Date: Thu, 29 Apr 1999 16:27:00 -0400
From: "Douglas C. Ravenel" <drav@math.rochester.edu>
Subject: Re: Tangora response


We did indeed have this discussion in January, and you can find
all of it in Don's archive
<http://www.lehigh.edu/~dmd1/postings.html> under the heading
"Question about homotopy groups of spheres ".


Here is the note about it that I sent to Tangora on 2/2/99, which
he says he could not find.


Marty,

This is the way I remember it. Fix a prime p. By "size" of
a group, I mean the base p log of the order of its p-component.
Then empirically:

        1. The size of the stable k-stem grows linearly with k.

        2. Hence as you say the cumulative size of all stable
stems up to k grows quadratically with k.

        3. When you use an "algorithm" based on the EHP sequence,
the amount of data you need to store (using only leading terms of
cycles in the homotopy analog of the Lambda algebra), grows
cubically with k.  This is because in order to compute the k stem
you have to feed in (as Hopf invariants) a certain fraction lower
homotopy groups, most of which are stable.  Thus the derivative
of this function of k is the quadratic function of 2.

Heuristically, the storage required for the EHP spectral sequence
is the double integral of the size of the stable k-stem, so it is
a cubic function of k.

Doug


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On Thu, 29 Apr 1999, DON DAVIS wrote:

> Date: Thu, 29 Apr 1999 14:30:57 CDT
> From: "Martin C. Tangora (312) 996-3064" <Tangora@uic.edu>
> Subject:     Re: (job and) question
>
> > Date: Thu, 29 Apr 1999 13:10:02 -0400 (EDT)
> > From: Jie Wu <jiewu@math.upenn.edu>
> > Subject: Size of homotopy groups
> >
> > The following questions were asked by Wolfgang Ziller:
> >
> > Are there any descriptions about the size (order) t(k) of \pi_k(S^n)?
> >
> > Is  t(k) bounded by an exponential function on k, a polynomial function on
> > k or even better?
>
> We had this thread in January 1999.  At the time I was going to post
> the following references but was confused about one point, wrote to
> Doug Ravenel about it, and never got back to the list.  Now I can't
> find Doug's reply.  Must be getting old.  Anyway, Hans-Werner Henn
> has a paper on the growth of homotopy groups and he shows that it's
> exponential.  I proved that the same growth rate holds for the E_1 term
> of the Adams spectral sequence, by which I mean the lambda algebra;
> so it seems most plausible that the E_2 term obeys the same law.