Date: Fri, 22 Jan 1999 16:51:56 -0500 (EST)
From: "Douglas C. Ravenel" <drav@harpo.math.rochester.edu>
Subject: Homotopy groups of spheres


Here is my two cents worth on the recent discussion about
the problem of computing the homotopy groups of spheres.

I believe that Anick's result has very little to do with it.  As
I recall, he was concerned with the problem of computing the
rational homotopy of a CW-complex $X$ with cells in dimensions 2
and 4.  We know that the homotopy type of $X$ is determined by
the number of cells in each dimension along with some simple data
about how the 4-cells are attached to the 2-cells.  He showed
that the time needed to compute the rational homotopy behaves
badly as a function of the number of cells, by showing that it is
equivalent to other computational problems known to have such bad
behavior.

The rational homotopy groups of spheres are well known, and
Anick's result has no obvious (to me) relation to the still
unsolved problem of computing the torsion of the homotopy groups
of spheres.  In 1957 Ed Brown showed that these groups
were finitely computable, but the algorithm he offered to prove
this was not one that anyone would want to use.

The discovery of the Lambda Algebra (described in \S3.3 of my
green book) led to a practical algorithm for computing the
$E_2$-term of the relevant unstable Adams spectral sequence, an
algebraic approximation to the actual homotopy groups. These have
been implemented by Curtis-Goerss-Mahowald-Milgram for $p=2$ and
by Tangora for odd primes, both around 1985.  It is amusing to
note that as soon as the 2-primary algorithm was discovered,
George Whitehead used it to compute up to the 40-stem by hand in
1969, and it took more than 15 years for the computer programmers
to catch up with him.

Experience indicates that the time needed for these algorithms to
run grows exponentially with the stem, but I know of no theorem
to this effect.  Empirical evidence also suggests that the size
of the answer grows cubically with the stem, but again nothing
has been proved.  The discrepancy between these two growth rates
is intriguing.

The striking (meaning fascinating or infuriating, depending on
your point of view) thing about the problem is that it is very
easy to state yet very difficult to solve.  The history of the
subject shows that each computational breakthough leads to an
increased insight into how hard the problem actually is.  I do
not expect to see a complete solution in my lifetime and would
not be surprised if my unborn grandchildren do not live to see
one either.

Doug

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