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diff --git a/Master/texmf-dist/doc/latex/spectralsequences/examples/HFPSS-KO.tex b/Master/texmf-dist/doc/latex/spectralsequences/examples/HFPSS-KO.tex new file mode 100644 index 00000000000..7409a2a77de --- /dev/null +++ b/Master/texmf-dist/doc/latex/spectralsequences/examples/HFPSS-KO.tex @@ -0,0 +1,88 @@ +%% +%% 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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