%% %% Package: spectralsequences.sty version 1.0 %% Author: Hood Chatham %% Email: hood@mit.edu %% Date: 2017-06-18 %% License: Latex Project Public License %% %% 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{article} \usepackage[landscape,margin=2cm]{geometry} \usepackage{spectralsequences} \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 style=center, y axis gap=0.425cm, tick step=4, classes=Z2class, differentials=->, degree={-1}{#1-1}, scale=0.85, 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, ] \draw[background,xshift=-0.5cm,yshift=-0.51cm,step=1cm,gray,very thin] (\xmin+0.01,\ymin+0.01) grid (\xmax+0.9,\ymax+0.9); \pgfmathsetmacro\xitstart{int(int(\xmin/8)*8-16)} \pgfmathsetmacro\xitgap{int(\xitstart+4)} \pgfmathsetmacro\xitend{int(\xmax+2)} \pgfmathsetmacro\xmaxpp{int(\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) } } \pgfmathsetmacro\xitstart{int(\xitgap)} \pgfmathsetmacro\xitgap{int(\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=0.28cm,yshift=-0.4cm](-\antidiag,\antidiag)--(-1,1)--(-0.4,0)--(\xmax+2,0)--(\xmax+2,\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}