How to draw a braid of long exact sequences

That brings back memories. But here is the basic structure. From the xy guide you will learn how to make the arrows curve more (if needed)

\documentclass[a4paper]{memoir}
\usepackage[all]{xy}
\usepackage{amsmath}
\begin{document}

\[
\xymatrix{
A \ar[rd] \ar@/^/[rr] &
&
C \ar[rd] \ar@/^/[rr] &
&
E \ar[rd] \ar@/^/[rr] &
&
G \\
&
B \ar[ur] \ar[dr] &
&
D \ar[ur] \ar[dr]&
&
F \ar[ur] \ar[dr]&
\\
A \ar[ur] \ar@/_/[rr]^J &
&
C \ar[ur] \ar@/_/[rr]^{\sigma_*} &
&
E \ar[ur] \ar@/_/[rr] &
&
G \\
}
\]

\end{document}

Remarks

The same as @daleif's answer, just with tikz-cd instead of xy-pic. Have fun!

Implementation

\documentclass[tikz]{standalone}
\usepackage{amsmath,amssymb,tikz-cd}
\begin{document}
\begin{tikzcd}[cells={nodes={minimum width=1.5cm}}]% to make it more symmetric
    L_{m+1}(\mathbb{Z}) \arrow[bend left]{rr} \arrow{rd}
    &
    & \Theta_m \arrow[bend left]{rr}{0} \arrow{rd}
    &
    & \pi_m(BO) \arrow[bend left]{rr} \arrow{rd}
    &
    & \Omega^{fr}_{m-1}
    \\
%
    & \Theta^{fr}_{m} \arrow{ru} \arrow{rd}
    &
    & A_m \arrow{ru} \arrow{rd}
    &
    & \Theta^{fr}_{m-1} \arrow{ru} \arrow{rd}
    &
    \\
%
    \pi_m(O) \arrow[bend right]{rr}{J} \arrow{ru}
    &
    & \Omega^{fr}_m \arrow[bend right]{rr}{\sigma^*} \arrow{ru}
    &
    & L_m(\mathbb{Z}) \arrow[bend right]{rr} \arrow{ru}
    &
    & \Theta_{m-1}
\end{tikzcd}
\end{document}

Output

Another tikz option, but using matrix of nodes.

% arara: pdflatex
\documentclass{standalone}

\usepackage{tikz}
\usetikzlibrary{matrix}
\usepackage{amssymb}

% make the arrows look like stealth fighter jets
\tikzset{>=stealth}
\begin{document}

\begin{tikzpicture}

    \matrix[name=M, matrix of nodes,cells={nodes={minimum width=1.5cm}}, column sep=1.5cm, row sep=1cm]{
        $L_{m+1}(\mathbb{Z})$ &                & $\Theta_m$      &       & $\pi_m(BO)$       &                    & $\Omega_{m-1}^{fr}$ \\
                              &$\Theta_m^{fr}$ &                 & $A_m$ &                   & $\Theta_{m-1}^{fr}$& \\ 
        $\pi_m(O)$            &                & $\Omega_m^{fr}$ &       & $L_m(\mathbb{Z})$ &                    & $\Theta_{m-1}$ \\
    };
    % straight edges
    \foreach \start/\end in {
        M-1-1/M-2-2,
        M-3-1/M-2-2,
        M-2-2/M-1-3,
        M-2-2/M-3-3,
        M-1-3/M-2-4,
        M-3-3/M-2-4,
        M-2-4/M-1-5,
        M-2-4/M-3-5,
        M-1-5/M-2-6,
        M-3-5/M-2-6,
        M-2-6/M-1-7,
    M-2-6/M-3-7}
    {
        \draw[->] (\start) edge (\end);
    }
    % curved arrows
    \draw[->] (M-1-1) to[out=30,in=150] (M-1-3);
    \draw[->] (M-1-3) to[out=30,in=150] node[pos=0.5,above]{$0$} (M-1-5);
    \draw[->] (M-1-5) to[out=30,in=150]  (M-1-7);
    \draw[->] (M-3-1) to[out=-30,in=-150] node[pos=0.5,above]{$J$} (M-3-3);
    \draw[->] (M-3-3) to[out=-30,in=-150] node[pos=0.5,above]{$\sigma^*$} (M-3-5);
    \draw[->] (M-3-5) to[out=-30,in=-150] (M-3-7);
\end{tikzpicture}

\end{document}