Classifying space

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically in <a href="/facts/Homotopy_theory/MJwYyRny">homotopy theory</a>, a classifying space BG of a <a href="/facts/Topological_group/cAgNMMRU">topological group</a> G is the quotient of a <a href="/facts/Weakly_contractible/5dwc35pz">weakly contractible</a> space EG (i.e., a topological space all of whose <a href="/facts/Homotopy_group/xjogL95Y">homotopy groups</a> are trivial) by a proper <a href="/facts/Free_action/Vh9VtUUw">free action</a> of G. It has the property that any G <a href="/facts/Principal_bundle/bbB2D94B">principal bundle</a> over a <a href="/facts/Paracompact/rjTrvF7j">paracompact</a> manifold is isomorphic to a <a href="/facts/Pullback_bundle/nBdt2Cx7">pullback</a> of the principal bundle 
 
 
 
 E
 G
 →
 B
 G
 
 
 {\displaystyle EG\to BG}
 
. As explained later, this means that classifying spaces <a href="/facts/Representable_functor/dILwN4oj">represent</a> a set-valued <a href="/facts/Functor/l2u3rIBE">functor</a> on the <a href="/facts/Homotopy_category/JaEnPakj">homotopy category</a> of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of <a href="/facts/Topological_space/v2ut7CbK">topological spaces</a>, such as <a href="/facts/Sierpi%C5%84ski_space/Ha6nckJV">Sierpiński space</a>. This notion is generalized by the notion of <a href="/facts/Classifying_topos/PyjMgsyW">classifying topos</a>. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. 
For a <a href="/facts/Discrete_group/HIjkpJhS">discrete group</a> G, BG is a <a href="/facts/Connected_space/5CUTxfoA">path-connected</a> <a href="/facts/Topological_space/v2ut7CbK">topological space</a> X such that the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> of X is isomorphic to G and the higher <a href="/facts/Homotopy_groups/xjogL95Y">homotopy groups</a> of X are <a href="/facts/Trivial_group/XPjQkQFm">trivial</a>; that is, BG is an <a href="/facts/Eilenberg%E2%80%93MacLane_space/mrux6oMJ">Eilenberg–MacLane space</a>, specifically a K(G, 1).

Classifying space open-in-new

Classifying space