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%%
%% Description: The Serre spectral sequence for $\HF_3$ cohomology of $K(\F_3,n-1)\to pt \to K(\F_3,n)$ for $n=2,3,4$, with Bockstein SS's for going up to $\Zbb_3$ coefficients.
%%
%% This is a computation of the Z_3 cohomology of K(F_3,n). This was part of a way-too-hard problem on Haynes Algebraic Topology II problem set which asked to compute
%% the smallest k>0 such that H_{n+k}(K(--,n);Z_p) : Ab --> Ab is not the zero functor and to compute what functor it is. One of the main ideas is to use the Bockstein
%% spectral sequence, because it handles the badness of integral Kunneth for us too.
%%
\documentclass{article}
\usepackage[margin=0.2in,top=0.4in,landscape]{geometry}
\usepackage{spectralsequences}
\usepackage{amssymb}
\def\Z{\mathbb{Z}}
\begin{document}
\sseqset{
cohomological Serre grading,
title={Page \page},
class placement transform={scale=1.3, rotate=90},
class pattern=linear,
classes={draw=none},
class label handler = {\sseqnormalizemonomial{#1}},
bocksteinSS/.style={
page=0, y range={-1}{0},
no differentials, no labels, no y ticks, no title,
yscale=1.5, x axis extend end=40pt, y axis extend end = 40pt, x axis gap =50pt,
class label handler = {},
}
}
\begin{sseqdata}[name=K(Z/3;2),
x range={0}{8}, xscale=2.2, yscale=1.5,
x label={$H^*(K(\Z/3,2),\Z/3)$},
y label={$H^*(K(\Z/3,1),\Z/3)$},
right clip padding=20pt, x axis gap=20pt,
]
\foreach \x/\xlabel in
{ 0/1, 2/\iota_2, 3/\beta\iota_2, 4/\iota_2^2, 5/\iota_2\beta\iota_2, 6/\iota_2^3,
7/\iota_2^2\beta\iota_2, 7/P^1\beta\iota_2, 8/\iota_2^4, 8/\beta P^1\beta\iota_2 }
\foreach \y/\ylabel in
{ 0/1, 1/\alpha, 2/x, 3/\alpha x, 4/x^2, 5/\alpha x^2, 6/x^3} {
\class["\ylabel\xlabel"](\x,\y)
}
\class(9,0)\class(9,0)
\class(9,1)
\class(9,2)
\class(9,3)
\class(9,4)
\class(9,5)
\class(10,0)\class(10,0)\class(10,0)
\class(10,2)\class(10,2)
\class(10,4)\class(10,4)
\class(11,0)\class(11,0)
\class(11,2)
\class(12,0)
\class(13,0)
\class(14,0)
\class(15,0)
%\class["(\beta\iota_2)^2"](6,0)
\d2(0,1) %d2(a)=i
\d3(0,2) %d3(x) = b(i)
\d2(0,3) % d2(ax) = ix
\d3(0,4) % d2(x^2) = xb(i)
\d2(0,5) % d2(ax^2) = ix^2
%\d7
\d2(2,1) % d2(ai) = i^2
\d2(2,3) % d2(axi) = xi^2
\d2(2,5) % d2(ax^2i)=x^2i^2)
\d2(3,1) % d2(abi)=ib(i)
\d2(3,3)
\d2(3,5)
\d2(4,1) % d2(ai^2)=i^3
\d2(4,3)
\d2(4,5)
\d2(5,1,,1) %d2(ai_2bi_2) = i_2^2 bi_2
\d2(5,3,,1)
\d2(5,5,,1)
\d2(6,1,,1) %d2(ai_2^3) = i_2^4
\d2(6,3,,1)
\d2(6,5,,1)
\d2(7,1,1) % d2(ai_2^2bi_2) = i_2^3 bi_2
\d2(7,3,1)
\d2(7,5,1)
\d2(7,1,2) % d2(aP^1bi_2) = i_2 P^1bi_2
\d2(7,3,2)
\d2(7,5,2)
\d3(7,2,2) % d3(x P^1bi_2) = bi_2 P^1bi_2
\d3(7,4,2)
\d2(8,1,1,2) % d2(a i_2^4) = i_2^5
\d2(8,3,1,2)
\d2(8,5,1,2)
\d2(8,1,2,2) % d2(a bP^1bi_2) = i_2 bP^1bi_2
\d2(8,3,2,2)
\d2(8,5,2,2)
\d3(8,2,2) % d3(x bP^1bi_2) = bi_2 bP^1bi_2
\d3(8,4,2)
\d5(3,4,,2) % d5(x^2i_2) = bP^1bi_2 -- Kudo differential
\d7(0,6,,2) % d7(x^3) = P^1bi_2
\d7(2,6,,2)
\d7(3,6,,3)
\d7(4,6,,2)
\d7(5,6)
\d7(6,6)
\d7(7,6,1)
\d7(7,6,2)
\d7(8,6,1)
\d7(8,6,2)
\end{sseqdata}
\printpage[name=K(Z/3;2),page=2]
\newpage
\printpage[name=K(Z/3;2),page=3]
\newpage
\printpage[name=K(Z/3;2),page=5]
\newpage
\printpage[name=K(Z/3;2),page=6]
\newpage
\printpage[name=K(Z/3;2),page=7]
\newpage
\begin{sseqpage}[
name=K(Z/3;2), bocksteinSS,
x axis extend end = 30pt, y axis gap=30pt,
]
\begin{scope}[background]
\node[anchor=west] at (-2,0) { H^*(K(\Z/3,2),\Z/3)};
\node[anchor=west] at (-2,-1) { H^*(K(\Z/3,2),\Z)};
\end{scope}
\draw[->] (2,0) to[bend left=30, "\beta^1"] (3,0);
\draw[->] (4,0) to[bend left=30, "\beta^1"{pos = 0.55}] (5,0);
\draw[->] (6,0) to[bend right=30, "\beta^2"'] (7,0,1);
\draw[->] (7,0,2) to[bend left=30, "\beta^1"] (8,0,2);
\draw[->] (8,0,1) to[bend right=30, "\beta^1"' {pos = 0.46}] (9,0);
\class["\Z\{1\}"](0,-1)
\class["\Z/3\{\beta\iota_2\}"](3,-1)
\class["\Z/3\{\beta\iota_2^2\}"](5,-1)
\class["\Z/9\{\beta\iota_2^3\}"](7,-1)
\class["\Z/3\{\beta P^1\iota_2\}"](8,-1)
\end{sseqpage}
\newpage
\begin{sseqdata}[name=K(Z/3;3),xscale=2.2,yscale=1.7,y axis gap=40pt,x axis gap=30pt, x range={0}{9},
x label={$H^*(K(\Z/3,2),\Z/3)$},
y label={$H^*(K(\Z/3,1),\Z/3)$}
]
\foreach \x/\ymax/\xlabel in { 0/6/1, 3/6/\iota_3, 4/3/{{\beta\iota_3}} }
\foreach \y/\ylabel in { 0/1, 2/\iota_2, 3/{{\beta\iota_2}}, 4/\iota_2^2, 5/\iota_2{\beta\iota_2}, 6/\iota_2^3 } {
\class["\ylabel\xlabel"](\x,\y)
}
\sseqset{class label handler = {}}
\class["P^1\beta\iota_2"](0,7)
\class["\iota_2^2\beta\iota_2"](0,7)
\class["\beta P^1\beta\iota_2"](0,8)
\class["\iota_2^4"](0,8)
\class["P^1\iota_3"](7,0)
\class["\iota_3\beta\iota_3"](7,0)
\class["\beta P^1\iota_3"](8,0)
\class["(\beta\iota_3)^2"](8,0)
\class["P^1\beta\iota_3"](8,0)
\class["\beta P^1\beta\iota_3"](9,0)
\d3(0,2)
\d4(0,3)
\d3(0,4)
\d3(0,5)
\d7(0,6,,1)
\d3(0,7,2)
\d8(0,7,1,3)
\d3(0,8,2)
\d9(0,8,1)
\d5(3,4,,1)
\d3(4,2,,2)
\d4(4,3,,2)
\end{sseqdata}
\printpage[name=K(Z/3;3),page=3]
\newpage
\printpage[name=K(Z/3;3),page=4]
\newpage
\printpage[name=K(Z/3;3),page=5]
\newpage
\printpage[name=K(Z/3;3),page=7]
\newpage
\printpage[name=K(Z/3;3),page=9]
\newpage
\begin{sseqpage}[ name=K(Z/3;3), bocksteinSS ]
\begin{scope}[background]
\node[anchor=west] at (-2.2,0) { H^*(K(\Z/3,3),\Z/3)};
\node[anchor=west] at (-2.2,-1) { H^*(K(\Z/3,3),\Z)};
\end{scope}
\draw[->] (3,0) to[bend left=30, "\beta^1"] (4,0);
\draw[->] (7,0,1) to[bend right=30, "\beta^1"' {pos=0.6}] (8,0,1);
\draw[->] (7,0,2) to[bend left=30, "\beta^1" {pos=0.43}] (8,0,2);
\draw[->] (8,0,3) to[bend left=30, "\beta^1"{pos=0.37}] (9,0);
\class["\Z\{1\}"](0,-1)
\class["\Z/3\{\beta\iota_3\}"](4,-1)
\class["\Z/3\{\beta P^1\iota_3\}"](8,-1)
\class["\Z/3\{\beta(\iota_3\beta\iota_3)\}"](8,-1)
\class["\Z/3\{\beta P^1\beta\iota_3\}"](9,-1)
\end{sseqpage}
\begin{sseqdata}[name=K(Z/3;4),
xscale=2,yscale=1.5,y axis gap=20pt,x axis gap=30pt, x range={0}{10},
x label = {$H^*(K(\Z/3,3),\Z/3)$},
y label = {$H^*(K(\Z/3,4),\Z/3)$},
class label handler = {}
]
\class["1"](0,0)
\class["\iota_3"](0,3)
\class["\beta\iota_3"](0,4)
\class["P^1\iota_3"](0,7)
\class["\beta P^1\iota_3"](0,8)
\class["\iota_3\beta\iota_3"](0,7)
\class["(\beta\iota_3)^2"](0,8)
\class["P^1\beta\iota_3"](0,8)
\class["\beta P^1\beta\iota_3"](0,9)
\class["\iota_4"](4,0)
\class["\iota_3\iota_4"](4,3)
\class["\beta\iota_3\iota_4"](4,4)
\class["\beta\iota_4"](5,0)
\class["\iota_3\beta\iota_4"](5,3)
\class["\beta\iota_3\beta\iota_4"](5,4)
\class["\iota_4^2"](8,0)
\class["P^1\iota_4"](8,0)
\class["\iota_4 \beta\iota_4"](9,0)
\class["P^1\beta\iota_4"](9,0)
\class["\beta P^1\iota_4"](9,0)
\class["\beta P^1\beta\iota_4"](10,0)
\d4(0,3) % d4(i_3) = i_4
\d4(0,7,2)
\d4(4,3,,1)
\d4(5,3,,1)
\d5(0,4)
\d8(0,7,1,2)
\d9(0,8,1,2)
\d5(0,8,2)
\d9(0,8,3,3)
\d10(0,9)
%\d
\end{sseqdata}
\printpage[name=K(Z/3;4),page=4]
\newpage
\printpage[name=K(Z/3;4),page=5]
\newpage
\printpage[name=K(Z/3;4),page=8]
\newpage
\printpage[name=K(Z/3;4),page=9]
\newpage
\printpage[name=K(Z/3;4),page=10]
\newpage
\begin{sseqpage}[ name=K(Z/3;4), bocksteinSS, yscale=1.3 ]
\begin{scope}[background]
\node[anchor=west] at (-2.2,0) { H^*(K(\Z/3,4),\Z/3)};
\node[anchor=west] at (-2.2,-1) { H^*(K(\Z/3,4),\Z)};
\end{scope}
\draw[->] (4,0) to[bend left=30, "\beta^1"] (5,0);
\draw[->] (8,0,1) to[bend right=30, "\beta^1"' {pos=0.6}] (9,0,1);
\draw[->] (8,0,2) to[bend left=30, "\beta^1" {pos=0.6}] (9,0,3);
\draw[->] (9,0,2) to[ "\beta^1"] (10,0);
\class["\Z\{1\}"](0,-1)
\class["\Z/3\{\beta\iota_4\}"](5,-1)
\class["\Z/3\{\beta P^1\iota_4\}"](9,-1)
\class["\Z/3\{\beta(\iota_4^2)\}"](9,-1)
\class["\Z/3\{\beta P^1\beta\iota_4\}"](10,-1)
\end{sseqpage}
\end{document}
|
