Subject: Steenrod algebra question
Date: Thu, 22 Aug 2002 14:22:57 -0400 (Eastern Daylight Time)
From: "Nicholas J. Kuhn" <njk4x@tetra.mail.virginia.edu>

Looking at a new preprint by Tran Ngoc Nam reminded me of a conjecture
that occurred to me a few years ago.  Some of you may find this
interesting (and/or know how to prove it) ...

Let A be the mod 2 Steenrod algebra, and let alpha(t) be the usual
thing: the number of 1's in the 2adic expansion of t.

Conjecture  If alpha(t)>s, then Ext_A^{s,t}(Z/2,Z/2) = 0.

Remark 1  Back when this first occurred to me, Bob Bruner checked that
this was consistent with many dimensions of experimental evidence.

Remark 2 Via Bill Singer's transfer map, this is consistent with the
Peterson Conjecture (Reg Wood's Theorem) on the degrees of A-module
indecomposables in polynomial algebras.  Indeed, Wood's theorem implies
the conjecture in dimensions when Singer's map is an isomorphism, e.g.
for s < 4.   Of course, when Singer's map is known to be an
isomorphism (NOT always the case), this was proved by calculating both
domain and range enroute.

Nick Kuhn