From: Norio IWASE
Date: Sat, 4 Nov 2006 07:48:24 +0900
Dear Doug,
I can not see what is the problem... if we name the identity map as
i : \Sigma^j X = \Sigma^{j+1} X', the new \gamma' and \gamma can be
expressed as i\gamma' and \gamma i^{-1} and we have no contradiction:
by Toda's formula, we should have
\theta(1) = \theta(1 1) = 1\theta(1) + 1\theta(1) = 2\theta(1)
which implies \theta(1) = 0 and hence we should also have
0 = \theta(1) = \theta(i^{-1}i) = \theta(i^{-1})i - i^{-1}\theta(i).
The new maps i\gamma' and \gamma i^{-1} should satisfy
\theta(i\gamma') = \theta(i)\gamma' - i\theta(\gamma') and
\theta(\gamma i^{-1})
= \theta(\gamma)i^{-1} + (-1)^j\gamma\theta(i^{-1}).
Then we will have the stable relation
\theta((\gamma i^{-1}) (i\gamma'))
= \theta(\gamma i^{-1})(i\gamma')
+ (-1)^{j+1}(\gamma i^{-1})\theta(i\gamma')
= \theta(\gamma)(\gamma') + (-1)^j\gamma\theta(i^{-1})i\gamma'
+ (-1)^j\gamma\theta(\gamma') - (-1)^j\gamma i^{-1}\theta(i)\gamma'
which coincides in stable category with the original answer
\theta(\gamma)(\gamma') + (-1)^j\gamma\theta(\gamma'),
since \theta(i^{-1})i - i^{-1}\theta(i) = 0. So it looks consistent,
if we are working in stable category.
Best Regards, Norio
_____________________________________________________________________
From: "Hans-J. Baues"
Date: Mon, 6 Nov 2006 13:31:28 +0100 (CET)
Dear Doug,
let R be a connective ring spectrum and let A be the algebra of homotopy
groups of R. Then
the desuspension M of A is an A-bimodule with the usual sign convention.
If the unit 1 of A is torsion then a derivation theta from A to M (of
degree 0)
is defined which is unique up to inner derivations.
In your case R is the endomorphism spectrum of a one point union U of Z_p
spaces.
Your remark shows that R depends on U, that is,
U=XwedgeYwedgeW and U=sigma(X)wedgeYwedgeW
yield different ring spectra R.
Best regards
Hans
Prof. Dr. H.-J. Baues
Max-Planck-Institut fuer Mathematik
Vivatsgasse 7, D-53111 Bonn,
Postal Address: P.O.Box: 7280, D-53072 Bonn,
Germany
E-mail: baues@mpim-bonn.mpg.de
Phone : +49 228 402 235