Date: Tue, 21 Nov 2000 15:01:16 -0600 (CST)
From: Jesus Gonzalez <jesus@math.cinvestav.mx>
Subject: Re: question abt stable stems

Here are some further comments on Rognes' question.

Jesus
----------------------

with respect to the upper bound for the p-exponent of the n-th stable
stem, Mahowald's theory of bo-resolutions (see for instance "The image of
the stable $J$-homomorphism" Topology 28 (1989), no. 1, 39--58) improves
the slope of the classical vanishing line in the 2-local Adams spectral
sequence. The idea is to set a sharper vanishing line in the bo-Adams
spectral sequence. For odd primes this is done in my paper "A vanishing
line in the BP<1> Adams spectral sequence" (Topology 39 No. 6 (2000)
1137-1153). Roughly speaking, the BP<1>-Adams spectral sequence has a
vanishing line of slope 1/p^2, in contrast with the 1/p slope vanishing
line in the classical Adams spectral sequence. There results a roughly
n/p^2 bound (except for Im-J) for the p-exponent in the n-th stem.

Ravenel's comments on the polynomial growth of the beta family seem to be
in the border line.

Jesus Gonzalez
----------------------

> 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
>
>