Editing Classifying space (section)

==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'.