132 Consider a different choice of decomposition $L'$ in place of $L$ above. |
132 Consider a different choice of decomposition $L'$ in place of $L$ above. |
133 This leads to a cycle of 1-simplices. |
133 This leads to a cycle of 1-simplices. |
134 We want to find 2-simplices which fill in this cycle. |
134 We want to find 2-simplices which fill in this cycle. |
135 Choose a decomposition $M$ which has common refinements with each of |
135 Choose a decomposition $M$ which has common refinements with each of |
136 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
136 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
137 (We also also require that $KLM$ antirefines to $KM$, etc.) |
137 (We also require that $KLM$ antirefines to $KM$, etc.) |
138 Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick. |
138 Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick. |
139 (Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.) |
139 (Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.) |
140 |
140 |
141 \begin{figure}[t] \centering |
141 \begin{figure}[t] \centering |
142 \begin{tikzpicture} |
142 \begin{tikzpicture} |