Comparison of the geometric bar and W-constructions

We show that, for a simplicial group K,the realization of the W-construction of K is naturally homeomorphic to the universal bundle of its geometric realization. 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