%% %% 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 %% %% 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); \def\xmin{-12} \def\xmax{14} \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] \def\xmin{-12}\def\xmax{14} \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}