Date: Fri, 7 May 1999 16:28:44 -0400
From: Zbigniew Fiedorowicz <fiedorow@math.ohio-state.edu>
Subject: Re: more on group completion

Tibor Beke <beke@math.uu.nl> wrote

>\bar{W}(u) is a weak equivalence for any simplicial monoid.

This is obviously false as stated: suppose M is a discrete simplicial
monoid.

>Let U be the Grothendieck functor Monoids --> Groups ("group completion").
>Let M be a simplicial monoid whose \pi_0 is a group.  Then the natural
>inclusion u: M --> U(M) is a weak equivalence.

This is very likely to be false.

In 1979 Dusa MacDuff showed that any connected CW complex can be realized
as the classifying space of a discrete monoid (Topology, 18, 313-320). I don't h
ave her paper at hand, but I gave an alternative proof of this result
in a paper which appeared in Am. J. Math. in 1984.  My construction is
functorial: there is a functor X |---> M(X) taking connected CW spaces to
discrete monoids and there is a chain of natural homotopy equivalences
connecting X to BM(X).

While my construction doesn't have the property that M(*) = 1, I think
it could be arranged. GRANTED THIS, we can provide a counterexample to
the above conjecture as follows.

Start with a simplicial topological space X_* with the following properties:
   1) for all n, X_n is a simply connected CW space
   2) X_0 = *
   3) \Omega |X_*| is not contractible
   4) the degeneracies are cofibrations

Now apply the functor M() degreewise to X_* to obtain the simplicial monoid
M(X_*). Because M(X_0)=M(*)=1, \pi_0 = 1. Then it follows that
        |M(X_*)| \simeq \Omega B|M(X_*)| = \Omega |BM(X_*)|
Now by the basic property of M(), the |X_*| \simeq |BM(X_*)|. Thus
        |M(X_*)| \simeq \Omega |X_*|
is not contractible.

On the other hand for a discrete monoid M, U(M) = \pi_1(BM).
It follows that
   |U(M(X_*))| = |\pi_1(BM(X_*))| = |\pi_1(X_*)| = *

Zig Fiedorowicz