Classifying space

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

E

G

→

B

G

{\displaystyle EG\to BG}

. 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 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, 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

π

:

Y

⟶

X

{\displaystyle \pi \colon Y\longrightarrow X\ }

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.