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 groups 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 of the principal bundle
However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy.
In that case the projection map 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 freely.
(The 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.
The classifying property required of BG in fact relates to π.
We must be able to say that given any principal G-bundle over a space Z, there is a classifying map φ from Z to BG, such that
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.
Specifically, let EG be the weak simplicial complex whose n- simplices are the ordered (n+1)-tuples
The group G acts on EG by left multiplication, and only the identity e takes any simplex to itself.
[2] 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: the contravariant functor from the homotopy category to the category of sets, defined by 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.
[clarification needed] As was shown by the Bott periodicity theorem, the homotopy groups of BG are also of fundamental 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 foliations, and the classifying toposes for logical theories of the predicate calculus in intuitionistic logic that take the place of a 'space of models'.