Comparison of the geometric bar and W-constructions

TL;DR

This paper proves that for a simplicial group, the W-construction's realization is homeomorphic to its universal bundle, connecting algebraic and geometric structures through recursive descriptions and functor properties.

Contribution

It establishes a natural homeomorphism between the realization of the W-construction and the universal bundle for simplicial groups, using recursive and functorial techniques.

Findings

Realization of W-construction is homeomorphic to the universal bundle.

Realization functor preserves group actions and colimits.

Recursive descriptions facilitate the proof of the main result.

Abstract

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