# Explicit contraction for the universal simplicial bundle WG

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.

• How do these constructions relate to EG and BG? – Sean Tilson Jun 3 '12 at 4:08
• Hi Sean, I didn't get a chance to answer your question, but it is as Peter May says in his answer. EG is the geometric realisation of the functor W applied to the constant simplicial group, and for this G can be a vanilla group, a topological group, or a group internal to any finite-product category $C$ with a geometric realisation/homotopy colimit functor $sC \to C$. – David Roberts Jun 3 '12 at 22:13

## 1 Answer

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 $EG$.

• Thanks, Peter, I'll check that out. Perhaps I should have mentioned that MacLane gave a contracting homotopy of the chain complex (non-normalised) associated to W of a simplicial ring in his original lecture on W and Wbar. – David Roberts Jun 3 '12 at 21:28
• Lemma A.16, for anyone else who is reading. And recall that the memoir is available here:math.uchicago.edu/~may/PAPERS1965.html – David Roberts Jun 3 '12 at 22:11