How to draw a commutative diagram with a half circle added on its left?

Question

Here is the figure I wanna draw:

I know how to draw the commutative diagram without the extension on the left of my figure, here is the code:

\documentclass{book}
\usepackage{amsmath}
\usepackage{tikz-cd}
\usepackage{mathtools, amssymb}
 \usepackage{pst-node, auto-pst-pdf}
 

\begin{document}
\[
    \begin{tikzcd}[sep=large]
M \ar[r] \ar[d, "\mathrm{Pinch}" ' ] \arrow[dr, dashed, "\beta_1 + \beta_2 + \beta_3", sloped]
     &   * \ar[d] \\
M \vee M \vee M \ar[r, "{(\beta_1 , \beta_2 , \beta_3)}" ']
      &   N \arrow[r]
         
  & \dots
    \end{tikzcd}
\]
\end{document}

Any help on how to add this half circle in the figure?

EDIT:

I have a problem in the order of putting $E$ and $B$ to get 2 bended arrows as in my picture above.

Here is my new code:

\[
    \begin{tikzcd}[sep=large]
M \ar[r]  \dar[bend right]  % <--- 
\dar["\mathrm{Pinch}" ]
     &   * \ar[d] 
     &   E \\
M \vee M \vee M \ar[r, "{(\beta_1, \beta_2, \beta_3)}" ']
    &   N \\
    \end{tikzcd}
\]

EDIT:

Another trial, here is my new code:

\[
    \begin{tikzcd}[sep=large]
M \ar[r]  \dar[bend right] % <--- 
\dar["\mathrm{Pinch}" ]
     &   * \ar[d]\\ 
     S' \dar[bend right]\\
     M \vee M \vee M \ar[r, "{(\beta_1, \beta_2, \beta_3)}" ']
    &   N \\
    \end{tikzcd}
\]

Answer 1

• Your MWE does not reproduce showed image
• if you like to bend some arrow, just add to its option bend right or bend left, for example, based on your MWE:

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

\begin{document}
\[
    \begin{tikzcd}[sep=large]
M \ar[r] \dar["\mathrm{Pinch}" ] 
  \ar[dr, dashed, "\beta_1 + \beta_2 + \beta_3", sloped]
  \dar[bend right]  % <--- 
     &   * \ar[d]   \\
M \vee M \vee M \ar[r, "{(\beta_1, \beta_2, \beta_3)}" ']
    &   N \dar      \\
    & \dots
    \end{tikzcd}
\]
\end{document}

Answer 2

You can add it by doubling the number of rows.

\documentclass{book}
\usepackage{amsmath}
\usepackage{tikz-cd}
\usepackage{mathtools, amssymb}

\begin{document}
\[
\begin{tikzcd}[column sep=large]
&[-2em]
  M \arrow[r]
    \arrow[dd, "\mathrm{Pinch}"']
    \arrow[ddr, dashed, "\beta_1 + \beta_2 + \beta_3", sloped]
    \arrow[dl,bend right]
& * \arrow[dd]
\\
E \arrow[dr,bend right]
\\
& M \vee M \vee M \arrow[r, "{(\beta_1 , \beta_2 , \beta_3)}"']
& N \arrow[r]
& \dotsb
\end{tikzcd}
\]
\end{document}

Note the slight reduction of the horizontal space between the first two columns.