Editing Principal bundle (section)

==Classification of principal bundles==

{{Main| Classifying space}}
Any topological group {{math|''G''}} admits a '''classifying space''' {{math|''BG''}}: the quotient by the action of {{math|''G''}} of some [[weakly contractible]] space, ''e.g.'', a topological space with vanishing [[homotopy group]]s. The classifying space has the property that any {{math|''G''}} principal bundle over a [[paracompact]] manifold ''B'' is isomorphic to a [[pullback bundle|pullback]] of the principal bundle {{math|''EG'' → ''BG''}}.<ref>{{Citation | last1=Stasheff | first1=James D. | title=Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) | publisher=[[American Mathematical Society]] | location=Providence, R.I. | year=1971 | chapter=''H''-spaces and classifying spaces: foundations and recent developments | pages=247–272}}, Theorem 2</ref> In fact, more is true, as the set of isomorphism classes of principal {{math|''G''}} bundles over the base {{math|''B''}} identifies with the set of homotopy classes of maps {{math|''B'' → ''BG''}}.