From: Tibor Beke <beke@math.uu.nl>
Subject: Re: response re htpy gps
Date: Wed, 20 Jan 1999 15:25:53 +0100 (MET)

>> To be brief: it is often stated that ``one of the major problems in
>> topology is the homotopy groups of spheres''. What is the well-defined
>> question here? For example, is there an explicit conjecture that needs
>> to be proved?
>
> I am sure I am biased, but it seems to me that the only "well defined"
> problem in this area that I can think of is dead. It was killed by David
> Anick's proof that the problem of computing the homotopy groups of
> spheres is of the  NP-complete type, which essentially means that
> algorithms are useless for this purpose.

Certainly, but just as surely every negative result gives rise to good
questions.  To stick with the homotopy groups of spheres (without claiming
that I submit one of those questions...)

-- Find infinite families of homotopy groups of spheres with "nice
definitions".

The paradigm is the image of J (which I think most mathematicians attest to
as a beautiful piece of mathematics).  Let me hammer away at the notion of
"nice definition" to suggest that it's possible to stay inside
well-definedness.  It is a result of Dan Kan (I think) that the function
n,k -> \pi_k(S_n) is recursive (may even be primitive recursive).  It's
complemented by Anick's result that its real-time complexity is very high.
There are "partial factors" of this function (ie. "infinite families") that
are of much lower complexity, however.  I don't know the complexity of the
Bernoulli numbers, but they may well belong to "Kalmar elementary
functions" or some other sub-recursive family.  If you wish to stick to
models of computation, then the order to give probably ought to be "find
families that can be defined with limited programming syntax" rather than
"find families that can be computed fast" as, it seems, computations will
eventually require exponential time.

Incidentally, I do not claim that interesting infinite families of homotopy
groups should coincide with ones that have simple subrecursive definitions,
and God forbid that I should suggest that the beauty of Adams' result is
that it allows this kind of formalist interpretation.  But once I let
beauty out of the bag (as I think philosophers of mathematics should) I'd
like to add that in my view, mathematics is the study of structures.  As
has often been said, two of these are the intuition of natural numbers and
formal computation, and of spatio-temporal geometry.  Being able to link
notions of one to the other has usually been seen as a piece of "beautiful
mathematics" (eg. Gauss' integral formula for the genus, locating Bernoulli
numbers in the homotopy groups of spheres, extending the notion of
cohomology from spaces to groups).  Whatever one's concept of mathematics,
I think it should allow for the fact that -- in addition to seeking
correspondences between them -- often the simple *identification* of a
structure counts as a turning point of mathematics (Riemann-Weyl's notion
of "manifold"; abstraction of the notion of "group", ca. 1880; Hilbert's
distillation of first-order logic, ca.1920; Emmy Noether's insistence that
homology is a *group*; the notion of category;  and so on).  I don't think
well-definedness of questions is a major driving force in contemporary
mathematics, or in any contemporary theoretical science in fact (though it
is probably more of a driving force with us than in theoretical physics or
biology).

Tibor Beke