233 \begin{proof} |
233 \begin{proof} |
234 We will imitate the proof of Corollary 1.3 of \cite{MR0283802}. |
234 We will imitate the proof of Corollary 1.3 of \cite{MR0283802}. |
235 |
235 |
236 Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$. |
236 Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$. |
237 After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$ |
237 After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$ |
238 such that $g\circ f(P)$ is a small neighborhood of the |
238 such that $g^{-1}\circ f(P)$ is a small neighborhood of the |
239 identity in $\Homeo(X)$. |
239 identity in $\Homeo(X)$. |
240 The sense of ``small" we mean will be explained below. |
240 The sense of ``small" we mean will be explained below. |
241 It depends only on $\cU$ and some auxiliary covers. |
241 It depends only on $\cU$ and some auxiliary covers. |
242 |
242 |
243 We assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$. |
243 We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$. |
244 |
244 |
245 |
245 Since $X$ is compact, we may assume without loss of generality that the cover $\cU$ is finite. |
246 |
246 Let $\cU = \{U_\alpha\}$, $1\le \alpha\le N$. |
247 \nn{...} |
247 |
248 |
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 |
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255 Theorem 5.1 of \cite{MR0283802}, applied $N$ times, allows us |
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256 to choose a homotopy $h:P\times [0,N] \to \Homeo(X)$ with the following properties: |
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257 \begin{itemize} |
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258 \item $h(p, 0) = f(p)$ for all $p\in P$. |
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259 \item The restriction of $h(p, i)$ to $U_0^i \cup \cdots \cup U_i^i$ is equal to the homeomorphism $g$, |
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260 for all $p\in P$. |
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261 \item For each fixed $p\in P$, the family of homeomorphisms $h(p, [i-1, i])$ is supported on $U_i^{i-1}$ |
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262 (and hence supported on $U_i$). |
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263 \end{itemize} |
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264 To apply Theorem 5.1 of \cite{MR0283802}, the family $f(P)$ must be sufficiently small, |
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265 and the subdivision mentioned above is chosen fine enough to insure this. |
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266 |
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267 By reparametrizing, we can view $h$ as a homotopy (rel boundary) from $h(\cdot,0) = f: P\to\Homeo(X)$ |
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268 to the family |
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269 \[ |
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270 h(\bd(P\times [0,N]) \setmin P\times \{0\}) = h(\bd P \times [0,N]) \cup h(P \times \{N\}) . |
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271 \] |
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272 We claim that the latter family of homeomorphisms is adapted to $\cU$. |
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273 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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274 Via an inductive assumption and a preliminary homotopy, we have arranged above that $f(\bd P)$ is |
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275 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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276 of the $U_\alpha$'s for each $j$. |
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277 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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278 of the $U_\alpha$'s, specifically, the support of $f(Q_j)$ union $U_i$. |
249 \end{proof} |
279 \end{proof} |
250 |
280 |
251 |
281 |
252 |
282 |
253 \begin{lemma} \label{extension_lemma_c} |
283 \begin{lemma} \label{extension_lemma_c} |