From: "Justin R. Smith" <jsmith@mcs.drexel.edu>
Subject: Re: more questions, & comment
Date: Fri, 22 Jan 1999 11:18:09 -0000

Although Milgram's model can be iterated indefinitely (well, until the space
fails to be connected), Baues's model cannot be iterated more than once.
Baues's model is simpler (or has fewer generators) and Milgram's --- in fact
he gets a cell decomposition for Omega X whose cellular chain complex is the
cobar construction of that of X.

To iterate this more than once, one needs m-coalgebras (or operads) as in my
paper "Iterating the cobar construction".

>
>Then Baues apparently has a paper treating the case of $\Omega ^2 X$.
>(cf also a paper of Getzler?). After that, we fall back into the same type
>of situation as for homotopy groups of spheres: apparently, people don't
>seem to be satisfied that we have a combinatorial model for $\Omega ^nX$,
>but as far as I can tell, a simple iteration of Milgram (or Baues) will
>give that, for example in the case where $X$ has only one $i$-cell for
>$i\leq n$. For the iteration, use a contracting method given by the HB
>article of Carlsson-Milgram (nb is that  paragraph in their article
>correct?).
>

>The problem with the iteration of course is that at each stage one has to
>make some choices so it isn't really all that canonical; also, it doesn't
>give rise to lots of neat and cool new shapes of polyhedra or anything like
>that, so in this sense one might say that it doesn't really answer the
>question. The problem is that it is difficult to say that something doesn't
>really answer a question, when one doesn't know what the question is!

>
>---Carlos Simpson
>
____________________________________________________________
Date: Fri, 22 Jan 1999 12:51:52 -0500 (EST)
From: "John R. Klein" <klein@math.wayne.edu>
Subject: Re: 6 more comments

I always had the (perhaps false) impression that research mathematicians
only concern themselves with the questions like "What is Mathematics?"
after their own research fizzles out. Is everyone who posts to this list
coasting into an early retirement?

John R. Klein