Comparison of the geometric bar and W-constructions Original Research Article

For a simplicial group K, the realization of the W-construction image of K is shown to be naturally homeomorphic to the universal bundle E¦K¦→ B¦K¦ of its geometric realization ¦K¦. The argument involves certain recursive descriptions of the W-construction and classifying bundle and relies on the facts that the realization functor carries an action of a simplicial group to a geometric action of its realization and preserves reduced cones and colimits.