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