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311 \item[(G)] $A_\beta(q,1) = A(q',1)$, for all $q,q' \in Q_\beta$, on $X \setmin V_\beta^{N-i+1}$; |
311 \item[(G)] $A_\beta(q,1) = A(q',1)$, for all $q,q' \in Q_\beta$, on $X \setmin V_\beta^{N-i+1}$; |
312 \item[(H)] $A_\beta(q, 1) = g$ on $(U_i^i \cup W_{i-1}^{i-\frac12})\setmin V_\beta^{N-i+1}$; and |
312 \item[(H)] $A_\beta(q, 1) = g$ on $(U_i^i \cup W_{i-1}^{i-\frac12})\setmin V_\beta^{N-i+1}$; and |
313 \item[(I)] the support of $A_\beta$ is contained in $U_i^{i-1} \cup V_\beta^{N-i+1}$. |
313 \item[(I)] the support of $A_\beta$ is contained in $U_i^{i-1} \cup V_\beta^{N-i+1}$. |
314 \end{itemize} |
314 \end{itemize} |
315 |
315 |
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316 Next we define $B_\beta$. |
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317 Theorem 5.1 of \cite{MR0283802} implies that we can choose a homotopy $B_\beta:Q_\beta\times I\to \Homeo(X)$ |
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318 such that |
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319 \begin{itemize} |
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320 \item[(J)] $B_\beta(\cdot, 0) = A_\beta(\cdot, 1)$; |
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321 \item[(K)] $B_\beta(q,1) = g$ on $W_i^i$; |
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322 \item[(L)] the support of $B_\beta(\cdot,1)$ is contained in $V_\beta^{N-i}$; and |
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323 \item[(M)] the support of $B_\beta$ is contained in $U_i^i \cup V_\beta^{N-i}$. |
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324 \end{itemize} |
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325 |
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326 All that remains is to define the ``glue" $C$ which interpolates between adjacent $\beta$ and $\beta'$. |
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327 First consider the $k=2$ case. |
316 |
328 |
317 |
329 |
318 |
330 |
319 \nn{resume revising here} |
331 \nn{resume revising here} |
320 |
332 |