Subject: A response to John Palmieri'S question Date: Wed, 04 Jul 2001 16:15:27 +0000 From: Norihiko Minami Concerning Ext over the Steenrod algebra with Sq^0 inverted, I had the following conjecture: New Doomsday Conjecture: For each $s,$ there exists some integer $n(s)$ such that no nontrivial element in the image of \[ ({Sq}^0)^{n(s)} Ext_{A}^{s, *}(Z/2,Z/2) \subseteqq Ext_{A}^{2^{n(s)}s, *}(Z/2,Z/2) \] is a nontrivial permanent cycle. Of course, we may also formulate the odd primary analogue, but the conjecture itself appears to be extremely difficult. Actually, a positive answer to this conjecture even for the case $s=2$ would imply there are just finitely many Kervaire invariant one elements. More may be found in my papers 99i:55023 Minami, Norihiko The iterated transfer analogue of the new doomsday conjecture. Trans. Amer. Math. Soc. 351 (1999), no. 6, 2325--2351. 99g:55018 Minami, Norihiko On the Kervaire invariant problem. Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997), 229--253, Contemp. Math., 220, Amer. Math. Soc., Providence, RI, 1998. 96d:55010 Minami, Norihiko The Adams spectral sequence and the triple transfer. Amer. J. Math. 117 (1995), no. 4, 965--985. Nori Minami > Subject: question for topology list > Date: 27 Jun 2001 17:50:04 -0700 > From: palmieri@math.washington.edu (John H. Palmieri) > > 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 > >