Date: Fri, 22 Jan 1999 16:38:30 GMT
From: carlos@picard.ups-tlse.fr (Carlos Simpson)
Subject: Re: 6 more comments

There seems to be a phase transition between two phases of mathematics,
one where the word ``study'' is well-defined, the other where it is not.
In the second  phase, the criterion of ``beauty'' comes in.
In particular, I agree that it is not *necessary* that there be well-defined
questions in a particular subject.

But to get back to topology, is it really true that nobody has any well-defined
questions about e.g. homotopy groups of spheres? For example, with what
degree of accuracy is it known how to bound the number of elements of
$\pi _i(S^n)$?

Another related question is
``combinatorial models for loop space''. For $\Omega X$ one has the Milgram
model---which already requires a fair amount of digging into the
``Handbook'' to find out about, for example Milgram states there that he
originally had in mind his model for $\Omega X$ but when it got written up
it was only for $\Omega S X$ (and that consequently he has periodically
gotten a number of theses in the mail giving the model for $\Omega X$...).

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: Thu, 21 Jan 1999 08:33:25 -0500 (EST)
From: James Stasheff <jds@math.unc.edu>
Subject: Re: 6 more comments

I vote for : study of structure and pattern with attention to beauty
and also to inexplicable relevance/usefulness in the real world
.oooO   Jim Stasheff            jds@math.unc.edu
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