Classifying Space - Formalism

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 Eilenberg–MacLane space construction. In homotopy theory the definition of a topological space BG, the classifying space for principal G-bundles, is given, together with the space EG which is the total space of the universal bundle over BG. That is, what is provided is in fact a continuous mapping

Assume that the homotopy category of CW complexes is the underlying category, from now on. 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 γ is the pullback of a bundle of π along φ. In less abstract terms, the construction of γ by 'twisting' should be reducible via φ to the twisting already expressed by the construction of π.

For this to be a useful concept, there evidently must be some reason to believe such spaces BG exist. In abstract terms (which are not those originally used around 1950 when the idea was first introduced) this is a question of whether the contravariant functor from the homotopy category to the category of sets, defined by

h(Z) = set of isomorphism classes of principal G-bundles on Z

is a representable functor. The abstract conditions being known for this (Brown's representability theorem) the result, as an existence theorem, is affirmative and not too difficult.