Date: Sat, 30 Jan 1999 08:06:31 GMT
From: carlos@picard.ups-tlse.fr (Carlos Simpson)
Subject: Re: Wilkerson response

>What is pi_453(S^{127})?  Is there a computer program that will tell
>you the answer?  What would its algorithm be?

>
>As for \pi_453(S^127) itself, I'll bow to the real experts, but
>I'd be surprised if it's known ( say, its 2 torsion ?) or if there is
>a computer program that could attack it in a reasonable amount of time.
>Bob B. ? Don D. ?  Volunteers?
>

Repeating oneself is probably an indicator of impending early retirement,
but here goes...


There *is* a perfectly good algorithm for computing the homotopy groups of a
finite simplicial complex assuming that it is known to be simply connected,
given by E. Brown in Annals of Mathematics, 1957. Roughly the idea is as
follows: he shows how to inductively construct the ``whitehead tower''
of $X$, namely a sequence of spaces X_k with maps
\phi _k : X_k -----> K(\pi _k , k)
such that the fiber of \phi _k is X_{k+1}. At each stage,
\pi _k is calculated as H_k(X_k); thus \pi _k=\pi _k(X). The space X_{k+1}
is constructed as a fibration over X_k with fiber K(\pi _k, k+1). This
fibration is constructed explicitly by explicitly constructing the cocycle
which classifies it. In the second part of the paper, he shows how to do
this construction in an ``approximate'' way so that it is finitely
computable;
thus he gets an algorithmic computation of \pi _k(X).

This is the algorithm which Doug Ravenel said was ``not one that anybody
would want to use''. That phrase is not a ``well-defined'' one, and this is
why I asked a while ago if anybody had a ``well-defined'' question about
computing homotopy groups.

One way to make it well-defined would be to quantify the notion of ``not
one that anybody would want to use'' by looking at running spacetime for
the algorithm.

I don't know if anybody has explicitly calculated the running time for E.
Brown's algorithm. If anybody knows the answer to this question, that would
certainly be a propos for the current flurry of toplist email.

It seems clear that everybody who knows about E. Brown's algorithm, is
(as a matter of opinion) *sure* that, for example, the spacetime necessary
to use it to calculate pi_453(S^{127}) is greater than the available
spacetime of the universe (assuming a model where the universe eventually
recollapses).
This distinction seems to pertain more to physics than to mathematics, though.

The next ``algorithm'' would be as follows. Kan gives an explicit construction
whereby a simplicial set is transformed into a simplicial free group; the
latter being the loop space of the former. This reduces the problem to one
of computing the homotopy groups of a simplicial free group. Then Curtis
showed that
if you replace the simplicial free group \{ G_k\} by the collection of
lower-central-series quotients \{ G_k /\Gamma _r G_k \}, the homotopy
groups are the same up to i= log_2(r). This reduces the problem to a
problem of computing
the homotopy groups of a simplicial group whose components are lcs quotients
of free groups. If F_k is a simplicial group, the homotopy is
\pi _k = N(G)_k / \partial _0(N(G)_{k+1})
where N(G) is the ``normalized subcomplex'' defined in the same way as for
a simplicial abelian group. Now, it seems to me that this computation
problem should be finitely computable, but I don't know whether an
algorithm has been developped. (anybody?)

The third algorithm is ``iterating the Milgram model''. The Milgram model is
a creation whereby for each simplex \Delta ^k, you put in a cube
\Box ^{k-1} as a generator of a topological monoid. The topological monoid
(actually I am simplifying here to the case where there is only one 0-cell)
is therefore made up of cubes (products of the generating cubes).
If there is no nondegenerate 1-simplex in X, then the number of cubes in
each dimension is finite. This gives a finite geometric model for \Omega X.
One can iterate this. I wrote up a short note last year, iterating
a procedure which I now feel is the same as this Milgram model (I wasn't
aware of the Milgram model as such, when I wrote up the note). Although I
haven't
had time to look up Justin Smith's paper, judging from the title
``Iterating the cobar construction'', that must be what he says. Thus, I
now feel that there is nothing at all new in my note, but if anybody
wants to look at it it is on q-alg/9710011 (my apologies to Clarence,
I also didn't know about Hopf at the time).

To get back to our discussion, some further remarks are in order. To the
question of calculating the homotopy groups of an arbitrary simplicial
complex,
it does seem that Anick's result (as recounted by Ravenel) is relevant,
and apparently gives a lower bound on the computational complexity.
Thus, for an arbitrary complex this raises the question of whether any
known algorithm attains (in order of growth, say) the same computational
complexity
as Anick's lower bound.

As far as I understand it, the question raised by Ravenel for the homotopy
groups of spheres, is whether there might not be a special algorithm which
works only for spheres and which gives a much better growth rate (cubic
rather than exponential). Other than that, I didn't understand in his
email, exactly what *is* the question which he doesn't expect to be solved
in his lifetime.

It seems to me that these meta-mathematical questions are relevant to
Ronnie Brown's original question, on several levels. To the question ``what
do topologists do?'', one common answer is that they are looking for
(among many other things) the homotopy groups of spheres. However, in view
of E. Brown 1957, this doesn't seem to me to be accurate at all.
A number of people have written of problems of ``public relations''; here
seems to be a major public relations disaster---the fact that this problem
was
solved in 1957 seems to be  unknown to most people working in most other
fields of mathematics (it was certainly unknown to me up until about a year
ago),
and even to many topologists.
How to put this into teaching? When a graduate or undergraduate learns about
homotopy groups, they seem to be these really mysterious objects which one
could never calculate. In particular, there seems to be a great divide
between
$\pi _1$, for which we have Van Kampen's theorem and which can therefore be
calculted, and the higher $\pi _i$ which seem to be totally inaccessible.
This impression is false. While it is probably impossible to teach the
details of E. Brown's algorithm to a first year graduate algebraic topology
class,
it seems to me to be absolutely essential to mention its existence, pretty
much the day after you define the homotopy groups.
Public relations starts at home!

I'll stop here for now---
Carlos