Editing Classifying space (section)

==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>