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 for which all its homotopy groups are trivial) by a 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 eg â bg., theorem 2 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). ==motivation== an example for g infinite cyclic 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 :\pi: 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. taking x as the quotient space circle, we can regard the projection Ï 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 Ï has a universal property amongst principal c-bundles; that any principal c-bundle in a definite way 'comes from' Ï. ==formalism== a more formal statement takes into account that g may be a topological group (not simply a discrete group), and that group actions of g are taken to be continuous; in the absence of continuous actions the classifying space concept can be dealt with, in homotopy terms, via the eilenbergâmaclane space construction
Algebraic_topology Fiber_bundles Homotopy_theory Representable_functors