Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

$\newcommand\Ext{\mathrm{Ext}} \newcommand\Z{\mathbb{Z}} \newcommand\G{\mathbb{G}}$ The reference for this question will be the paper by Henn, Karamanov and Mahowald - "The homotopy of the $K(2)$-local Moore spectrum at the prime 3 revisted" (so everything is at $n=2,p=3$).

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)

$$ E_1^{p,q,t}=\Ext_{\mathbb Z_3[[\G_2^1]]}^q(C_p,M_t)\Longrightarrow H^{p+q}(\G_2^1,M_t) $$ for $M=(E_2)_*(V(0)) = (E_2)_*/(3)$ and then use this to calculate $H^*(\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 $\delta^1((\omega^2 u^{-4})^{3k+2} \beta)$?)

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.