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Let $\G_2$ be the extended Morava stabiliser group, and let $\G_2^1$ be the kernel of the reduced norm $\G_2 \to \Z_3$. Then Henn,Karamanov and Mahowald use a permutation resolution of $\mathbb{G}_2^1$ to construct a spectral sequence (the algebraic spectral sequence)


E_1^{p,q,t}=\Ext_{\mathbb Z_3[[\G_2^1]]}^q(C_p,M_t)\Longrightarrow H^{p+q}(\G_2^1,M_t) $$ forM=(E_2)_*(V(0)) = (E_2)_/(3) and then use this to calculateH^(\G_2,(E_2)_/(3)), which in turn is used to calculate \pi_(L_{K(2)}V(0)) (via the Adams-Novikov spectral sequence) It's a very technical paper, and suffice to say I can't define all the terms I'm going to use in an introduction. But I am specifically interested in some of the differentials in the Adams-Novikov spectral sequence. In particular, let's just start with the first differential d_5(\Delta_k \tilde{\alpha} \beta) = \pm \Delta_{k-1}\beta^4 v_1. (The calculation is on pp. 32-33) Consider the short exact sequence$$ 0 \to \Sigma^4 (E_2)_*/(3) \stackrel{v_1}{\to} (E_2)_*/(3) \to (E_2)_*(3,u_1) \to 0$$It turns out that \Delta_k \tilde{\alpha} \beta is in the kernel of multiplication by v_1 and so must be in the image of the Bockstein \delta_{\G_2}^1 in H^*(\G_2,-) and \delta_{\G_2^1}^1 in H^*(\G_2^1,-) associated to the above short exact sequence (after appropriate suspension). So I go and look up the calculation of H^*(\G_2^1,(E_2)_*/(3,u_1)) and work out (by degree) what possible classes it can be to get that$$\delta_{\G_2^1}^1((\omega^2 u^{-4})^{3k+2}\beta) = \pm \Sigma^4 \Delta_k \tilde{\alpha}\beta,$$and in fact the same is true for \delta^1_{\G_2^1}. What I am interested in is how the geometric boundary theorem is then used to get the result. From Ravenel (2.3.4) this applies to a cofiber sequence$$W \stackrel{f}{\to} X \stackrel{g}{\to} Y \stackrel{h}{\to} \Sigma W with $E_*(h)=0$. If such a situation applies then we have maps $\delta_r$ for $2 \le r \le \infty$ such that $\delta_2 = \delta^1$ in our language (I think), but more importantly $\delta_r d_r = d_r \delta_r$.

Thus I think the calculation of $d_5(\Delta_k \tilde{\alpha} \beta)$ should basically be something akin to $d_5(\Delta_k \tilde{\alpha} \beta) = d_5\delta(-) = \delta d_5(-)$, where $d_5(-)$ is something we can calculate via knowledge of differentials in $H^*(\G_2,(E_*)/(3,u_1)).$

Is this the right idea? I am slightly confused by the $\Sigma^4$ floating around and more confused by the fact that HKM seem to be doing some additional calculations (e.g. why do I care about $\delta^1((\omega^2 u^{-4})^{3k+2} \beta)$?)