Two responses to Rognes question abt stable stems.  The question appears
at the bottom of the second response.......DMD
_____________________________________________

Subject: Re: question abt stable stems
Date: Mon, 20 Nov 2000 11:58:08 -0500
From: Clarence Wilkerson <wilker@math.purdue.edu>

A couple of papers of H.W. Henn seem relevant. These deal with
the finite case, so the stable groups are still in question by these
methods.

1) 7m:55017 55Q05 (55P35)
Henn, Hans-Werner(D-HDBG)
On the growth of homotopy groups.
Manuscripta Math. 56 (1986), no. 2, 235--245.

and
2)85g:55019 55Q40
Bödigheimer, Carl-Friedrich; Henn, Hans-Werner
A remark on the size of $\pi \sb{q}(S\sp{n})$.
Manuscripta Math. 42 (1983), no. 1, 79--83.
_______________________________________________
Date: Mon, 20 Nov 2000 15:02:29 -0500 (EST)
From: "Douglas C. Ravenel" <drav@math.rochester.edu>
Subject: Re: question abt stable stems

I am not aware of any result in this direction.  Empirical
evidence suggest that the log of the order of the p-component of
the k-stem grows linearly with k on average in the know range.
Very roughly speaking one has products elements in the beta
family with powers of $\beta_1$, ie the cokernel of J has the
same growth behavior as the polynomial algebra on two generators.
You can see this in Hatcher's pictures at
http://www.math.cornell.edu/~hatcher/stemfigs/stems.html.

I know of no way to prove or even know if this phenomenon
persists to higher dimensions.

You might want to look at the discussion of homotopy groups of
spheres that occured here last year; see
http://www.lehigh.edu/~dmd1/p1999.html.


Doug Ravenel



On Mon, 20 Nov 2000, DON DAVIS wrote:

> Date: Mon, 20 Nov 2000 17:08:31 +0100
> From: rognes@math.uio.no
> Subject: Question about stable stems
>
> Is there a known upper bound on the order of the n-th stable stem,
> as a function of n ?  The vanishing line in the mod p Adams spectral
> sequence gives an upper bound on the p-exponent of the n-th stable stem,
> but is there a known upper bound on its p-rank ?
>
> - John Rognes
>
>



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