Tikz-cd and the Mayer-Vietoris sequence

Question

In this post I have found a very useful answer to made the Mayer-Vietoris sequence for homology using tikz-cd. However, I need some help creating a diagram like this:

I want to draw it using the same package, but I cannot modify correctly the code given in the post to draw the reverse arrow for the connecting homomorphism. This is the code that I have tried to modify, but have had no success.

\documentclass{article}
\usepackage{tikz-cd}

\tikzset{
  curarrow/.style={
  rounded corners=8pt,
  execute at begin to={every node/.style={fill=red}},
    to path={-- ([xshift=50pt]\tikztostart.center)
    |- (#1) node[fill=white] {$\scriptstyle d^*$}
    -| ([xshift=-50pt]\tikztotarget.center)
    -- (\tikztotarget)}
    }
}

\begin{document}

   \begin{tikzcd}[arrow style=math font,cells={nodes={text height=2ex,text depth=0.75ex}}]
       H^{k+1}(M) \arrow[r, "i^{*}"] & H^{k+1}(U) \oplus H^{k+1}(V) \arrow[r] & \cdots \\
       H^{k}(M) \arrow[r, "i^{*}"] & H^{k}(U) \oplus H^{k}(V) \arrow[r, "j^{*}"] \arrow[draw=none]{u}[name=Y, shape=coordinate]{} \arrow[draw=none]{d}[name=Z,shape=coordinate]{} & H^{k}(U \cap V) \arrow[curarrow=Y]{ull}{} \\
       \cdots \arrow[r] & H^{k-1}(U) \oplus H^{k-1}(V) \arrow[r, "j^{*}"] & H^{k-1}(U \cap V) \arrow[curarrow=Z]{ull}{}
   \end{tikzcd}

\end{document}

Answer 1

You can obtain this diagram with pstricks and a psmatrix environment. For the connecting homomorphism, I add intermediate empty nodes at each end, between consecutives rows, and connect them with a composite path:

\documentclass{article}
\usepackage{pst-node,  auto-pst-pdf}

\begin{document}

 \[  \def\psrowhookiii{\psset{mrow=r}}
  \begin{psmatrix}[rowsep=1.35]
  \pnode[-0.5,0]{A}\\
  \pnode[1,1]{A2}H^{q+1}_c(M) & H^{q + 1}_c(U) \oplus H^{q + 1}_c(V) & H^{q + 1}_c(U \cap V)\pnode[0,-0.8]{B2}\\
   \pnode[0,1]{A3}  H^q_c(M) & H^q_c(U) \oplus H^q_c(V) & H^q_c(U \cap V)\pnode[-1,-0.8]{B3}\\
       & &   \pnode{B}
       \psset{arrows=->, arrowinset=0.12, nodesep=4pt,  linearc=0.4}
       \ncline{2,3}{2,2} \ncline{2,2}{2,1}\ncbar[angle=180, ]{2,1}{A2}%
       \ncbar[arm=0.7, angle=180, arrows=-, nodesepB=0]{3,1}{A3}\ncline[arrows=-, nodesep=0]{A3}{B2}\nbput[labelsep=1.5pt]{d^*}%
       \ncbar[arm=0.76, angle=0, arrows=->, nodesepA=0]{B2}{2,3}%
       \ncline{3,3}{3,2} \ncline{3,2}{3,1}\ncbar[arm=0.76, angle=0, arrows=->, nodesepA=0]{B3}{3,3}%
    \end{psmatrix} \]

\end{document} 

Answer 2

I created a working modification of the code block you posted:

\documentclass{article}
\usepackage{tikz-cd}

\tikzset{
  curarrow/.style={
  rounded corners=8pt,
  execute at begin to={every node/.style={fill=red}},
    to path={-- ([xshift=-50pt]\tikztostart.center)
    |- (#1) node[fill=white] {$\scriptstyle d_*$}
    -| ([xshift=50pt]\tikztotarget.center)
    -- (\tikztotarget)}
    }
}

\begin{document}

    \begin{tikzcd}[arrow style=math font,cells={nodes={text height=2ex,text depth=0.75ex}}]
       \cdots & H_{k-1}(U) \oplus H_{k-1}(V) \arrow[l] \arrow[draw=none]{d}[name=Y, shape=coordinate]{} & \arrow[l] H_{k-1}(U \cap V) \\
       H_{k}(M) \arrow[curarrow=Y]{urr}{} & H_{k}(U) \oplus H_{k}(V) \arrow[l] \arrow[draw=none]{d}[name=Z,shape=coordinate]{} & H_{k}(U \cap V) \arrow[l] \\
       H_{k+1}(M) \arrow[curarrow=Z]{urr}{} & H_{k+1}(U) \oplus H_{k+1}(V) \arrow[l] & \cdots \arrow[l]
   \end{tikzcd}

\end{document}

The changes I made:

• in the code block you posted, following the middle middle position H^{k}(U) \oplus H^{k}(V), there were two coordinate-setting arrows

  \arrow[draw=none]{u}[name=Y, shape=coordinate]{} \arrow[draw=none]{d}[name=Z,shape=coordinate]{}

  These coordinates are used to set the position of the boundary map labels: the Y coordinate is set between the middle middle and middle top (via an upward pointing arrow), and the Z coordinate is set between the middle middle and middle bottom (via a downward pointing arrow). I want to send an arrow from the middle left to the upper right, but the coordinate Y must be defined before I use it, so I moved the definition to the middle top, pointing down to the middle middle.

• I modified the definition of the curarrow style slightly. In the original, the path starts by moving to the right by 50pt, turning around, then overshooting to the left by 50pt before ending; these are reflected in the xshifts. Since the orientation of the arrow is reversed, I reversed the signs on the xshifts to reflect the change.

• Obviously the arrows are reversed. Since the original code block produced Mayer-Vietoris for cohomology and the arrows have been reversed, I also took the liberty of switching the groups to homology groups. This is probably not what you wanted, sorry.