Subject: Re: two postings From: Tom Goodwillie Date: Fri, 3 Nov 2006 09:51:46 -0500 > > 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 You did say the *graded* group of maps. If X=\Sigma X' then there will be a canonical isomorphism between the degree j part of Map(X,Y) and the degree j+1 part of Map(X',Y), but what's the problem? (Presumably this isomorphism commutes with \theta only up to some sign.) > 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 >