Әлсіз жирылғыш кеңістіктің топтың еркін әрекетімен алынған бөлігі

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 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 As explained later, this means that classifying spaces 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 spaces, 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, roughly speaking, a 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, that is, BG is an Eilenberg–MacLane space, or a K(G, 1).

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

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. Taking X as the 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.

This still leaves the question of doing effective calculations with BG; for example, the theory of characteristic classes is essentially the same as computing the cohomology groups of BG, at least within the restrictive terms of homotopy theory, for interesting groups G such as Lie groups (H. Cartan's theorem). As was shown by the Bott periodicity theorem, the homotopy groups 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 −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 Grassmannians, a much more hands on approach to the theory is possible for cases such as the unitary groups 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 foliations, and the classifying toposes for logical theories of the predicate calculus in intuitionistic logic that take the place of a 'space of models'.