All details in the question are for the case $p=2$ though I expect the answer shouldn't be that different for odd primes.
Adams showed (i think it was him) the following statement:
- The element $h_j$ in the $1$-line of the ASS (of the sphere) persists (i.e. survives to $E_{\infty}$) iff the corresponding cohomology operation $Sq^{2^{j}}$ acts non-trivialy on the cohomology of some $2$-cell complex.
According to [Theorem $7.8.$] in Browder's article "The Kervaire Invariant of Framed Manifolds and its Generalization" which he assigns to Adams but which i was unable to find a proof of in the references he provides, The following statement is also true:
- An element $h_ih_j$ in the 2-line persists and evaluates to 1 on a secondary cohomology operation $\psi$ iff $\psi$ acts non-trivially on the cohomology of some 2-cell complex.
Its an algebraic consequence of the minimal resolution of $\mathbb{F}_2$ over the steenrod algebra that the $s$-line in the ASS can be interpreted as "relations (between relations)$^{s-1}$" and thus its natural to think of these as corresponding to some sort of $s$-order cohomology operations even if not in the most precise manner. The above statements are much stronger though as their consequences are about the stable homotopy groups of spheres.
Question 1: Is there a systematic connection between actions of $s$-order cohomology operations on the cohomology of $2$-cell complexes and persistence of elements in the $s$-line of the ASS? If so what is it?
In the end what i'm interested in is whether this can be used "systematically" to draw conclusions about differentials in Adams spectral sequences. I'll explain. It seems to me that the way Browder uses this theorem is by applying it to the $d_2$ in the ASS of a certain bordism spectrum he constructs to conclude that it vanishes at somewhete under certain conditions using the fact that he can calculate explicitly $d_1$ in the relevant ranges. I suspect that at least in the $E_2$-page the vanishing of the secondary cohomology operation associated to $d_1 \circ d_1=0$ always implies the vanishing of $d_2$ as the $d_2$ can be expressed as a certain Toda bracket which is "bounded from above" by the associated secondary cohomology operation. For higher differetials I have no idea...
Question 2: Is there a systematic technique lurking here for proving vanishing of $d_n$'s in Adams spectral sequences which exploits this interplay between differentials, Toda brackets, higher order cohomology operations, and the $E_2$ of the ASS of the sphere? If so what's the jist of it?
Finally let me add that i'm pretty convinced that there's an altogether different way to phrase this question which makes everything less confusing, comments about rephrasing and equivalent formulations (even if they are not full answers) would be very welcome.
