229 Lemma \ref{basic_adaptation_lemma} holds for continuous homeomorphisms |
229 Lemma \ref{basic_adaptation_lemma} holds for continuous homeomorphisms |
230 in item 4. |
230 in item 4. |
231 \end{lemma} |
231 \end{lemma} |
232 |
232 |
233 \begin{proof} |
233 \begin{proof} |
234 We will imitate the proof of Corollary 1.3 of \cite{MR0283802}. |
234 The proof is similar to the proof of Corollary 1.3 of \cite{MR0283802}. |
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235 |
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236 Since $X$ is compact, we may assume without loss of generality that the cover $\cU$ is finite. |
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237 Let $\cU = \{U_\alpha\}$, $1\le \alpha\le N$. |
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238 |
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239 We will need some wiggle room, so for each $\alpha$ choose a large finite number of open sets |
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240 \[ |
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241 U_\alpha = U_\alpha^0 \supset U_\alpha^1 \supset U_\alpha^2 \supset \cdots |
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242 \] |
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243 so that for each fixed $i$ the sets $\{U_\alpha^i\}$ are an open cover of $X$, and also so that |
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244 the closure $\ol{U_\alpha^i}$ is compact and $U_\alpha^{i-1} \supset \ol{U_\alpha^i}$. |
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245 \nn{say specifically how many we need?} |
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246 |
235 |
247 |
236 Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$. |
248 Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$. |
237 After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$ |
249 After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$ |
238 such that $g^{-1}\circ f(P)$ is a small neighborhood of the |
250 such that $g^{-1}\circ f(P)$ is contained in a small neighborhood of the |
239 identity in $\Homeo(X)$. |
251 identity in $\Homeo(X)$. |
240 The sense of ``small" we mean will be explained below. |
252 The sense of ``small" we mean will be explained below. |
241 It depends only on $\cU$ and some auxiliary covers. |
253 It depends only on $\cU$ and the choice of $U_\alpha^i$'s. |
242 |
254 |
243 We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$. |
255 We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$. |
244 |
256 |
245 Since $X$ is compact, we may assume without loss of generality that the cover $\cU$ is finite. |
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246 Let $\cU = \{U_\alpha\}$, $1\le \alpha\le N$. |
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247 |
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248 We will need some wiggle room, so for each $\alpha$ choose open sets |
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249 \[ |
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250 U_\alpha = U_\alpha^0 \supset U_\alpha^1 \supset \cdots \supset U_\alpha^N |
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251 \] |
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252 so that for each fixed $i$ the sets $\{U_\alpha^i\}$ are an open cover of $X$, and also so that |
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253 the closure $\ol{U_\alpha^i}$ is compact and $U_\alpha^{i-1} \supset \ol{U_\alpha^i}$. |
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254 |
257 |
255 Theorem 5.1 of \cite{MR0283802}, applied $N$ times, allows us |
258 Theorem 5.1 of \cite{MR0283802}, applied $N$ times, allows us |
256 to choose a homotopy $h:P\times [0,N] \to \Homeo(X)$ with the following properties: |
259 to choose a homotopy $h:P\times [0,N] \to \Homeo(X)$ with the following properties: |
257 \begin{itemize} |
260 \begin{itemize} |
258 \item $h(p, 0) = f(p)$ for all $p\in P$. |
261 \item $h(p, 0) = f(p)$ for all $p\in P$. |