Question

For a simplicial group $G$, there is a universal bundle $WG \to \overline{W}G$ in the category of simplicial sets, detailed in for example May's book (djvu).

Now $WG$ has a simple enough description in terms of $G$ that I would expect one could construct a contracting homotopy directly. Has this been done in the literature?

The proof in May's book (and in the original sources) that $WG$ is contractible goes via showing that $WG$ is 'of type (W)', and that such simplicial sets are Kan, simply-connected and have trivial homology.

Answer 1

Pages 75-81 of Appendix A of ``On the theory and applications of differential torsion products'', Memoirs AMS 142 (1974), by V.K.A.M. Gugenheim and myself, gives a detailed treatment of the $W$-construction for simplicial augmented algebras over a commutative ring $R$. Not the answer to your question, but if I remember rightly, it should lift to an answer when suitably specialized; more precisely, the chain homotopy of Lemma A.16 (up to signs coming from variant choices) should specialize to one coming from a contracting homotopy as desired. When $G$ is a group regarded as a constant simplicial group, Lemma A.15 lifts to give an isomorphism between $WG$ and the simplicial set $E_*G$ whose realization is $EG$.