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
>