\documentclass[crop,tikz]{standalone} \usepackage{tikz} \usepackage{dsfont} \usepackage{amsmath, amsthm, amssymb} \usetikzlibrary{matrix, cd} \usepackage{etoolbox} \newcommand*\Z{\mathds{Z}} \newcommand*\CP{\mathbb{CP}} \newcommand*\ee[2]{H^{#1}(\CP^2) \otimes_{\Z} H^{#2}(S^1)} \newcommand*\zt[2]{#1 \otimes_{\Z} #2} \newcommand*\HT[2]{H^{#1}(\CP^2) \otimes_{\Z} #2} \newcommand*\HCP[1]{H^{#1}(\CP^2)} \newcommand\mygrid[3][]{ % this definition uses \foreach inside a matrix \let\mymatrixcontent\empty % see http://tex.stackexchange.com/q/60394/18228 \newcommand{\row}{% \foreach \j in {1,...,#2}{ \foreach \i in {1,...,#3} {% \begingroup\edef\x{\endgroup \noexpand\gappto\noexpand\mymatrixcontent{|[draw,minimum size=1cm,#1]| \&}}\x }% \gappto\mymatrixcontent{\\}% } } \row \matrix(m)[matrix of nodes,ampersand replacement=\&,row sep=-\pgflinewidth,column sep=-\pgflinewidth]{ \mymatrixcontent }; \foreach \x[count=\i from 0] in {1,...,#2}\node[left] at (m-\x-1.west) {$\i$}; \foreach \y[count=\j from 0] in {1,...,#3}\node[above] at (m-1-\y.north) {$\j$}; } \newcommand*\ZZ{|[draw,circle]| \Z_2} \usepackage{graphicx} \usepackage{caption} \begin{document} % Writing All E2 terms \begin{tikzpicture} \matrix (m) [matrix of math nodes, nodes in empty cells,nodes={minimum width=5ex, minimum height=5ex,outer sep=-5pt}, column sep=1ex,row sep=1ex]{ % S^1& & & & & & & & \\ % 0& \ee{0}{5}& \ee{1}{5}& \ee{2}{5}& \ee{3}{5}& \ee{4}{5}& \ee{5}{5}& &\\ 0& \ee{0}{4}& \ee{1}{4}& \ee{2}{4}& \ee{3}{4}& \ee{4}{4}& \ee{5}{4}& &\\ 0& \ee{0}{3}& \ee{1}{3}& \ee{2}{3}& \ee{3}{3}& \ee{4}{3}& \ee{5}{3}& &\\ 0& \ee{0}{2}& \ee{1}{2}& \ee{2}{2}& \ee{3}{2}& \ee{4}{2}& \ee{5}{2}& &\\ 0& \ee{0}{1}& \ee{1}{1}& \ee{2}{1}& \ee{3}{1}& \ee{4}{1}& \ee{5}{1}& &\\ 0& \ee{0}{0}& \ee{1}{0}& \ee{2}{0}& \ee{3}{0}& \ee{4}{0}& \ee{5}{0}& &\\ \quad\strut % & 0& 1& 2& 3& 4& 5& \CP_2& \strut \\}; % \draw[thick] (m-8-1.east) -- (m-1-1.east) ; \draw[thick] (m-8-1.north) -- (m-8-9.north) ; \end{tikzpicture} \vspace{1cm} % Plugging in known groups \begin{tikzpicture} \matrix (m) [matrix of math nodes, nodes in empty cells,nodes={minimum width=5ex, minimum height=5ex,outer sep=-5pt}, column sep=1ex,row sep=1ex]{ % S^1& & & & & & & & \\ % 0& \zt{\Z}{0}& \HT{1}{0}& \HT{2}{0}& \HT{3}{0}& \HT{4}{0}& \HT{5}{0}& &\\ 0& \zt{\Z}{0}& \HT{1}{0}& \HT{2}{0}& \HT{3}{0}& \HT{4}{0}& \HT{5}{0}& &\\ 0& \zt{\Z}{0}& \HT{1}{0}& \HT{2}{0}& \HT{3}{0}& \HT{4}{0}& \HT{5}{0}& &\\ 0& \zt{\Z}{0}& \HT{1}{0}& \HT{2}{0}& \HT{3}{0}& \HT{4}{0}& \HT{5}{0}& &\\ 0& \zt{\Z}{\Z}& \HT{1}{\Z}& \HT{2}{\Z}& \HT{3}{\Z}& \HT{4}{\Z}& \HT{5}{\Z}& &\\ 0& \zt{\Z}{\Z}& \HT{1}{\Z}& \HT{2}{\Z}& \HT{3}{\Z}& \HT{4}{\Z}& \HT{5}{\Z}& &\\ \quad\strut % & 0& 1& 2& 3& 4& 5& \CP_2& \strut \\}; % \draw[thick] (m-8-1.east) -- (m-1-1.east) ; \draw[thick] (m-8-1.north) -- (m-8-9.north) ; \end{tikzpicture} % Simplifying \begin{tikzpicture} \matrix (m) [matrix of math nodes, nodes in empty cells,nodes={minimum width=5ex, minimum height=5ex,outer sep=-5pt}, column sep=1ex,row sep=1ex]{ % S^1& & & & & & & & \\ % 0& 0& 0& 0& 0& 0& 0& &\\ 0& 0& 0& 0& 0& 0& 0& &\\ 0& 0& 0& 0& 0& 0& 0& &\\ 0& 0& 0& 0& 0& 0& 0& &\\ 0& \Z& \HCP{1}& \HCP{2}& \HCP{3}& \HCP{4}& \HCP{5}& &\\ 0& \Z& \HCP{1}& \HCP{2}& \HCP{3}& \HCP{4}& \HCP{5}& &\\ \quad\strut % & 0& 1& 2& 3& 4& 5& \CP_2& \strut \\}; % \draw[thick] (m-8-1.east) -- (m-1-1.east) ; \draw[thick] (m-8-1.north) -- (m-8-9.north) ; \end{tikzpicture} \begin{tikzpicture} \matrix (m) [matrix of math nodes, nodes in empty cells,nodes={minimum width=5ex, minimum height=5ex,outer sep=-5pt}, column sep=1ex,row sep=1ex]{ % S^1& & & & & & & & \\ % 0& 0& 0& 0& 0& 0& 0& &\\ 0& 0& 0& 0& 0& 0& 0& &\\ 0& 0& 0& 0& 0& 0& 0& &\\ 0& 0& 0& 0& 0& 0& 0& &\\ 0& \Z& \HCP{1}& \HCP{2}& \HCP{3}& \HCP{4}& \HCP{5}& \cdots&\\ 0& \Z& \HCP{1}& \HCP{2}& \HCP{3}& \HCP{4}& \HCP{5}& \cdots&\\ \quad\strut % & 0& 1& 2& 3& 4& 5& \CP_2& \strut \\}; % \draw[thick] (m-8-1.east) -- (m-1-1.east) ; \draw[thick] (m-8-1.north) -- (m-8-9.north) ; \draw[-stealth] (m-6-2.south east) -- (m-7-4.north west); \draw[-stealth] (m-6-3.south east) -- (m-7-5.north west); \draw[-stealth] (m-6-4.south east) -- (m-7-6.north west); \draw[-stealth] (m-6-5.south east) -- (m-7-7.north west); \draw[-stealth] (m-6-6.south east) -- (m-7-8.north west); \end{tikzpicture} \begin{tikzpicture} \draw[step=0.5cm,color=gray] (-1,-1) grid (1,1); \node at (-0.75,+0.75) {A}; \node at (-0.25,+0.75) {B}; \node at (+0.25,+0.75) {C}; \node at (+0.75,+0.75) {D}; \node at (-0.75,+0.25) {E}; \node at (-0.25,+0.25) {F}; \node at (+0.25,+0.25) {G}; \node at (+0.75,+0.25) {H}; % ... \node at (+0.75,-0.75) {Q}; \end{tikzpicture} \tikz\draw grid(4,4)foreach[count=~]\l in{1,...,16}{ ({.5+mod(~-1,4},{3.5-div(~-1,4})node{$H^1 \mathbb{CP}^\l$} }; \begin{tikzpicture}[scale=.5,] \mygrid[scale=.6]{6}{8} % two extra columns just for demonstration \foreach\j[count=\i,remember=\i as \x(initially 3)] in{1,...,6}{ \filldraw(m-\i-\j)circle[radius=2pt]; \filldraw(m-\x-\j)circle[radius=2pt]; \filldraw(m-\j-\x)circle[radius=2pt]; \filldraw(m-6-4)circle[radius=2pt]; } \end{tikzpicture} \begin{tikzcd} & & & & 0 \arrow[lllldd] \\ & & & & \\ H_2S^1 \arrow[rr] \arrow[rr, "{(i^*, -j^*)}"] & & H_2 M \oplus H_2 D^2 \arrow[rr, "l^* - r^*"] & & H_2 \mathbb{RP}^2 \arrow[lllldd, "\delta_2"'] \\ & & & & \\ H_1S^1 \arrow[rr, "{(i^*, -j^*)}"] & & H_1 M \oplus H_1 D^2 \arrow[rr, "l^*-r^*"] & & H_1 \mathbb{RP}^2 \arrow[lllldd, "\delta_1"'] \\ & & & & \\ H_0 S^1 \arrow[rr, "{(i^*, -j^*)_0}"] & & H_0 M \oplus H_0 D^2 \arrow[rr, "(l^* - r^*)_0"] & & H_0 \mathbb{RP}^2 \arrow[lllldd, "\delta_0"] \\ & & & & \\ 0 & & & & \end{tikzcd} \begin{tikzcd} & & & & 0 \arrow[lllldd, out=0, in=-180, "0"'] \\ & & & & \\ 0 \arrow[rr] \arrow[rr, "{(0, 0)}"] & & 0 \oplus 0 \arrow[rr, "0"] & & \delta M\mathbb{RP}^2 \arrow[lllldd, "\delta_2"', out=0, in=-180] \\ & & & & \\ \mathbb{Z} \arrow[rr, "{x\mapsto (2x, 0)}"] & & \mathbb{Z} \oplus 0 \arrow[rr, "{(x,0) \mapsto x}"] & & H_1 \mathbb{RP}^2 \arrow[lllldd, "\delta_1"', out=0, in=-180] \\ & & & & \\ \mathbb{Z} \arrow[rr, "{(i^0, -j^0)}"] & & \mathbb{Z} \oplus \mathbb{Z} \arrow[rr, "l^0 - r^0"] & & H_0 \mathbb{RP}^2 \arrow[lllldd, "\delta_0"', out=0, in=-180] \\ & & & & \\ 0 & & & & \end{tikzcd} \end{document}