology theory on based CW-complexes comes from an [[spectrum (topology)|omega-spectrum]].

===Relation with homology===
For a fixed abelian group <math> G </math> there are maps on the [[stable homotopy theory | stable homotopy groups]] 
:<math> \pi_{q+n}^s(X \wedge K(G,n)) \cong \pi_{q+n+1}^s(X \wedge \Sigma K(G,n)) \to \pi_{q+n+1}^s(X \wedge K(G,n+1)) </math>
induced by the map <math> \Sigma K(G,n) \to K(G,n+1)</math>. Taking the direct limit over these maps, one can verify that this defines a reduced homology theory 
:<math>h_q(X) = \varinjlim _{n} \pi_{q+n}^s(X \wedge K(G,n)) </math>
on CW complexes. Since <math> h_q(S^0) = \varinjlim \pi_{q+n}^s(K(G,n)) </math> vanishes for <math> q \neq 0</math>, <math> h_* </math>  agrees with reduced singular homology <math>\tilde{H}_*(\cdot,G) </math> with coefficients in G on CW-complexes.

=== 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 <math>n</math> if <math>a\colon G \to G'</math> is any homomorphism of abelian groups, then there is a [[empty set|non-empty]] set

: <math>K(a,n) = \{[f]: f\colon K(G,n) \to K(G',n), H_n(f) = a\},</math>

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

=== Relation with Postnikov/Whitehead towers ===
Every connected CW-complex <math> X </math>  possesses a [[Postnikov tower]], that is an inverse system of spaces:

:<math>\cdots \to X_3 \xrightarrow{p_3} X_2 \xrightarrow{p_2}  X_1 \simeq K(\pi_1(X), 1) </math>
such that for every <math> n </math>:
#there are commuting maps <math> X \to X_n </math>, which induce isomorphism on <math> \pi_i </math> for <math> i \leq n</math> ,
#<math> \pi_i(X_n) = 0 </math> for <math> i > n </math>,
#the maps <math> X_n \xrightarrow{p_n} X_{n-1} </math> are fibrations with fiber <math> K(\pi_n(X),n)</math>.

Dually there exists a [[Whitehead tower]], which is a sequence of CW-complexes: 

:<math>\cdots \to X_3 \to X_2 \to X_1 \to X </math>
such that for every <math>n</math>: 
# the maps <math> X_n \to X </math> induce isomorphism on <math>\pi_i </math> for <math>i > n</math>,
# <math>X_n</math> is [[n-connected space|n-connected]],
# the maps <math> X_n \to X_{n-1}</math> are fibrations with fiber <math> K(\pi_n(X), n-1)</math>.

With help of [[Serre spectral sequence| Serre spectral sequences]] computations of higher [[homotopy group]]s of spheres can be made. For instance <math> \pi_4(S^3) </math> and <math> \pi_5(S^3) </math> using a Whitehead tower of <math> S^3 </math> can be found here,<ref>Xi Yin [http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/AT.pdf "On Eilenberg-MacLanes Spaces"] {{Webarchive|url=https://web.archive.org/web/20210929112440/http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/AT.pdf |date=2021-09-29 }}, Retrieved 2021-06-14</ref> more generally those of <math> \pi_{n+i}(S^n) \ i \leq 3 </math> using a Postnikov systems can be found here. <ref>Allen Hatcher [https://pi.math.cornell.edu/~hatcher/AT/ATch5.pdf Spectral Sequences], Retrieved 2021-04-25</ref>

=== Cohomology operations ===
For fixed natural numbers ''m,n'' and abelian groups ''G,H'' exists a bijection between the set of all [[cohomology operation]]s <math>\Theta :H^m(\cdot,G) \to H^n(\cdot,H) </math> and <math> H^n(K(G,m),H) </math> defined by <math> \Theta \mapsto \Theta(\alpha) </math>, where <math> \alpha \in H^m(K(G,m),G) </math> is a [[fundamental class]]. 

As a result, cohomology operations cannot decrease the degree of the cohomology groups and degree preserving cohomology operations are corresponding
to coefficient homomorphism <math> \operatorname{Hom}(G,H) </math>. This follows from the [[Universal coefficient theorem | Universal coefficient theorem for cohomology]] and the [[n-connected space | (m-1)-connectedness]] of <math> K(G,m) </math>.

Some interesting examples for cohomology operations are [[Steenrod algebra |Steenrod Squares and Powers]], when <math> G=H</math> are [[Cyclic group|finite cyclic groups]]. When studying those the importance of the cohomology of <math> K(\Z /p ,n) </math> with coefficients in <math> \Z /p </math> becomes apparent quickly;<ref>Cary Malkiewich [https://web.archive.org/web/20170815173103/http://www.math.uiuc.edu/~cmalkiew/steenrod.pdf "The Steenrod algebra"], Retrieved 2021-06-14</ref> some extensive tabeles of those groups can be found here. <ref>[http://doc.rero.ch/record/482/files/Clement_these.pdf Integral Cohomology of Finite Postnikov Towers] </ref>

=== Group (co)homology ===
One can define the [[ group cohomology |group (co)homology]] of G with coefficients in the group A as the singular (co)homology of the Eilenberg–MacLane space <math> K(G,1) </math> with coefficients in A.

=== Further 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 [[Postnikov system#Whitehead tower|Whitehead tower]] arising from the [[short exact sequence]]
:<math>0\to K(\Z,2)\to \operatorname{String}(n)\to \operatorname{Spin}(n)\to 0</math>
with <math>\operatorname{String}(n)</math> the [[string group]], and <math>\operatorname{Spin}(n)</math> the [[spin group]]. The relevance of <math>K(\Z,2)</math> lies in the fact that there are the homotopy equivalences
:<math>K(\mathbb{Z},1) \simeq U(1) \simeq B\Z</math>
for the [[classifying space]] <math>B\Z</math>, and the fact <math>K(\Z,2) \simeq BU(1)</math>. Notice that because the complex spin group is a group extension
:<math>0\to K(\Z,1) \to \operatorname{Spin}^\Complex(n) \to \operatorname{Spin}(n) \to 0</math>,
the String group can be thought of as a "higher" complex spin group extension, in the sense of [[Higher group|higher group theory]] since the space <math>K(\Z,2)</math> is an example of a higher group. It can be thought of the topological realization of the [[groupoid]] <math>\mathbf{B}U(1)</math> whose object is a single point and whose morphisms are the group <math>U(1)</math>. Because of these homotopical properties, the construction generalizes: any given space <math>K(\Z,n)</math> can be used to start a short exact sequence that kills the homotopy group <math>\pi_{n+1}</math> in a [[topological group]].

==See also==
*[[Classifying space]], for the case <math> n = 1 </math>
*[[Brown representability theorem]], regarding representation spaces
*[[Moore space (algebraic topology)|Moore space]], the homology analogue
*[[Hopf–Whitney theorem]], application to calculate homotopy classes

==Notes==
{{reflist|group="note"}}
{{reflist}}

==References==

=== Foundational articles ===
*{{citation|first1=Samuel|last1= Eilenberg|author-link1=Samuel Eilenberg| first2=Saunders|last2= MacLane|author-link2=Saunders MacLane|title=   Relations between homology and homotopy groups of spaces |journal= [[Annals of Mathematics]] |volume=  46  |series=(Second Series)|year=1945|issue=3|  pages=480–509|doi= 10.2307/1969165|jstor= 1969165|mr=0013312}}
* {{cite journal|first1=Samuel|last1= Eilenberg|author-link1=Samuel Eilenberg| first2=Saunders|last2= MacLane|author-link2=Saunders MacLane|title=Relations between homology and homotopy groups of spaces. II |journal= [[Annals of Mathematics]] |series=(Second Series)|volume=  51|issue=3  |year=1950|  pages=514–533|doi=10.2307/1969365|jstor= 1969365|mr=0035435}}
*{{Cite journal|first1=Samuel|last1= Eilenberg|author-link1=Samuel Eilenberg| first2=Saunders|last2= MacLane|author-link2=Saunders MacLane|year=1954|title=On the groups <math>H(\Pi,n)</math>. III. Operations and obstructions| url=https://www.jstor.org/stable/1969849|journal=[[Annals of Mathematics]]|volume=60|issue=3|pages=513–557|doi=10.2307/1969849|jstor= 1969849|mr=0065163}}

=== Cartan seminar and applications ===
The Cartan seminar contains many fundamental results about Eilenberg–MacLane spaces including their homology and cohomology, and [[Postnikov system|applications]] for calculating the homotopy groups of spheres.

* http://www.numdam.org/volume/SHC_1954-1955__7/ {{Webarchive|url=https://web.archive.org/web/20220425232321/http://www.numdam.org/volume/SHC_1954-1955__7/ |date=2022-04-25 }}

=== Computing integral cohomology rings ===

* [[arxiv:1312.5676|Derived functors of the divided power functors]]
* [http://doc.rero.ch/record/482/files/Clement_these.pdf Integral Cohomology of Finite Postnikov Towers]
* [https://mathoverflow.net/questions/24754/cohomology-of-the-eilenberg-maclane-spaces-kg-n (Co)homology of the Eilenberg-MacLane spaces K(G,n)]

=== Other encyclopedic references ===
* [https://encyclopediaofmath.org/wiki/Eilenberg-MacLane_space Encyclopedia of Mathematics]
*{{nlab|id=Eilenberg-Mac+Lane+space|title=Eilenberg-Mac Lane space}}

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[[Category:Homotopy theory]]