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\begin{document}
\title{Lots of diagrams}
\author{Daryl  DeFord}
\address{Department of Mathematics\\
Dartmouth College}
\email{ddeford@math.dartmouth.edu}


\date{\today}


\maketitle
\section{Commutative Diagrams}
\subsection{}
\bcd
0\arrow{r}&C'\arrow[swap]{r}{u}&C\arrow[swap]{r}{v}&C''\arrow{r}\arrow[bend right,dotted,swap]{l}{w}&0\\
\ &\ &\ &\mathscr{B}\arrow[swap]{u}{\iota}\arrow{ul}{\pi^{-1}}
\ecd \\ 
\subsection{}
\subsection{}
\bcd 
0\arrow[bend left]{d}{id}\arrow[bend right, swap]{d}{0}\arrow{r}&M\arrow{dl}\arrow[bend left]{d}{id}\arrow[bend right, swap]{d}{0}\arrow{r}{i}&N\arrow[bend left]{d}{id}\arrow[bend right, swap]{d}{0}\arrow{dl}{r}\arrow{r}{\pi}&P\arrow[bend left]{d}{id}\arrow[bend right, swap]{d}{0}\arrow{dl}{s}\arrow{r}&0\arrow{dl}\arrow[bend right, swap]{d}{0}\arrow[bend left]{d}{id}\\
0\arrow{r}&M\arrow{r}{i}&N\arrow{r}{\pi}&P\arrow{r}&0\\
\ecd \\ 
\subsection{}

\bcd
0\arrow{d}{0=id}\arrow{r}&ker(i)\arrow{r}{}\arrow{d}{0=id}&ker(\pi)/im(i)\arrow{d}{0=id}\arrow{r}{}& P/im(\pi)\arrow{d}{0=id}\arrow{r}&0\arrow{d}{0=id}\\
0\arrow{r}&ker(i)\arrow{r}{}&ker(\pi)/im(i)\arrow{r}{}& P/im(\pi)\arrow{r}&0
\ecd \\ 
\subsection{}
 
\bcd
X\arrow{r}{i_X}\arrow{dr}{f_X}&X\coprod Y\arrow[dotted]{d}{h}& Y\arrow{dl}{f_Y}\arrow{l}{i_Y}\\
\ &Z&\ 
\ecd \\ 
\subsection{}


\bcd
\ &\ &\ &\ &D'_{n+1}\arrow{d}{\partial_D}\arrow{r}{i}&D_{n+1}\arrow{r}{\pi}\arrow{d}{\partial_D}&D''_{n+1}\arrow{d}{\partial_D}\\
C'_{n+1}\arrow{urrrr}{f}\arrow{d}{\partial_C}\arrow[swap]{r}{i}&C_{n+1}\arrow{urrrr}{f}\arrow[swap]{r}{\pi}\arrow{d}{\partial_C}&C''_{n+1}\arrow{urrrr}{f}\arrow{d}{\partial_C} &\ &D'_{n}\arrow{d}{\partial_D}\arrow{r}{i}&D_{n}\arrow{r}{\pi}\arrow{d}{\partial_D}&D''_{n}\arrow{d}{\partial_D}\\
C'_{n}\arrow{urrrr}{f}\arrow{d}{\partial_C}\arrow[swap]{r}{i}&C_{n}\arrow{urrrr}{f}\arrow[swap]{r}{\pi}\arrow{d}{\partial_C}&C''_{n}\arrow{urrrr}{f}\arrow{d}{\partial_C} &\ &D'_{n-1}\arrow{r}{i}&D_{n-1}\arrow{r}{\pi}&D''_{n-1}\\
C'_{n-1}\arrow{urrrr}{f}\arrow[swap]{r}{i}&C_{n-1}\arrow{urrrr}{f}\arrow[swap]{r}{\pi}&C''_{n-1}\arrow{urrrr}{f} &\ &\ &\ &\ &\ &\ 
\ecd \\ 
\subsection{}

\bcd
0\arrow[dotted]{r}&ker(f')\arrow[dotted]{r}\arrow[hookrightarrow]{d}{i}&ker(f)\arrow[dotted]{r}{c}\arrow[hookrightarrow]{d}{i} &ker(f'')\arrow[hookrightarrow]{d}{i}&\ \\
0\arrow{r}&C'\arrow[hookrightarrow]{r}{i}\arrow{d}{f'}&C\arrow{d}{f}\arrow[twoheadrightarrow]{r}{\pi}&C''\arrow{d}{f''}\arrow{r}&0\\
0\arrow{r} &D'\arrow[hookrightarrow]{r}{i}\arrow[twoheadrightarrow]{d} &D \arrow[twoheadrightarrow]{d}\arrow[twoheadrightarrow]{r}{\pi} &D''\arrow[twoheadrightarrow]{d}\arrow{r}&0 \\
\ &coker(f')\arrow[dotted]{r}{d}&coker(f)\arrow[dotted]{r} &coker(f'')\arrow[dotted]{r}&0 \\
\ecd \\ 
\subsection{}

\bcd
0\arrow[dotted]{r}&ker(f)\arrow[dotted]{r}\arrow[hookrightarrow]{d}{i}&ker(g\circ f)\arrow[dotted]{r}\arrow[hookrightarrow]{d}{i_1\times 0} &ker(g)\arrow[hookrightarrow]{d}{i}&\ \\
0\arrow{r}&M\arrow[hookrightarrow]{r}{i\times f}\arrow{d}{f}&M\oplus N\arrow{d}{g\circ f\circ \pi_1\times i\circ \pi_2}\arrow[twoheadrightarrow]{r}{-f\circ\pi_1+\pi_2}&N\arrow{d}{g}\arrow{r}&0\\
0\arrow{r} &N\arrow[hookrightarrow]{r}{g\times i}\arrow[twoheadrightarrow]{d} &P\oplus N \arrow[twoheadrightarrow]{d}\arrow[twoheadrightarrow]{r}{-\pi_1+g\circ\pi_2} &P\arrow[twoheadrightarrow]{d}\arrow{r}&0 \\
\ &coker(f)\arrow[dotted]{r}&coker(g\circ f)\arrow[dotted]{r} &coker(g)\arrow[dotted]{r}&0 \\
\ecd \\ 
\subsection{}

\bcd
\ &\arrow[dotted]{d}{\partial}&\arrow[dotted]{d}{\partial}&\arrow[dotted]{d}{\partial}\\
0\arrow{r}&Z_{n+1}\arrow{d}{\partial}\arrow[hookrightarrow]{r}{i}&C_{n+1}\arrow{d}{\partial}\arrow[twoheadrightarrow]{r}{\partial}&B_{n}\arrow{d}{\partial}\arrow{r}&0\\
0\arrow{r}&Z_n\arrow{d}{\partial}\arrow[hookrightarrow]{r}{i}&C_n\arrow{d}{\partial}\arrow[twoheadrightarrow]{r}{\partial}&B_{n-1}\arrow{d}{\partial}\arrow{r}&0\\
0\arrow{r}&Z_{n-1}\arrow[dotted]{d}{\partial}\arrow[hookrightarrow]{r}{i}&C_{n-1}\arrow[dotted]{d}{\partial}\arrow[twoheadrightarrow]{r}{\partial}&B_{n-2}\arrow{r}\arrow[dotted]{d}{\partial}&0\\
\ &\ &\ &\ &\ 
\ecd \\ 
\subsection{}



\bcd
\arrow[dotted]{r}{i_*}&H_n(C_*)\arrow{r}{\partial_*=0}&H_n(B_*)\arrow{r}{\Delta}&H_{n-1}(Z_*)\arrow{r}{i_*}&H_{n-1}(C_*)\arrow[dotted]{r}{\partial_*=0}&\ 
\ecd \\ 
\subsection{}

\bcd
\arrow[dotted]{r}{i_*}&H_n(C_*)\arrow{r}{\partial_*=0}&0\arrow{r}&H_n(B_*)\arrow{r}{\Delta}&H_{n-1}(Z_*)\arrow{r}{i_*}&H_{n-1}(C_*)\arrow{r}{\partial_*=0}&0\arrow[dotted]{r} &
\ecd \\ 
\subsection{}

\bcd
\  &Y\arrow{dd}{i}\\
X\arrow{ur}{f}\arrow[hookrightarrow]{dr}{j}&\ \\
\ &\ M_f
\ecd \\ 
\subsection{} 

\bcd 
\tilde{H}_1(\mathbb{S}^1)\arrow[equal]{r}&H_1(\mathbb{S}^1,D^1_+)\arrow[equal]{d}&\ \\
\ &H_1(D^1_-,\mathbb{S}^0)\arrow[equal]{r}& \tilde{H}_0(\mathbb{S}^0)
\ecd \\ 
\subsection{}

\bcd
\ & y\arrow{d}{\partial}& z\arrow[dotted]{l}\\
i^{-1}(dy)&dy\arrow[dotted]{l}&\ 
\ecd \\ 
\subsection{}

\bcd
\Sigma\mathbb{S}^n\arrow{d}{\varphi}\arrow{r}{\Sigma f}&\Sigma\mathbb{S}^n\arrow{d}{\varphi}\\
\mathbb{S}^{n+1}\arrow{r}{\Sigma f'}&\mathbb{S}^{n+1}
\ecd \\ 
\subsection{}

\bcd[row sep=1em]
\ &0\arrow{d}&&0\arrow{d}&0\arrow{d}&&\ \\
\ \arrow[dotted]{r}&\tilde{H_1}(X)\arrow{d}\arrow{r}{j_*}&H_1(X,A)\arrow[equal]{d}\arrow{r}{\overline{\partial}}&\tilde{H_0}(A)\arrow{d}\arrow{r}{i_*}&\tilde{H_0}(X)\arrow{d}\arrow{r}{j_*}&H_0(X,A)\arrow[equal]{d}\arrow{r}&0\\
\ \arrow[dotted]{r}&H_1(X)\arrow{d}\arrow{r}{j_*}&H_1(X,A)\arrow{r}{\partial}&H_0(A)\arrow{d}\arrow{r}{i_*}&H_0(X)\arrow{d}\arrow{r}{j_*}&H_0(X,A)\arrow{r}&0\\
\ &H_0(*)&\ &H_0(*)\arrow[equal]{r}&H_0(*)&\ &\ 
\ecd \\ 
\subsection{}

\bcd
\ \arrow[dotted]{r}&H'_p\arrow{r}{i}\arrow{d}{f'}&H_p\arrow{d}{f}\arrow{r}{\pi}&H_p''\arrow{d}{f''}\arrow{r}{\partial}&H_{p-1}'\arrow{d}{f'}\arrow[dotted]{r}&\ \\
\ \arrow[dotted]{r}&G'_p\arrow{r}{j}&G_p\arrow{r}{\rho}&G_p''\arrow{r}{\partial'}&G_{p-1}'\arrow[dotted]{r}&\ 
\ecd \\ 
\subsection{}

\bcd
\ \arrow[dotted]{r}&H'_p\arrow{r}{\Delta}&H_p\oplus G_p'\arrow{r}{\nabla}&G_p\arrow{r}{\overline{\partial}}&H_{p-1}'
\arrow[dotted]{r}& \ 
\ecd \\ 
\subsection{}

\bcd
\ \arrow[dotted]{r}&ker(\partial_{q+1})\oplus im(\partial_{q+1})\arrow{r}&ker(\partial_{q})\oplus im(\partial_{q})\arrow{r}&ker(\partial_{q-1})\oplus im(\partial_{q-1})\arrow[dotted]{r}&
\ecd \\ 
\subsection{}

\bcd[row sep=.15em]
\ \arrow[dotted]{r}&ker(\partial_{q+1})\arrow{r}&0&\ \\
\ &\oplus&\\
0\arrow{r}&im(\partial_{q+1})\arrow[hookrightarrow]{r}&ker(\partial_q)\arrow{r}& 0\\
\ &\ &\oplus& \ \\
\ &0\arrow{r}&im(\partial_{q})\arrow[hookrightarrow]{r}&ker(\partial_{q-1})\arrow{r}& 0\\
\ecd \\ 
\subsection{}

\bcd
H_q(X,A)\arrow{r}{d_{(X,A)}}\arrow[swap]{d}{j_*}&H_{q-1}(A)\arrow{d}{j_*}\\
H_q(X,A)\arrow[swap]{r}{d_{(X,A,B)}}&H_{q-1}(A,B)
\ecd \\ 
\subsection{}

\bcd
H_q(X,A)\arrow{r}{d_{(X,A)}}\arrow[swap]{dr}{d_{(X,A,B)}}&H_{q-1}(A)\arrow{d}{j_*}\\
\ &H_{q-1}(A,B)
\ecd \\ 
\subsection{}

\bcd 
0\arrow{r}&C_*(X)\arrow{r}{p}&C_*(X)\arrow{r}{\pi}&C_*(X)/pC_*(X)\arrow{r}&0
\ecd \\ 
\subsection{}

\bcd 
\left(\ \right. 0\arrow{r}&pC_*(X)\arrow[hookrightarrow]{r}{\iota}&C_*(X)\arrow{r}{\pi}&C_*(X)/pC_*(X)\arrow{r}&0 \left. \right)
\ecd \\ 
\subsection{}

\bcd 
\arrow[dotted]{r}&H_q(X)\arrow{r}{p_*}&H_q(X)\arrow{r}{\pi_*}&H_q(C_*(X)/pC_*(X))\cong H_q(X;\pz)\arrow{r}{d}&H_{q-1}(X)\arrow[dotted]{r}&\ 
\ecd \\ 
\subsection{}

\bcd 
0\arrow{r}\arrow{d}&H_q(X)/pH_q(X)\arrow{d}{\sim}\arrow{r}{i}&H_q(X,\pz)\arrow{r}{d}\arrow{d}{\hat{f}_*}&_p(H_{q-1}(X))\arrow{r}\arrow{d}{\sim}&0\arrow{d}\\
0\arrow{r}&H_q(Y)/pH_q(Y)\arrow{r}{i}&H_q(Y,\pz)\arrow{r}{d}&_p(H_{q-1}(Y))\arrow{r}&0
\ecd \\ 
\subsection{}

\bcd
\ \arrow[dotted]{r}&H_q(A)\arrow{r}\arrow{d}{f|_{A*}}&H_q(X)\arrow{r}\arrow{d}{f_*}&H_q(X,A)\arrow{r}\arrow{d}{f_*}&H_{q-1}(A)\arrow{r}\arrow{d}{f|_{A*}}&H_{q-1}(X)\arrow[dotted]{r}\arrow{d}{f_*}&\ \\
\ \arrow[dotted]{r}&H_q(B)\arrow{r}&H_q(Y)\arrow{r}&H_q(Y,B)\arrow{r}&H_{q-1}(B)
\arrow{r}&H_{q-1}(Y)\arrow[dotted]{r}&\ 
\ecd \\ 
\subsection{}

\small{\bcd
\ \arrow[dotted]{r}&H_q(X\times \mathbb{S}^{n-1},X\times *)\arrow{r}&H_q(X\times D_n^{+\varepsilon},X\times D_1^{+\varepsilon})\oplus H_q(X\times D_n^{-\varepsilon},X\times D_1^{-\varepsilon})\arrow{dll}\\
H_q(X\times \mathbb{S}^n,X\times *)\arrow{r}&H_{q-1}(X\times \mathbb{S}^{n-1},X\times *)\arrow{r}& H_{q-1}(X\times D_n^{+\varepsilon},X\times D_1^{+\varepsilon})\oplus H_{q-1}(X\times D_n^{-\varepsilon},X\times D_1^{-\varepsilon})
\ecd \\ 
\subsection{}}


\bcd[column sep=huge]
\ \arrow[dotted]{r}&E_{n+1}\arrow{r}{d_{n+1}}\arrow[bend left]{d}{id}\arrow[bend right,swap]{d}{0}\arrow[dotted,swap]{dl}{h_{n+1}}&E_{n}\arrow{r}{d_{n}}\arrow[bend left]{d}{id}\arrow[swap,bend right]{d}{0}\arrow[swap]{dl}{h_{n}}&E_{n-1}\arrow[bend left]{d}{id}\arrow[bend right,swap]{d}{0}\arrow[swap]{dl}{h_{n-1}}\arrow[dotted]{r}&\ \\
\ \arrow[dotted]{r}&E_{n+1}\arrow[swap]{r}{d_{n+1}}&E_{n}\arrow[swap]{r}{d_{n}}&E_{n-1}\arrow[swap]{r}{d_{n-1}}\arrow[dotted]{r}&\ 
\ecd \\ 
\subsection{}

\bcd
0 \arrow[dotted]{r}&\bigoplus\limits_{\binom{n+1}{m+1}}\mathbb{Z}\arrow{r}&\bigoplus\limits_{\binom{n+1}{m}}\mathbb{Z}\arrow[dotted]{r}&\cdots\arrow[dotted]{r}&\cdots\bigoplus\limits_{\binom{n+1}{1}}\mathbb{Z}\arrow{r}&0
\ecd \\ 
\subsection{}

\bcd
\ \arrow[dotted]{r}&0\arrow{r}&\mathbb{Z}\arrow{r}{A}&\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}\arrow{r}{B}&\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}\arrow{r}&0
\ecd \\ 
\subsection{}

\bcd
 \ \arrow[dotted]{r}&0\arrow{r}&\mathbb{Z}\arrow{r}{C}&\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}\arrow{r}{D}&\mathbb{Z}\arrow{r}&0
\ecd \\ 
\subsection{}

\section{Galois Diagrams}

\bcd
\ &F(\sqrt{m},\sqrt{n})\arrow[swap, dash]{dl}{?_1}\arrow[dash]{d}{?_2}\arrow[dash]{dr}{?_3}&\ \\
F(\sqrt{m})&F(\sqrt{n})&F(\sqrt{mn})\\
\ & F\arrow[dash]{ul}{2}\arrow[dash]{u}{2}\arrow[swap, dash]{ur}{2} & \
\ecd \\ 
\subsection{}

\bcd
\ &\mathbb{Q}(\sqrt{m_1},\ldots,\sqrt{m_{k+1}})\arrow[dash]{dl}{?}\arrow[dash]{dr}{?}\arrow[dash]{d}{?}&\ \\
\mathbb{Q}(\sqrt{m_1},\ldots,\sqrt{m_k})&\mathbb{Q}(\sqrt{m_{k+1}},\ldots,\sqrt{m_k})&\mathbb{Q}(\sqrt{m_1m_{k+1}},\ldots,\sqrt{m_k})\\
\ &\mathbb{Q}(\sqrt{m_2},\ldots,\sqrt{m_k})\arrow[dash]{ul}{2}\arrow[dash]{ur}{2}\arrow[dash]{u}{2}\arrow[dash,bend left]{uu}{4}&\ \\
\mathbb{Q}(\sqrt{m_1}) \arrow[dash]{uu}{2^{k-1}}&\mathbb{Q}(\sqrt{m_{k+1}})\arrow[dash]{uu}{2^{k-1}}&\mathbb{Q}(\sqrt{m_1m_{k+1}})\arrow[dash]{uu}{2^{k-1}}\\
\ &\mathbb{Q}\arrow[dash]{ul}{2}\arrow[dash]{ur}{2}\arrow[dash]{u}{2}
\arrow[dash]{uuul}{2^k}\arrow[dash]{uuur}{2^k}\arrow[dash,bend right,swap]{uuu}{2^k}\arrow[dash,bend left]{uu}{2^{k-1}}&\ 
\ecd \\ 
\subsection{}

\bcd
\ &\mathbb{Q}(\sqrt{m_1},\sqrt{m_2}\ldots,\sqrt{m_{k+1}})\arrow[swap, dash]{dl}{?_1}\arrow[dash]{d}{?_2}\arrow[dash]{dr}{?_{k+1}}&\ \\
\mathbb{Q}(\sqrt{m_2},\sqrt{m_3}\ldots,\sqrt{m_{k+1}})&\mathbb{Q}(\sqrt{m_1},\sqrt{m_3}\ldots,\sqrt{m_{k+1}})\cdots&\cdots\mathbb{Q}(\sqrt{m_1},\sqrt{m_2}\ldots,\sqrt{m_{k}})\\
\mathbb{Q}(\sqrt{m_{k+1}})\arrow[dash]{u}{} & \mathbb{Q}\arrow[dash]{l}{2}\arrow[dash,swap]{r}{2}\arrow[dash]{ul}{2^k}\arrow[dash]{u}{2^k}\arrow[swap, dash]{ur}{2^k} & \mathbb{Q}(\sqrt{m_1})\arrow[dash]{u}{}
\ecd \\ 
\subsection{}

\bcd
\ &\mathbb{Q}(\alpha)\arrow[dash]{d}{}\\
\ &\mathbb{Q}(\beta)\arrow[dash]{d}{}\\
\ &\mathbb{Q}\arrow[bend left,dash]{uu}{11}
\ecd \\ 
\subsection{} 

\bcd
\ &F(\alpha)\arrow[dash,swap]{dl}{\leq n}\arrow[dash]{dr}{?}&\ \\
F&\ &\mathbb{Q}(\alpha)\\
\ &\mathbb{Q}\arrow[dash,swap]{ur}{n}\arrow[dash]{ul}{2} &\ \\
\ecd \\ 
\subsection{}


\bcd
\ &F(\alpha)\arrow[dash,swap]{dl}{n}\arrow[dash]{dr}{2}\arrow[dash,dotted]{dd}{2n}&\ \\
F&\ &\mathbb{Q}(\alpha)\\
\ &\mathbb{Q}\arrow[dash,swap]{ur}{n}\arrow[dash]{ul}{2} &\ \\
\ecd \\ 
\subsection{}

\bcd
\ &\ &D_8\arrow[dash]{d}\arrow[dash]{dl}\arrow[dash]{dr}&\ &\ \\
\ &\{1,\sigma^3,\tau,\sigma^3\tau\}\arrow[dash]{dr}\arrow[dash]{dl}\arrow[dash]{d}&\{1,\sigma,\sigma^2,\sigma^3\}\arrow[dash]{d}&\{1,\sigma^2,\sigma\tau,\sigma^3\tau\}\arrow[dash]{dr}\arrow[dash]{dl}\arrow[dash]{d}&\ \\
\{1,\tau\}&\{1,\sigma^2\tau\}&\{1,\sigma^2\}&\{1,\sigma\tau\}&\{1,\sigma^3\tau\}\\
\ &\ &\{1\}\arrow[dash]{ul}\arrow[dash]{ull}\arrow[dash]{u}\arrow[dash]{ur}\arrow[dash]{urr}&\ &\ 
\ecd \\ 
\subsection{}


\bcd
\ &\ &\q\arrow[dash]{d}\arrow[dash]{dl}\arrow[dash]{dr}&\ &\ \\
\ &\q(\sqrt{2})\arrow[dash]{dr}\arrow[dash]{dl}\arrow[dash]{d}&\q(i)\arrow[dash]{d}&\q(\sqrt{-2})\arrow[dash]{dr}\arrow[dash]{dl}\arrow[dash]{d}&\ \\
\q(\sqrt[4]{2})&\q(i\sqrt[4]{2})&\q(\sqrt{2},i)&\q(\sqrt[4]{2}+i\sqrt[4]{2}&\q(\sqrt[4]{2}-i\sqrt[4]{2})\\
\ &\ &\q\arrow[dash]{ul}\arrow[dash]{ull}\arrow[dash]{u}\arrow[dash]{ur}\arrow[dash]{urr}&\ &\ 
\ecd \\ 
\subsection{}

\bcd
L\arrow[dash]{d}{\frac{n}d}\\
K(\alpha)\arrow[dash]{d}{d}\\
K
\ecd \\ 
\subsection{}

\bcd
\ &\q(\alpha)\q(\beta)\arrow[dash,swap]{dl}{\leq2}\arrow[dash]{dr}{\leq2}&\ \\
\q(\alpha)&\ &\q(\beta)\\
\ &\q(\sqrt{5})\arrow[dash,swap]{ul}{\leq2}\arrow[dash]{ur}{\leq2}&\ \\
\ &\q\arrow[dash]{u}{2}\arrow[dash,swap]{uur}{\leq4}\arrow[dash]{uul}{\leq4}&\ 
\ecd \\ 
\subsection{}
 \end{document}


