%% %% 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, 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 = {$H^*(K(\Z/3,4),\Z/3)$}, y label = {$H^*(K(\Z/3,3),\Z/3)$}, 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}