For a topological group , a universal principal bundle has contractible total space and classifies numerable principal -bundles by pullback. Over a paracompact base, isomorphism classes correspond to homotopy classes of maps to . For a discrete group, its classifying space of a discrete group is a ; for the circle, .
Give a discrete group a contractible free CW complex . The quotient has universal cover , so it has fundamental group and no higher homotopy groups. It is thus an Eilenberg–MacLane space , even when is nonabelian. The cohomology of with the appropriate local coefficients computes group cohomology.
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In mathematics, particularly in topology and algebraic topology, a **classifying space** is a specific type of topological space that allows one to classify certain types of mathematical structures up to isomorphism using principal bundles. The concept is most commonly associated with fiber bundles, especially vector bundles and principal G-bundles, where \( G \) is a topological group.