%%
%% Description: The homotopy fixed point spectral sequence for a maximal finite subgroup of $\mathbb{G}_{p-1}$ acting on $E_{p-1}$ with $p=3$
%%
%% This is the homotopy fixed point spectral sequence for EO_2 at the prime 3. The maximal finite subgroup of the Morava stabilizer for E_{p-1} is
%% of size 2p(p-1)^2 = 24, and so there is a norm element v in degree 24. There's also a bunch of trace classes on the zero line, but the trace map
%% E_* --> H^*( G ; E_* ) is induced by the trace map E_n --> EO_n, so all of these classes are permanent cycles. They are hard to compute and we don't draw them.
%% We also have classes \alpha and \beta coming from the stabilizer action, which are the images of \alpha_1 and \beta_1 in the ANSS.
%% By looking at cobar representatives, we can see that v*\beta_1 is the image of \beta_{3/3}.
%% Thus, the Toda differential in the ANSS d_3(\beta_{3/3}) = \alpha \beta^3 forces also that d_3(v) = \alpha \beta^2. Likewise, the Toda "Kudo" differential
%% d_9( \alpha \beta_{3/3}^2 ) = \beta^7 gives us upon dividing by \beta twice that d_9(\alpha v^2) = \beta^5. At this point, there are no possible differentials.
%% We see that v^3 survives so EO_n* is 72 = 2p^2(p-1)^2 periodic. The picture is exactly the same at other odd primes. At 2, this degenerates to the
%% HFPSS for KO = KU^{hC_2} (see example_KUHFPSS).
%%
\documentclass[tooltips]{spectralsequence-example}
\begin{document}
\begin{sseqdata}[name=EO(2),Adams grading,
y range={0}{14},x range={0}{160},
xscale=0.15, x tick step=5,
classes={fill, tooltip={(\xcoord,\ycoord)}},
title=Page \page
]
\foreach \v in {0,...,8}{
\foreach \b in {0,...,11}{
\foreach \a in {0,1}{
\class(3*\a + 10*\b+24*\v,\a+2*\b)
\ifnum\b>0\relax
\structline(3*\a+10*\b-10+24*\v,\a+2*\b-2) (3*\a + 10*\b+24*\v,\a+2*\b)
\fi
}
\structline(10*\b+24*\v,2*\b)(3 + 10*\b + 24*\v,2*\b+1)
\ifnum \v = \numexpr\v/3*3\relax
\else
\ifnum\b<9\relax
\d5(10*\b+24*\v,2*\b)
\fi
\fi
}
}
% v^2ab^2 is in degree 2*24 + 3 + 2*10 = 71, 5
% b^{pn+1} = b^{7} is in degree 7*10 = 70,14
\foreach \v in {2,5}{
\foreach \b in {0,...,6}{
\d9(\v*24 + 3 +10*\b,1+2*\b)
}
}
\end{sseqdata}
\printpage[name=EO(2),page=0,title={\phantom{Page 5}}]
\newpage
\printpage[name=EO(2),page=5]
\newpage
\printpage[name=EO(2),page=9]
\newpage
\begin{sseqpage}[name=EO(2),page=10,title={Page $\infty$}]
\classoptions["a" left](3,1)
\classoptions["b" right](10,2)
\classoptions["ab" left](13,3)
\classoptions["b^2" right](20,4)
\classoptions["b^3" right](30,6)
\classoptions["b^4" right](40,8)
\classoptions["v^3" right](72,0)
% Doug's additions:
\classoptions["{\left<a,a,b^2\right>}" below](27,1)
\structline[dashed](27,1)(30,6)
\structline[dashed](37,3)(40,8)
\structline[dashed](99,1)(102,6)
\structline[dashed](109,3)(112,8)
\end{sseqpage}
\end{document}
