2021 plain ball case. A marked pinched product $\pi: E \to M$ is a pinched product (that is, locally modeled on degeneracy maps) which restricts to a map between the markings which is also a pinched product, and in a neighborhood of the markings is the product of the map between the markings with an interval. 

  2021 plain ball case. A marked pinched product $\pi: E \to M$ is a pinched product (that is, locally modeled on degeneracy maps) which restricts to a map between the markings which is also a pinched product, and in a neighborhood of the markings is the product of the map between the markings with an interval.  (See Figure \ref{fig:marked-pinched-products}.)

  2022 \nn{figure, 2 examples}

  2022 

  2023 \begin{figure}[ht]

  2024 \begin{equation*}

  2025 \begin{tikzpicture}

  2026 \draw (0,2) -- (2,2.5);

  2027 \draw (0,2) -- (2,1.5);

  2028 \draw[line width=2pt] (2,1.5) -- (2,2.5);

  2029 \draw (0,0) -- (2,0) node[circle,fill,inner sep=2pt] {};

  2030 \draw[->] (1,1.5) -- (1,0.25);

  2031 \end{tikzpicture}

  2032 \qquad \qquad \qquad

  2033 \begin{tikzpicture}

  2034 \draw (2,2.5) -- (0,2.5) -- (0,1.5) -- (2,1.5);

  2035 \draw[line width=2pt] (2,1.5) -- (2,2.5);

  2036 \draw (0,0) -- (2,0) node[circle,fill,inner sep=2pt] {};

  2037 \draw[->] (1,1.2) -- (1,0.25);

  2038 \end{tikzpicture}

  2039 \end{equation*}

  2040 \caption{Two examples of marked pinched products.}

  2041 \label{fig:marked-pinched-products}

  2042 \end{figure}

  2043 

  2025 \nn{should  give figure}

  2046  (See Figure \ref{fig:decomposing-marked-pinched-products}.)

  2047 \begin{figure}[ht]

  2048 \begin{equation*}

  2049 \begin{tikzpicture}

  2050 \draw (0,2) -- (2,2.5);

  2051 \draw (0,2) -- (2,1.5);

  2052 \draw[dashed] (1.333,2.333) -- (1.333,1.666);

  2053 \draw[line width=2pt] (2,1.5) -- (2,2.5);

  2054 \draw (0,0) -- (2,0) node[circle,fill,inner sep=2pt] {};

  2055 \draw[->] (1,1.5) -- (1,0.25);

  2056 \end{tikzpicture}

  2057 \qquad \qquad \qquad

  2058 \begin{tikzpicture}

  2059 \draw (0,2) -- (2,2.5);

  2060 \draw (0,2) -- (2,1.5);

  2061 \draw[dashed] (0.666,2.166) -- (2,1.833);

  2062 \draw[line width=2pt] (2,1.5) -- (2,2.5);

  2063 \draw (0,0) -- (2,0) node[circle,fill,inner sep=2pt] {};

  2064 \draw[->] (1,1.5) -- (1,0.25);

  2065 \end{tikzpicture}

  2066 \end{equation*}

  2067 \caption{Two examples of decompositions of marked pinched products.}

  2068 \label{fig:decomposing-marked-pinched-products}

  2069 \end{figure}

  2070