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diff --git a/Master/texmf-dist/doc/latex/spectralsequences/examples/HFPSS-EO2_3.tex b/Master/texmf-dist/doc/latex/spectralsequences/examples/HFPSS-EO2_3.tex new file mode 100644 index 00000000000..b90f129512f --- /dev/null +++ b/Master/texmf-dist/doc/latex/spectralsequences/examples/HFPSS-EO2_3.tex @@ -0,0 +1,75 @@ +%% +%% 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{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}
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