From 787b1d312f205766480ce986a9f98548cc396dae Mon Sep 17 00:00:00 2001 From: Karl Berry Date: Mon, 19 Jun 2017 21:07:40 +0000 Subject: spectralsequences (19jun17) git-svn-id: svn://tug.org/texlive/trunk@44639 c570f23f-e606-0410-a88d-b1316a301751 --- .../spectralsequences/examples/example_KF3n.tex | 330 +++++++++++++++++++++ 1 file changed, 330 insertions(+) create mode 100644 Master/texmf-dist/doc/latex/spectralsequences/examples/example_KF3n.tex (limited to 'Master/texmf-dist/doc/latex/spectralsequences/examples/example_KF3n.tex') diff --git a/Master/texmf-dist/doc/latex/spectralsequences/examples/example_KF3n.tex b/Master/texmf-dist/doc/latex/spectralsequences/examples/example_KF3n.tex new file mode 100644 index 00000000000..dad6f608b58 --- /dev/null +++ b/Master/texmf-dist/doc/latex/spectralsequences/examples/example_KF3n.tex @@ -0,0 +1,330 @@ +%% +%% Package: spectralsequences.sty version 1.0 +%% Author: Hood Chatham +%% Email: hood@mit.edu +%% Date: 2017-06-18 +%% License: Latex Project Public License +%% +%% File: example_KF3n.tex +%% +%% 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=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 = {}, + } + %class labels=above, +} + +\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_3\}"](2,-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} \ No newline at end of file -- cgit v1.2.3