Subject: Question about Toda's Z_p-spaces
From: "Douglas C. Ravenel"
Date: Thu, 2 Nov 2006 18:11:54 -0500 (EST)
Dear Topologists,
In Toda's 1971 paper "Algebra of stable homotpy of Z_p-spaces and
applications," he defines a derivation \theta on the graded group of maps
from one Z_p-space (or module spectrum over the mod p Moore spectrum) to
another, and proves (Theorem 2.2)
\theta(\gamma \gamma')
= \theta(\gamma)\gamma'
+(-1)^{deg \gamma}\gamma\theta(\gamma')
I do not see how this degree is well defined. He has Z_p-spaces
W, X and Y with maps
\gamma':\Sigma^{i+j}W \to \Sigma^j X
and
\gamma :\Sigma^j X \to Y,
which means that the degree of \gamma is j.
But suppose X=\Sigma X' and we rewrite the maps as
\gamma':\Sigma^{i+j}W \to \Sigma^{j+1} X'
and
\gamma :\Sigma^{j+1}X' \to Y,
so now the degree of \gamma is j+1 and the formula above gives a different
answer.
What am I missing here?
Doug
Douglas C. Ravenel
Department of Mathematics |819 Hylan Building
University of Rochester |(585) 275-4415
Rochester, New York 14627 |FAX (585) 273-4655
Email: doug@math.removethis.rochester.&this.edu
Personal home page: http://www.math.rochester.edu/people/faculty/doug/