Computing Homotopy Fixed Point Spectral Sequences related to Morava E thories

Question

Given a finite subgroup of $G$ sitting inside the Morava stabilizer group $S_n$, we can form the homotopy fixed point spectrum $E_n^{hG}$. There is a spectral sequence with $E_2^{s,t} = H^s(G;\pi_t(E_n)) \Rightarrow \pi_{t-s}(E_n^{hG})$.

For resolving further differentials in this spcetral sequence I have seen a way by claiming some elements in the $E_2$ page is actually the image of certain elements of $\pi_*(S)$ under Hurewicz map and then by the known multiplicative relation(for example $h_1^3 = h_0^2 h_2$) we shall know that some elements in this spectral sequence must die and try to figure out who kills it or who it kills.

Question: what is the general way, or is there any general way of proving this kind of claim? i.e. how to prove that something in HFPSS $E_2$ is actually corresponding to the image of an element from $\pi_*(S)$?

I only know some classical results that might be related to this: If we start with $Ext_{BP_*BP}$ and keep track of the element we are interested in inside the chromatic spectral sequence, we will end up with continuous group cohomology of $S_n$(after tensoring with $\mathbb{F}_{p^n}$ at some point) thus cohomology of finite subgroup of it. Then we can look at how elements $t_i\in BP_*BP$ works as functions from $S_n$ to $\mathbb{F}_{p^n}$ to tell the image of $t_i$ in group cohomology and finally detecting elements of $\pi_*(S)$.

The second question comes from a concrete example of this. By classical result, we know that the non-exist elements $\beta_{p^j/p^j}$ and their monomials for $p \geq 5$ can be detected in the way above, in group cohomology of a finite group of order $p$ inside $S_{p-1}$ with coefficient $\mathbb{F}_{p^{p-1}}$. And my second question is the following:

Question: I believe that if I want to rephrase the detection of $\beta_{p^j/p^j}$ in modern language, it will becomes things like "If $\beta_{p^j/p^j}$ survives to $\pi_*(S)$, then it can be detected in the $C_p$ homotopy fixed point of $E_{p-1} $ for $p \geq 5$". So is this true? If so, if I can draw the $E_2$ of this spectral sequence, could I actually identify elements in $H^*(C_p;\mathbb{F}_{p^{p-1}})$ (where the classical detection happens) inside this $E_2$? Or is there a suitably defined map from $E_2$ to $H^*(C_p;\mathbb{F}_{p^{p-1}})$ which maps image of $\beta_{p^j/p^j}$ to non-zero elements?

Any help or hint is really appreciated.

Answer 1

In Ravenel's paper on the Arf invariant, he shows (p. 439) that there is a composite of maps $$ \mathrm{Ext}_{BP_*BP}(BP_*,BP_*) \to H^*_c(\mathbb{S}_n,E_*) \to H^*(C_p,E_*/\frak{m}), $$ under which the images of $\alpha_1,\beta_1$ and $\beta_{p,p}$ are non-zero. You can use these to determine the Hurewicz image of classes under the composite $$ \mathrm{Ext}_{BP_*BP}(BP_*,BP_*) \to H^*_c(\mathbb{S}_n,E_*) \to H^*(G,E_*). $$

Since this is probably the argument you outline in the first paragraph, it is useful to point out another method, via the functoriality of the Greek letter construction; this is the argument used in Proposition 7 of this paper by Goerss, Henn, and Mahowald.

I'm afraid I don't fully understand the second part of the question; if a class survives in the $BP$-Adams spectral sequence, and is detected in the $C_p$-HFPSS spectral sequence under the map $$ \mathrm{Ext}_{BP_*BP}(BP_*,BP_*) \to H^*(C_p,E_*), $$ then it must also survive in the latter spectral sequence.