Subject: question for topology list
Date: 27 Jun 2001 17:50:04 -0700
From: palmieri@math.washington.edu (John H. Palmieri)
To: Don Davis <dmd1@lehigh.edu>

Hi Don,

Here's a question for the topology list:

Has anyone come across the following thing?  I would call it the
Steenrod algebra with the Frobenius map inverted.  Let p=2.  Consider

      F_2 [xi_1, xi_2, xi_3, ...
           plus all square roots of the xi_i's, all 4th roots, all 8th
           roots, etc.]

Put the usual coproduct on this:

  xi_n^{2^j} -> Sum_i xi_{n-i}^2^{i+j} tensor xi_i^2^j,

except this is valid for all integers j, not just non-negative ones.
Then I think this becomes a Hopf algebra, using Milnor's formula for
the antipode.  It's Z[1/2]-graded, zero in negative degrees.  You can
of course do the same thing when p is odd, using only the reduced
powers part of the Steenrod algebra.

So, what is its Ext algebra?  It should be Ext over the Steenrod
algebra with Sq^0 inverted, but what's that?

--
J. H. Palmieri
Dept of Mathematics, Box 354350    mailto:palmieri@math.washington.edu
University of Washington           http://www.math.washington.edu/~palmieri/
Seattle, WA 98195-4350