249 such that $g^{-1}\circ f(P)$ is contained in a small neighborhood of the |
249 such that $g^{-1}\circ f(P)$ is contained in a small neighborhood of the |
250 identity in $\Homeo(X)$. |
250 identity in $\Homeo(X)$. |
251 The sense of ``small" we mean will be explained below. |
251 The sense of ``small" we mean will be explained below. |
252 It depends only on $\cU$ and the choice of $U_\alpha^i$'s. |
252 It depends only on $\cU$ and the choice of $U_\alpha^i$'s. |
253 |
253 |
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254 Our goal is to homotope $P$, rel boundary, so that it is adapted to $\cU$. |
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255 By the local contractibility of $\Homeo(X)$ (Corollary 1.1 of \cite{MR0283802}), |
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256 it suffices to find $f':P\to \Homeo(X)$ such that $f' = f$ on $\bd P$ and $f'$ is adapted to $\cU$. |
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257 |
254 We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU^N$. |
258 We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU^N$. |
255 So $\bd P = \sum Q_\beta$, and the support of $f$ restricted to $Q_\beta$ is $V_\beta^N$, the union of $k-1$ of |
259 So $\bd P = \sum Q_\beta$, and the support of $f$ restricted to $Q_\beta$ is $V_\beta^N$, the union of $k-1$ of |
256 the $U_\alpha^N$'s. Define $V_\beta^i \sup V_\beta^N$ to be the corresponding union of $k-1$ |
260 the $U_\alpha^N$'s. Define $V_\beta^i \sup V_\beta^N$ to be the corresponding union of $k-1$ |
257 of the $U_\alpha^i$'s. |
261 of the $U_\alpha^i$'s. |
258 |
262 |
259 Define |
263 Define |
260 \[ |
264 \[ |
261 W_j^i = U_1^i \cup U_2^i \cup \cdots \cup U_j^i . |
265 W_j^i = U_1^i \cup U_2^i \cup \cdots \cup U_j^i . |
262 \] |
266 \] |
263 |
267 |
264 We will construct a sequence of maps $f_i : P\to \Homeo(X)$, for $i = 0, 1, \ldots, N$, with the following properties: |
268 By the local contractibility of $\Homeo(X)$ (Corollary 1.1 of \cite{MR0283802}), |
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269 |
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270 We will construct a sequence of maps $f_i : \bd P\to \Homeo(X)$, for $i = 0, 1, \ldots, N$, with the following properties: |
265 \begin{itemize} |
271 \begin{itemize} |
266 \item[(A)] $f_0 = f$; |
272 \item[(A)] $f_0 = f|_{\bd P}$; |
267 \item[(B)] $f_i = g$ on $W_i^i$; |
273 \item[(B)] $f_i = g$ on $W_i^i$; |
268 \item[(C)] $f_i$ restricted to $Q_\beta$ has support contained in $V_\beta^{N-i}$; and |
274 \item[(C)] $f_i$ restricted to $Q_\beta$ has support contained in $V_\beta^{N-i}$; and |
269 \item[(D)] there is a homotopy $F_i : P\times I \to \Homeo(X)$ from $f_{i-1}$ to $f_i$ such that the |
275 \item[(D)] there is a homotopy $F_i : \bd P\times I \to \Homeo(X)$ from $f_{i-1}$ to $f_i$ such that the |
270 support of $F_i$ restricted to $Q_\beta\times I$ is contained in $U_i^i\cup V_\beta^{N-i}$. |
276 support of $F_i$ restricted to $Q_\beta\times I$ is contained in $U_i^i\cup V_\beta^{N-i}$. |
271 \nn{check this when done writing} |
277 \nn{check this when done writing} |
272 \end{itemize} |
278 \end{itemize} |
273 |
279 |
274 Once we have the $F_i$'s as in (D), we can finish the argument as follows. |
280 Once we have the $F_i$'s as in (D), we can finish the argument as follows. |
275 Assemble the $F_i$'s into a map $F: P\times [0,N] \to \Homeo(X)$. |
281 Assemble the $F_i$'s into a map $F: \bd P\times [0,N] \to \Homeo(X)$. |
276 View $F$ as a homotopy rel boundary from $F$ restricted to $P\times\{0\}$ (which is just $f$ by (A)) |
282 $F$ is adapted to $\cU$ by (D). |
277 to $F$ restricted to $\bd P \times [0,N] \cup P\times\{N\}$. |
283 $F$ restricted to $\bd P\times\{N\}$ is constant on $W_N^N = X$ by (B). |
278 $F$ restricted to $\bd P \times [0,N]$ is adapted to $\cU$ by (D). |
284 We can therefore view $F$ as a map $f'$ from $\Cone(\bd P) \cong P$ to $\Homeo(X)$ |
279 $F$ restricted to $P\times\{N\}$ is constant on $W_N^N = X$ by (B), and therefore is, {\it a fortiori}, also adapted |
285 which is adapted to $\cU$. |
280 to $\cU$. |
286 |
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287 The homotopies $F_i$ will be composed of three types of pieces, $A_\beta$, $B_\beta$ and $C$, % NOT C_\beta |
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288 as illustrated in Figure \nn{xxxx}. |
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289 ($A_\beta$, $B_\beta$ and $C$ also depend on $i$, but we are suppressing that from the notation.) |
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290 The homotopy $A_\beta : Q_\beta \times I \to \Homeo(X)$ will arrange that $f_i$ agrees with $g$ |
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291 on $U_i^i \setmin V_\beta^{N-i+1}$. |
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292 The homotopy $B_\beta : Q_\beta \times I \to \Homeo(X)$ will extend the agreement with $g$ to all of $U_i^i$. |
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293 The homotopies $C$ match things up between $\bd Q_\beta \times I$ and $\bd Q_{\beta'} \times I$ when |
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294 $Q_\beta$ and $Q_{\beta'}$ are adjacent. |
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295 |
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296 Assume inductively that we have defined $f_{i-1}$. |
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297 |
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298 Now we define $A_\beta$. |
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299 Choose $q_0\in Q_\beta$. |
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300 Theorem 5.1 of \cite{MR0283802} implies that we can choose a homotopy $h:I \to \Homeo(X)$ such that |
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301 \begin{itemize} |
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302 \item[(E)] the support of $h$ is contained in $U_i^{i-1} \setmin W_{i-1}^{i-\frac12}$; and |
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303 \item[(F)] $h(1) \circ f_{i-1}(q_0) = g$ on $U_i^i$. |
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304 \end{itemize} |
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305 Define $A_\beta$ by |
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306 \[ |
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307 A_\beta(q, t) = h(t) \circ f_{i-1}(q) . |
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308 \] |
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309 It follows that |
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310 \begin{itemize} |
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311 \item[(G)] $A_\beta(q,1) = A(q',1)$, for all $q,q' \in Q_\beta$, on $X \setmin V_\beta^{N-i+1}$; |
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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 |
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313 \item[(I)] the support of $A_\beta$ is contained in $U_i^{i-1} \cup V_\beta^{N-i+1}$. |
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314 \end{itemize} |
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315 |
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316 |
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317 |
281 |
318 |
282 \nn{resume revising here} |
319 \nn{resume revising here} |
283 |
320 |
284 Theorem 5.1 of \cite{MR0283802}, applied $N$ times, allows us |
321 |
285 to choose a homotopy $h:P\times [0,N] \to \Homeo(X)$ with the following properties: |
322 \nn{scraps:} |
286 \begin{itemize} |
323 |
287 \item $h(p, 0) = f(p)$ for all $p\in P$. |
324 Theorem 5.1 of \cite{MR0283802}, |
288 \item The restriction of $h(p, i)$ to $U_0^i \cup \cdots \cup U_i^i$ is equal to the homeomorphism $g$, |
325 |
289 for all $p\in P$. |
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290 \item For each fixed $p\in P$, the family of homeomorphisms $h(p, [i-1, i])$ is supported on |
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291 $U_i^{i-1} \setmin (U_0^i \cup \cdots \cup U_{i-1}^i)$ |
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292 (and hence supported on $U_i$). |
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293 \end{itemize} |
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294 To apply Theorem 5.1 of \cite{MR0283802}, the family $f(P)$ must be sufficiently small, |
326 To apply Theorem 5.1 of \cite{MR0283802}, the family $f(P)$ must be sufficiently small, |
295 and the subdivision mentioned above is chosen fine enough to insure this. |
327 and the subdivision mentioned above is chosen fine enough to insure this. |
296 |
328 |
297 By reparametrizing, we can view $h$ as a homotopy (rel boundary) from $h(\cdot,0) = f: P\to\Homeo(X)$ |
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298 to the family |
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299 \[ |
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300 h(\bd(P\times [0,N]) \setmin P\times \{0\}) = h(\bd P \times [0,N]) \cup h(P \times \{N\}) . |
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301 \] |
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302 We claim that the latter family of homeomorphisms is adapted to $\cU$. |
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303 By the second bullet above, $h(P\times \{N\})$ is the constant family $g$, and is therefore supported on the empty set. |
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304 Via an inductive assumption and a preliminary homotopy, we have arranged above that $f(\bd P)$ is |
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305 adapted to $\cU$, which means that $\bd P = \cup_j Q_j$ with $f(Q_j)$ supported on the union of $k-1$ |
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306 of the $U_\alpha$'s for each $j$. |
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307 It follows (using the third bullet above) that $h(Q_j \times [i-1,i])$ is supported on the union of $k$ |
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308 of the $U_\alpha$'s, specifically, the support of $f(Q_j)$ union $U_i$. |
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309 \end{proof} |
329 \end{proof} |
310 |
330 |
311 |
331 |
312 |
332 |
313 \begin{lemma} \label{extension_lemma_c} |
333 \begin{lemma} \label{extension_lemma_c} |