295 |
295 |
296 Assume inductively that we have defined $f_{i-1}$. |
296 Assume inductively that we have defined $f_{i-1}$. |
297 |
297 |
298 Now we define $A_\beta$. |
298 Now we define $A_\beta$. |
299 Choose $q_0\in Q_\beta$. |
299 Choose $q_0\in Q_\beta$. |
300 Theorem 5.1 of \cite{MR0283802} implies that we can choose a homotopy $h:I \to \Homeo(X)$ such that |
300 Theorem 5.1 of \cite{MR0283802} implies that we can choose a homotopy $h:I \to \Homeo(X)$, with $h(0)$ the identity, such that |
301 \begin{itemize} |
301 \begin{itemize} |
302 \item[(E)] the support of $h$ is contained in $U_i^{i-1} \setmin W_{i-1}^{i-\frac12}$; and |
302 \item[(E)] the support of $h$ is contained in $U_i^{i-1} \setmin W_{i-1}^{i-\frac12}$; and |
303 \item[(F)] $h(1) \circ f_{i-1}(q_0) = g$ on $U_i^i$. |
303 \item[(F)] $h(1) \circ f_{i-1}(q_0) = g$ on $U_i^i$. |
304 \end{itemize} |
304 \end{itemize} |
305 Define $A_\beta$ by |
305 Define $A_\beta$ by |
321 \item[(K)] $B_\beta(q,1) = g$ on $W_i^i$; |
321 \item[(K)] $B_\beta(q,1) = g$ on $W_i^i$; |
322 \item[(L)] the support of $B_\beta(\cdot,1)$ is contained in $V_\beta^{N-i}$; and |
322 \item[(L)] the support of $B_\beta(\cdot,1)$ is contained in $V_\beta^{N-i}$; and |
323 \item[(M)] the support of $B_\beta$ is contained in $U_i^i \cup V_\beta^{N-i}$. |
323 \item[(M)] the support of $B_\beta$ is contained in $U_i^i \cup V_\beta^{N-i}$. |
324 \end{itemize} |
324 \end{itemize} |
325 |
325 |
326 All that remains is to define the ``glue" $C$ which interpolates between adjacent $\beta$ and $\beta'$. |
326 All that remains is to define the ``glue" $C$ which interpolates between adjacent $Q_\beta$ and $Q_{\beta'}$. |
327 First consider the $k=2$ case. |
327 First consider the $k=2$ case. |
|
328 (In this case Figure \nn{xxxx} is literal rather than merely schematic.) |
|
329 Let $q = Q_\beta \cap Q_{\beta'}$ be a point on the boundaries of both $Q_\beta$ and $Q_{\beta'}$. |
|
330 We have an arc of Homeomorphisms, composed of $B_\beta(q, \cdot)$, $A_\beta(q, \cdot)$, |
|
331 $A_{\beta'}(q, \cdot)$ and $B_{\beta'}(q, \cdot)$, which connects $B_\beta(q, 1)$ to $B_{\beta'}(q, 1)$. |
|
332 |
|
333 \nn{Hmmmm..... I think there's a problem here} |
328 |
334 |
329 |
335 |
330 |
336 |
331 \nn{resume revising here} |
337 \nn{resume revising here} |
332 |
338 |
333 |
339 |
334 \nn{scraps:} |
340 \nn{scraps:} |
335 |
|
336 Theorem 5.1 of \cite{MR0283802}, |
|
337 |
341 |
338 To apply Theorem 5.1 of \cite{MR0283802}, the family $f(P)$ must be sufficiently small, |
342 To apply Theorem 5.1 of \cite{MR0283802}, the family $f(P)$ must be sufficiently small, |
339 and the subdivision mentioned above is chosen fine enough to insure this. |
343 and the subdivision mentioned above is chosen fine enough to insure this. |
340 |
344 |
341 \end{proof} |
345 \end{proof} |