118 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
118 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
119 the case. |
119 the case. |
120 (Consider the $x$-axis and the graph of $y = e^{-1/x^2} \sin(1/x)$ in $\r^2$.) |
120 (Consider the $x$-axis and the graph of $y = e^{-1/x^2} \sin(1/x)$ in $\r^2$.) |
121 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
121 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
122 refinements with both $K$ and $K'$. (For instance, in the example above, $L$ can be the graph of $y=x^2-1$.) |
122 refinements with both $K$ and $K'$. (For instance, in the example above, $L$ can be the graph of $y=x^2-1$.) |
123 This follows from Axiom \ref{axiom:vcones}, which in turn follows from the |
123 This follows from Axiom \ref{axiom:splittings}, which in turn follows from the |
124 splitting axiom for the system of fields $\cE$. |
124 splitting axiom for the system of fields $\cE$. |
125 Let $KL$ and $K'L$ denote these two refinements. |
125 Let $KL$ and $K'L$ denote these two refinements. |
126 Then 1-simplices associated to the four anti-refinements |
126 Then 1-simplices associated to the four anti-refinements |
127 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
127 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
128 give the desired chain connecting $(a, K)$ and $(a, K')$ |
128 give the desired chain connecting $(a, K)$ and $(a, K')$ |
129 (see Figure \ref{zzz4}). |
129 (see Figure \ref{zzz4}). |
130 (In the language of Axiom \ref{axiom:vcones}, this is $\vcone(K \du K')$.) |
130 (In the language of Lemma \ref{lemma:vcones}, this is $\vcone(K \du K')$.) |
131 |
131 |
132 \begin{figure}[t] \centering |
132 \begin{figure}[t] \centering |
133 \begin{tikzpicture} |
133 \begin{tikzpicture} |
134 \foreach \x/\label in {-3/K, 0/L, 3/K'} { |
134 \foreach \x/\label in {-3/K, 0/L, 3/K'} { |
135 \node(\label) at (\x,0) {$\label$}; |
135 \node(\label) at (\x,0) {$\label$}; |
145 \end{figure} |
145 \end{figure} |
146 |
146 |
147 Consider next a 1-cycle in $E(b, b')$, such as one arising from |
147 Consider next a 1-cycle in $E(b, b')$, such as one arising from |
148 a different choice of decomposition $L'$ in place of $L$ above. |
148 a different choice of decomposition $L'$ in place of $L$ above. |
149 %We want to find 2-simplices which fill in this cycle. |
149 %We want to find 2-simplices which fill in this cycle. |
150 By Axiom \ref{axiom:vcones} we can fill in this 1-cycle with 2-simplices. |
150 By Lemma \ref{lemma:vcones} we can fill in this 1-cycle with 2-simplices. |
151 Choose a decomposition $M$ which has common refinements with each of |
151 Choose a decomposition $M$ which has common refinements with each of |
152 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
152 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
153 (We also require that $KLM$ antirefines to $KM$, etc.) |
153 (We also require that $KLM$ antirefines to $KM$, etc.) |
154 Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick. |
154 Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick. |
155 (Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.) |
155 (Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.) |
188 \caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
188 \caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
189 \label{zzz5} |
189 \label{zzz5} |
190 \end{figure} |
190 \end{figure} |
191 |
191 |
192 Continuing in this way we see that $D(a)$ is acyclic. |
192 Continuing in this way we see that $D(a)$ is acyclic. |
193 By Axiom \ref{axiom:vcones} we can fill in any cycle with a V-Cone. |
193 By Lemma \ref{lemma:vcones} we can fill in any cycle with a V-Cone. |
194 \end{proof} |
194 \end{proof} |
195 |
195 |
196 We are now in a position to apply the method of acyclic models to get a map |
196 We are now in a position to apply the method of acyclic models to get a map |
197 $\phi:G_* \to \cl{\cC_F}(Y)$. |
197 $\phi:G_* \to \cl{\cC_F}(Y)$. |
198 We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is a 0-simplex |
198 We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is a 0-simplex |