Date: Sat, 30 Jan 1999 18:19:53 -0500 (EST)
From: "Douglas C. Ravenel" <drav@harpo.math.rochester.edu>
Subject: More about homotopy groups


Simpson's message seems to be begging for a response from me, so
here it is.

When I said Ed Brown's algorithm was unusable, I was speaking in
practical rather than well defined terms.  One criterion for the
practicality of such an algorithm is this: has it been
used to find any groups that one did not know before?  The
algorithms of Brown and Milgram, as far as I know, do not pass
this test, while the Curtis-Kan algorithm does.  I will say more
about the latter below.

The point of Brown's algorithm is that its existence shows that
homotopy groups are FINITELY computable.  This result is
philosophicaly satisfying but of little use to anyone who is
actually curious about the answer.  I doubt that anyone has or
would want to use it even for a modest calculation, say of
$\pi_{20}(S^9)$.  In computer science algorithms are considered
practical only if their running requirements grow algebraically
with the size of the problem.  There is an obvious finite
algorithm for the travelling salesman problem, namely find the
distance needed for each of the N! possible sequences of
destinations and pick the shortest one.  It is considered useless
due to the rapid growth of N! as a function of N.

The Curtis-Kan algorithm that Simpson mentioned was the subject
of the 6 author paper of Bousfield, Curtis, Kan et al that
appeared in Topology in the late 60s.  The main result is that
the mod p lower central series of Kan's simplical group leads to
a spectral sequence converging to the homotopy of a given
sphere, that can reasonbly be called the unstable Adams spectral
sequence.  Moreover the combinatorics surrounding it leads to a
practical (in the sense that I used that term above) algorithm
for finding its $E_2$-term.  It was this algorithm that was
implemented first by George Whitehead and later by the authors I
cited previously, and it did yield information that was not
previously known.

Finally, the problem I do not expect to see solved in my lifetime
is the complete and explicit determination of ALL homotopy groups
of spheres.

Doug

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