I apologize if this question is of the form "what are some interesting problems in bla" but I was wondering if anybody have studied the following set-up:
Suppose that I have a parametrized spectra $E$ over a base space $B$ (something like the set-up of ABGHR or May-Sigurdsson) then, fiberwise, I can look at spectral sequences associated to each fiber - for concreteness maybe we can just think about the Adams spectral sequence converging to the homotopy groups of $E_b$.
So the first question about this is:
1) Have people studied this set-up (in other words, what kind of questions do people ask about this set-up, if any)? References would be appreciated!
2) I suppose that there should be an action of $\pi_1(B)$ on the differentials/groups of the spectral sequence. Is there any way this could be interesting? tractable?