Date: Mon, 10 May 1999 14:29:01 -0400
From: Zbigniew Fiedorowicz <fiedorow@math.ohio-state.edu>
Subject: Re: 2 on gp completion

In response to Jim Stasheff's request for a summary of known
results and open questions on group completions:

Let M_* be a simplicial monoid. Then there are three possible
notions of group completion.

(1) A functor F: simplicial monoids ---> H-spaces
   together with a natural (up to homotopy) H-map
           |M_*| ----> F(M_*)
   such that the induced map in homology induces the localization
   of Pontrjagin homology rings
          H_*(|M_*|)  ---> H_*(|M_*|)[\pi^{-1}]
   where \pi=\pi_0(|M_*|). [This condition implies that \pi_0(F(M_*))
   is the Grothendieck group U(\pi).]

(2) The natural map
           |M_*| ---> \Omega B|M_*| \simeq |\Omega BM_*|
(Universal for A_\infty maps to 1-fold loop spaces.)

(3) The natural map
           M_* ---> U(M_*)
(Universal for homorphisms into simplicial groups.)

As far as I know, there is no general construction F(-) fulfilling (1).

According to McDuff and Segal (Inventiones, 1976), (2) fulfills the role of
(1) provided that \pi is central in H_*(|M_*|), or more generally if the
localization H_*(|M_*|)[\pi^{-1}] can be realized by a calculus of left
or right fractions (ie. Ore conditions).

For discrete monoids (3) obviously fulfills the role of (1).

McDuff's result (Topology, 1979) about realizing connected CW homotopy
types as classifying spaces of discrete monoids shows that (2) does not
fulfill (1) in general, and that the natural map
  \Omega B|M_*|  ---> |U(M_*)|
can fail (spectacularly!) to be an equivalence, contradicting the conjecture
that Puppe attributes to Kan-Moore.  The counterexample
I posted a couple of days ago, shows that
    \Omega B|M_*| ---> |U(M_*)|
can fail to be an equivalence even when \pi_0(|M_*|) is a group,
contradicting the weaker conjecture that Puppe attributes to Moore.

I think it's still open whether the natural map
    H_*(|M_*|)[\pi^{-1}] ---> H_*(\Omega B|M_*|)
is injective for all simplicial monoids.

Zig Fiedorowicz