Date: Mon, 1 Feb 1999 11:57:40 +0900
From: Andrzej Kozlowski <andrzej@tuins.ac.jp>
Subject: htpy gps of spheres

In the message that started this topic Carlos Simpson asked what to me
seems to be a very interesting question, which the experts seem to have
either ignored or been unable to answer. The question was: "...is there
an explicit conjecture that needs to be proved?
The closest to an answer to this question was provided in Doug Ravenel's
first message:

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.

Has anybody ever tried to make an explicit conjecture concerning, say,
the  two torsion, on the basis of the existing empirical data? It seems
to me such a conjecture would be very useful and even if turned out to be
practically unprovable it could play a role similar to that played by the
Riemann Hypothesis. It would give people a concrete, "well defined"
target for attempted proofs or disproofs. It would also make it at least
possible to try to predict the value of ?pi_453(S^127) etc. Moreover,
whenever one had a problem that reduced to the homotopy groups of spheres
one would also have a concrete conjecture. The only time I ever had
anything explicitely to do with the homotopy groups of spheres was when I
co-authored a paper, one of the results of which expressed the homotopy
groups of a certain space in terms of the homotopy groups of spheres. The
first referee did not like this. He wrote something like: "since nothing
at all is known about these groups I do not see why the authors claim
that their result tells us anything new". Having a concrete conjecture
would have at least provided a useful counter argument!

Andrzej Kozlowski
Toyama International University
JAPAN
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