111 Then 1-simplices associated to the four anti-refinements |
111 Then 1-simplices associated to the four anti-refinements |
112 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
112 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
113 give the desired chain connecting $(a, K)$ and $(a, K')$ |
113 give the desired chain connecting $(a, K)$ and $(a, K')$ |
114 (see Figure \ref{zzz4}). |
114 (see Figure \ref{zzz4}). |
115 |
115 |
116 \begin{figure}[!ht] |
116 \begin{figure}[t] \centering |
117 \begin{equation*} |
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118 \begin{tikzpicture} |
117 \begin{tikzpicture} |
119 \foreach \x/\label in {-3/K, 0/L, 3/K'} { |
118 \foreach \x/\label in {-3/K, 0/L, 3/K'} { |
120 \node(\label) at (\x,0) {$\label$}; |
119 \node(\label) at (\x,0) {$\label$}; |
121 } |
120 } |
122 \foreach \x/\la/\lb in {-1.5/K/L, 1.5/K'/L} { |
121 \foreach \x/\la/\lb in {-1.5/K/L, 1.5/K'/L} { |
123 \node(\la \lb) at (\x,-1.5) {$\la \lb$}; |
122 \node(\la \lb) at (\x,-1.5) {$\la \lb$}; |
124 \draw[->] (\la \lb) -- (\la); |
123 \draw[->] (\la \lb) -- (\la); |
125 \draw[->] (\la \lb) -- (\lb); |
124 \draw[->] (\la \lb) -- (\lb); |
126 } |
125 } |
127 \end{tikzpicture} |
126 \end{tikzpicture} |
128 \end{equation*} |
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129 \caption{Connecting $K$ and $K'$ via $L$} |
127 \caption{Connecting $K$ and $K'$ via $L$} |
130 \label{zzz4} |
128 \label{zzz4} |
131 \end{figure} |
129 \end{figure} |
132 |
130 |
133 Consider a different choice of decomposition $L'$ in place of $L$ above. |
131 Consider a different choice of decomposition $L'$ in place of $L$ above. |
137 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
135 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
138 (We also also require that $KLM$ antirefines to $KM$, etc.) |
136 (We also also require that $KLM$ antirefines to $KM$, etc.) |
139 Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick. |
137 Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick. |
140 (Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.) |
138 (Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.) |
141 |
139 |
142 \begin{figure}[!ht] |
140 \begin{figure}[t] \centering |
143 %\begin{equation*} |
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144 %\mathfig{1.0}{tempkw/zz5} |
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145 %\end{equation*} |
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146 \begin{equation*} |
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147 \begin{tikzpicture} |
141 \begin{tikzpicture} |
148 \node(M) at (0,0) {$M$}; |
142 \node(M) at (0,0) {$M$}; |
149 \foreach \angle/\label in {0/K', 45/K'L, 90/L, 135/KL, 180/K, 225/KL', 270/L', 315/K'L'} { |
143 \foreach \angle/\label in {0/K', 45/K'L, 90/L, 135/KL, 180/K, 225/KL', 270/L', 315/K'L'} { |
150 \node(\label) at (\angle:4) {$\label$}; |
144 \node(\label) at (\angle:4) {$\label$}; |
151 } |
145 } |
172 \draw[->] (K'L) to[bend left=10] (K'); |
166 \draw[->] (K'L) to[bend left=10] (K'); |
173 \draw[->] (K'L) to[bend right=10] (L); |
167 \draw[->] (K'L) to[bend right=10] (L); |
174 \draw[->] (KL) to[bend right=10] (K); |
168 \draw[->] (KL) to[bend right=10] (K); |
175 \draw[->] (KL) to[bend left=10] (L); |
169 \draw[->] (KL) to[bend left=10] (L); |
176 \end{tikzpicture} |
170 \end{tikzpicture} |
177 \end{equation*} |
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178 \caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
171 \caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
179 \label{zzz5} |
172 \label{zzz5} |
180 \end{figure} |
173 \end{figure} |
181 |
174 |
182 Continuing in this way we see that $D(a)$ is acyclic. |
175 Continuing in this way we see that $D(a)$ is acyclic. |