Editing Classifying space

{{Short description|Quotient of a weakly contractible space by a free action}}
In [[mathematics]], specifically in [[homotopy theory]], a '''classifying space''' ''BG'' of a [[topological group]] ''G'' is the quotient of a [[weakly contractible]] space ''EG'' (i.e., a topological space all of whose [[homotopy group]]s are trivial) by a proper [[free action]] of ''G''. It has the property that any ''G'' [[principal bundle]] over a [[paracompact]] manifold is isomorphic to a [[pullback bundle|pullback]] of the principal bundle <math>EG \to BG</math>.<ref>{{Citation | last1=Stasheff | first1=James D.|author-link=Jim Stasheff | title=Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) | publisher=[[American Mathematical Society]] | year=1971 | chapter=''H''-spaces and classifying spaces: foundations and recent developments | pages= 247–272 Theorem 2 |url=http://www.ams.org/books/pspum/022/ |mr=0321079 |chapter-url={{GBurl|p-wCCAAAQBAJ|p=247}} |doi=10.1090/pspum/022/0321079 |isbn=978-0-8218-9308-1
}}</ref> As explained later, this means that classifying spaces [[representable functor|represent]] a set-valued [[functor]] on the [[homotopy category]] of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of [[topological space]]s, such as [[Sierpiński space]]. This notion is generalized by the notion of [[classifying topos]]. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. 

For a [[discrete group]] ''G'', ''BG'' is a [[connected space|path-connected]] [[topological space]] ''X'' such that the [[fundamental group]] of ''X'' is isomorphic to ''G'' and the higher [[homotopy groups]] of ''X'' are [[trivial group|trivial]]; that is, ''BG'' is an [[Eilenberg–MacLane space]], specifically a ''K''(''G'', 1).

==Motivation==
An example of a classifying space for the [[infinite cyclic group]] ''G'' is the [[circle]] as ''X''. When ''G'' is a [[discrete group]], another way to specify the condition on ''X'' is that the [[universal cover]] ''Y'' of ''X'' is [[contractible]]. In that case the projection map

:<math>\pi\colon Y\longrightarrow X\ </math>

becomes a [[fiber bundle]] with structure group ''G'', in fact a [[principal bundle]] for ''G''. The interest in the classifying space concept really arises from the fact that in this case ''Y'' has a [[universal property]] with respect to principal ''G''-bundles, in the [[homotopy category]]. This is actually more basic than the condition that the higher homotopy groups vanish: the fundamental idea is, given ''G'', to find such a contractible space ''Y'' on which ''G'' acts ''[[Group action (mathematics)#Types of actions|freely]]''. (The [[weak equivalence (homotopy theory)|weak equivalence]] idea of homotopy theory relates the two versions.) In the case of the circle example, what is being said is that we remark that an infinite cyclic group ''C'' acts freely on the [[real line]] ''R'', which is contractible. Taking ''X'' as the [[Quotient space (topology)|quotient space]] circle, we can regard the projection π from ''R'' = ''Y'' to ''X'' as a [[helix]] in geometrical terms, undergoing projection from three dimensions to the plane. What is being claimed is that π has a universal property amongst principal ''C''-bundles; that any principal ''C''-bundle in a definite way 'comes from' π.

==Formalism==
A more formal statement takes into account that ''G'' may be a [[topological group]] (not simply a ''discrete group''), and that [[Group action (mathematics)|group action]]s of ''G'' are taken to be continuous; in the absence of continuous actions the classifying space concept can be dealt with, in homotopy terms, via the [[Eilenberg–MacLane space]] construction. In homotopy theory the definition of a topological space ''BG'', the '''classifying space''' for principal ''G''-bundles, is given, together with the space ''EG'' which is the '''total space''' of the [[universal bundle]] over ''BG''. That is, what is provided is in fact a [[continuous mapping]]

:<math>\pi\colon EG\longrightarrow BG. </math>

Assume that the homotopy category of [[CW complex]]es is the underlying category, from now on. The ''classifying'' property required of ''BG'' in fact relates to π. We must be able to say that given any principal ''G''-bundle

:<math>\gamma\colon Y\longrightarrow Z\ </math>

over a space ''Z'', there is a '''classifying map''' φ from ''Z'' to ''BG'', such that <math>\gamma</math>  is the [[pullback of a bundle|pullback]] of π along φ. In less abstract terms, the construction of <math>\gamma</math> by 'twisting' should be reducible via φ to the twisting already expressed by the construction of π.

For this to be a useful concept, there evidently must be some reason to believe such spaces ''BG'' exist. The early work on classifying spaces introduced constructions (for example, the [[bar construction]]), that gave concrete descriptions of ''BG'' as a [[simplicial complex]] for an arbitrary discrete group. Such constructions make evident the connection with [[group cohomology]]. 

Specifically, let ''EG'' be the [[Delta set|weak simplicial complex]] whose ''n-'' simplices are the ordered (''n''+1)-tuples <math>[g_0,\ldots,g_n]</math> of elements of ''G''. Such an ''n-''simplex attaches to the (n−1) simplices <math>[g_0,\ldots,\hat g_i,\ldots,g_n]</math> in the same way a standard simplex attaches to its faces, where <math>\hat g_i</math> means this vertex is deleted. The complex EG is contractible. The group ''G'' acts on ''EG'' by left multiplication, 
:<math>g\cdot[g_0,\ldots,g_n ]=[gg_0,\ldots,gg_n],</math> 
and only the identity ''e'' takes any simplex to itself. Thus the action of ''G'' on ''EG'' is a covering space action and the quotient map <math>EG\to EG/G</math> is the universal cover of the orbit space <math>BG = EG/G</math>, and ''BG'' is a <math>K(G,1)</math>.<ref>{{Cite book|last=Hatcher |first=Allen |author-link=Allen Hatcher|title=Algebraic topology|date=2002|publisher=[[Cambridge University Press]]|isbn=0-521-79160-X |pages=89|oclc=45420394}}</ref>

In abstract terms (which are not those originally used around 1950 when the idea was first introduced) this is a question of whether a certain functor is [[representable functor|representable]]: the [[contravariant functor]] from the homotopy category to the [[category of sets]], defined by

:''h''(''Z'') = set of isomorphism classes of principal ''G''-bundles on ''Z.''

The abstract conditions being known for this ([[Brown's representability theorem]]) ensure that the result, as an [[existence theorem]], is affirmative and not too difficult.

==Examples==
#The [[circle]] <math>S^1</math> is a classifying space for the [[infinite cyclic group]] <math>\Z.</math> The total space is <math>E\Z =\R. </math>
#The [[torus|''n''-torus]]  <math>\mathbb T^n</math> is a classifying space for <math>\Z^n</math>, the [[free abelian group]] of rank ''n''. The total space is <math>E\Z^n=\R^n.</math>
#The wedge of ''n'' circles is a classifying space for the [[free group]] of rank ''n''. 
#A [[closed manifold|closed]] (that is, [[compact space|compact]] and without boundary) connected [[Surface (topology)|surface]] ''S'' of [[Genus (mathematics)|genus]] at least 1 is a classifying space for its [[fundamental group]] <math>\pi_1(S).</math>
#A [[closed manifold|closed]] (that is, [[compact space|compact]] and without boundary) connected [[hyperbolic manifold]] ''M'' is a classifying space for its [[fundamental group]] <math>\pi_1(M)</math>.
#A finite locally connected [[CAT(0) space|CAT(0)]] [[cubical complex]] is a classifying space of its [[fundamental group]].
#The [[Real projective space#Infinite real projective space|infinite-dimensional projective space]] <math>\mathbb{RP}^\infty</math> (the direct limit of finite-dimensional projective spaces) is a classifying space for the cyclic group <math>\Z_2 = \Z /2\Z.</math> The total space is <math>E\Z_2 = S^\infty</math> (the direct limit of spheres <math>S^n.</math> Alternatively, one may use Hilbert space with the origin removed; it is contractible).
#The space <math>B\Z_n = S^\infty / \Z_n</math> is the classifying space for the [[cyclic group]] <math>\Z_n.</math> Here, <math>S^\infty</math> is understood to be a certain subset of the infinite dimensional Hilbert space <math>\Complex^\infty</math> with the origin removed; the cyclic group is considered to act on it by multiplication with roots of unity.
#The unordered [[Configuration space (mathematics)|configuration space]] <math>\operatorname{UConf}_n(\R^2)</math> is the classifying space of the [[Braid group|Artin braid group]] <math>B_n</math>,<ref>{{Cite book|title=Vladimir I. Arnold — Collected Works|last=Arnold|first=Vladimir I.|date=1969|publisher=Springer |pages=183–6|language=en|doi=10.1007/978-3-642-31031-7_18|chapter = The cohomology ring of the colored braid group|isbn = 978-3-642-31030-0}}</ref> and the ordered configuration space <math>\operatorname{Conf}_n(\R^2)</math> is the classifying space for the pure Artin braid group <math>P_n.</math>
#The (unordered) [[Configuration space (mathematics)|configuration space]] <math>\operatorname{UConf}_n(\R^\infty)</math>  is a classifying space for the symmetric group <math>S_n.</math><ref>{{Cite web|url=https://ncatlab.org/nlab/show/classifying+space|title=classifying space in nLab|website=ncatlab.org|access-date=2017-08-22}}</ref> 
#The infinite dimensional complex [[projective space]] <math>\mathbb{CP}^\infty</math> is the classifying space {{math|''BS''<sup>1</sup>}} for the circle {{math|''S''<sup>1</sup>}} thought of as a compact topological group.
#The [[Grassmannian]] <math> Gr(n, \R^\infty)</math> of ''n''-planes in  <math>\R^\infty</math> is the classifying space of the [[orthogonal group]] {{math|O(''n'')}}. The total space is <math>EO(n) = V(n, \R^\infty)</math>, the [[Stiefel manifold]] of ''n''-dimensional orthonormal frames in <math>\R^\infty.</math>

==Applications==
This still leaves the question of doing effective calculations with ''BG''; for example, the theory of [[characteristic class]]es is essentially the same as computing the [[cohomology group]]s of ''BG'', at least within the restrictive terms of homotopy theory, for interesting groups ''G'' such as [[Lie group]]s ([[H. Cartan's theorem]]).{{clarify|date=September 2014}}<!-- how BG makes sense for a non-Lie group? --> As was shown by the [[Bott periodicity theorem]], the [[homotopy group]]s of ''BG'' are also of fundamental interest. 

An example of a classifying space is that when ''G'' is cyclic of order two; then ''BG'' is [[real projective space]] of infinite dimension, corresponding to the observation that ''EG'' can be taken as the contractible space resulting from removing the origin in an infinite-dimensional [[Hilbert space]], with ''G'' acting via ''v'' going to &minus;''v'', and allowing for [[homotopy equivalence]] in choosing ''BG''. This example shows that classifying spaces may be complicated.

In relation with [[differential geometry]] ([[Chern–Weil theory]]) and the theory of [[Grassmannian]]s, a much more hands-on approach to the theory is possible for cases such as the [[unitary group]]s that are of greatest interest. The construction of the [[Thom complex]] ''MG'' showed that the spaces ''BG'' were also implicated in [[cobordism theory]], so that they assumed a central place in geometric considerations coming out of [[algebraic topology]]. Since [[group cohomology]] can (in many cases) be defined by the use of classifying spaces, they can also be seen as foundational in much [[homological algebra]].

Generalizations include those for classifying [[foliation]]s, and the [[classifying topos]]es for logical theories of the predicate calculus in [[intuitionistic logic]] that take the place of a 'space of models'.

==See also==
* [[Classifying space for O(n)]], ''B''O(''n'')
* [[Classifying space for U(n)]], ''B''U(''n'')
* [[Classifying space for SO(n)]]
* [[Classifying space for SU(n)]]
* [[Classifying stack]]
* [[Borel's theorem]]
* [[Equivariant cohomology]]

==Notes==
<references />

==References==
*{{cite book |first=J.P. |last=May |title=A Concise Course in Algebraic Topology |publisher=University of Chicago Press |date=1999 |isbn=978-0-226-51183-2 |url={{GBurl|g8SG03R1bpgC|p=3}} }}
*{{nlab|id=classifying+space|title=Classifying space}}
*{{Springer|id=C/c022440|title=Classifying space}}

[[Category:Algebraic topology]]
[[Category:Homotopy theory]]
[[Category:Fiber bundles]]
[[Category:Representable functors]]