Relations between several Adams spectral sequences (Q757893)

This paper expands \textit{H. R. Miller}'s work ``On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space'' [J. Pure Appl. Algebra 20, 287-312 (1981; Zbl 0459.55012)]. In this review we shall use the following notation: For a ring spectrum F denote by \(E_ r^{s,t}(X;F)\Rightarrow \pi_{t-s}(X)\) the usual F- based Adams spectral sequence for X. Here the double arrow is not meant to imply convergence. One has \(E_ 1^{s,t}(X;F)=\pi_ t(F\wedge \bar F^ s\wedge X)\) where \(\bar F\) is the cofibre of the unit map \(\iota\) : \(S^ 0\to E\) and \(E_ 2^{s,t}(-;F)=R^ s_ F\pi_ t\) is the s-th right derived functor (with respect to the class of F-injective spectra) of the functor \(\pi_ t\). For clarity the differentials will be written as \({}^ Fd_ r.\) Let G be a second ring spectrum and \(\lambda\) : \(F\to G\) a unit preserving map. Then Miller [loc. cit.] constructed a Mahowald spectral sequence \[ ^{Mah}E_ r^{*,*,*}(X;F,G)\Rightarrow E_ 2^{*,*}(X;G) \] by applying \(E_ 2^{*,*}(-;G)\) to the F-injective resolution of X. Its \(E_ 2\)-term \(^{Mah}E_ 2^{s,t,*}(-;F,G)\) is \(R^ s_ F(R^ t_ G\pi_*)\), the s-th right F-derived functor of \(R^ t_ G\pi_*\). A spectrum X is called (F,G)-primary iff for every \(\sigma\geq 0\) the G-based Adams spectral sequence \(E_ r^{s,t}(F\wedge \bar F^{\sigma}\wedge X;G)\Rightarrow \pi_{t-s}(F\wedge \bar F^{\sigma}\wedge X)\) converges and is trivial from \(E_ 2\) on. Under these assumptions Miller constructed a May spectral sequence \[ ^{May}E_ r^{*,*,*}(X;F,G)\Rightarrow E_ 2^{*,*}(X;F), \] where \(^{May}E_ 1^{s,t,u}(X;F,G)=^{Mah}E_ 2^{s,t,u}(X;F,G)\) for all s, t, u (here notation refers to the author's indexing which is different from Miller's [loc. cit.]). The author proves the following theorem about these spectral sequences (of which the second statement is already due to Miller). Theorem: Under the assumptions above 1. \(^{May}d_ 1\) and \(^{Mah}d_ 2\) commute, 2. if \(x\in E_ 2^{*,*}(X;G)\) is represented by \(\tilde x\in^{Mah}E_ 2^{*,*,*}(X;F,G)\) then \({}^ Gd_ 2(x)\) is represented (up to a certain sign) by \(^{May}d_ 2(\tilde x).\) 3. if \(x\in E_ 2^{*,*}(X;F)\) is represented by \(\tilde x\in^{May}E_ 2^{*,*,*}(X;F,G)\) then \({}^ Fd_ 2(x)\) is represented by \(^{Mah}d_ 2(\tilde x).\) 4. if \(y\in^{May}E_ 1^{*,*,*}(X;F,G)=^{Mah}E_ 2^{*,*,*}(X;F,G)\) is a permanent cycle in both spectral sequences representing \({}^ Fy\in E_ 2^{*,*}(X;F)\) and \({}^ Gy\in E_ 2^{*,*}(X;G)\), there is an element z representing \({}^ Fd_ 2(^ Fy)\) and \(\pm^ Gd_ 2(^ Gy)\) simultaneously. In fact the author proves a somewhat more general result (Theorem 7.2) by introducing the notation of \(E_ 2\)-functors and double \(E_ 2\)- functors. These are collections of interrelated functors from the homotopy category of spectra to the category of abelian groups, satisfying a collection of properties which allow for the construction of all these spectral sequences. The details are too complicated to state here; examples are the cases of ring spectra outlined above. The final section discusses some example calculations for the case of the Thom map BP\(\to H{\mathbb{Z}}/p\).