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%%
%% Description: The homotopy fixed point spectral sequence for $C_2$ acting on $KU$ by complex conjugation, compare the odd-primary case HFPSS-EO2_3
%%
%% File: example_KUHFPSS.tex
%%
%% I think this is one of Eric Peterson's favorite spectral sequences, and his enthusiasm for it has rubbed off on me a bit.
%% Here our group is C_2 which acts on the periodicity element by -1, so again we have a norm element v in degree 2|G| = 4.
%% There are easier ways to understand this, but I explained the EO_3 one in terms of the comparison map from the ANSS,
%% and that's interesting here too. In the ANSS at 2, there is a differential d3(\alpha_3) = \alpha_1^4. Here a cobar calculation
%% shows that \alpha v = \alpha_3, so dividing the differential by \alpha gives d3(v) = \alpha_1^3. Now there's no Kudo differential
%% because the prime is 2, and the spectral sequence immediately collapses.
%%
%% Second, a demonstration of the falsehood of ku^{hC_2} = ko -- ku^{hC_2} has an extra generator as a ring, which is in degree -4.
%% In particular, it's not even connective.
%%
\documentclass{spectralsequence-example}
\begin{document}
\sseqset{
Z2class/.sseq style={circle,inner sep=0.3ex,fill=black},
Zclass/.sseq style={fill=none,draw,inner sep=0.6ex},
2Zclass/.sseq style={fill=none,rectangle,draw,inner sep=0.6ex,outer sep=0.5ex}
}
\begin{sseqdata}[
name=KRHFPSS,
x range={-12}{14},
y range={0}{10},
y axis type=center,
y axis gap=0.425cm,
tick step=4,
classes=Z2class,
differentials=->,
degree={-1}{#1-1},
scale=1.45,
right clip padding=0.1cm,
top clip padding=0.05cm,
x axis extend start=0cm,
x axis extend end=0.33cm,
y axis extend end=0.3cm,
grid=go
]
% This is just to make sure the bounding box doesn't move around
\path[background] (\xmin-1,\ymin-1) rectangle (\xmax+1,\ymax+1);
\sseqparseint\xitstart{\xmin/8*8-16} % division is integer division (I think with rounding towards 0...) so /8*8 rounds up to the nearest multiple of 8?
\sseqparseint\xitgap{\xitstart+4}
\sseqparseint\xitend{\xmax+2}
\sseqparseint\xmaxpp{\xmax+2}
\foreach \x in {\xitstart,\xitgap,...,\xitend} {
\class[Zclass](\x,0)
\foreach \z in {0,...,\xmaxpp} {
\class(\x+\z+1,\z+1)
\structline(\x+\z,\z)(\x+\z+1,\z+1)
}
}
\sseqparseint\xitstart{\xitgap}
\sseqparseint\xitgap{\xitstart+8}
\foreach \x in {\xitstart,\xitgap,...,\xitend} {
\foreach\z in {0,...,\xmax}{
\d4(\x+\z,\z)
}
\replaceclass[2Zclass](\x,0)
}
\end{sseqdata}
\printpage[name=KRHFPSS,page=0]
\newpage
\printpage[name=KRHFPSS,page=5]
\newpage
\begin{sseqpage}[name=KRHFPSS,page=0,keep changes]
\pgfmathsetmacro\antidiag{min(-\xmin,\ymax+0.8)}
\clip[background,xshift=0.2cm,yshift=-0.33cm](-\antidiag,\antidiag)--(-1,1)--(-0.4,0)--(\xmax + 0.28,0)--(\xmax+0.28,\antidiag)--cycle;
\foreach \z in {2,6}{
\doptions[draw=none]4(-\z-1,\z-1)
\structlineoptions[draw=none](-\z-1,\z-1)(-\z,\z)
\structlineoptions[draw=none](-\z-3,\z+1)(-\z-2,\z+2)
\replaceclass(-\z-2,\z+2)
}
\structlineoptions[draw=none](-3,1)(-2,2)
\end{sseqpage}
\newpage
\printpage[name=KRHFPSS,page=5]
\end{document}
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