234 The proof is similar to the proof of Corollary 1.3 of \cite{MR0283802}. |
234 The proof is similar to the proof of Corollary 1.3 of \cite{MR0283802}. |
235 |
235 |
236 Since $X$ is compact, we may assume without loss of generality that the cover $\cU$ is finite. |
236 Since $X$ is compact, we may assume without loss of generality that the cover $\cU$ is finite. |
237 Let $\cU = \{U_\alpha\}$, $1\le \alpha\le N$. |
237 Let $\cU = \{U_\alpha\}$, $1\le \alpha\le N$. |
238 |
238 |
239 We will need some wiggle room, so for each $\alpha$ choose a large finite number of open sets |
239 We will need some wiggle room, so for each $\alpha$ choose $2N$ additional open sets |
240 \[ |
240 \[ |
241 U_\alpha = U_\alpha^0 \supset U_\alpha^1 \supset U_\alpha^2 \supset \cdots |
241 U_\alpha = U_\alpha^0 \supset U_\alpha^\frac12 \supset U_\alpha^1 \supset U_\alpha^\frac32 \supset \cdots \supset U_\alpha^N |
242 \] |
242 \] |
243 so that for each fixed $i$ the sets $\{U_\alpha^i\}$ are an open cover of $X$, and also so that |
243 so that for each fixed $i$ the set $\cU^i = \{U_\alpha^i\}$ is an open cover of $X$, and also so that |
244 the closure $\ol{U_\alpha^i}$ is compact and $U_\alpha^{i-1} \supset \ol{U_\alpha^i}$. |
244 the closure $\ol{U_\alpha^i}$ is compact and $U_\alpha^{i-\frac12} \supset \ol{U_\alpha^i}$. |
245 \nn{say specifically how many we need?} |
245 %\nn{say specifically how many we need?} |
246 |
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247 |
246 |
248 Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$. |
247 Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$. |
249 After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$ |
248 After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$ |
250 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 |
251 identity in $\Homeo(X)$. |
250 identity in $\Homeo(X)$. |
252 The sense of ``small" we mean will be explained below. |
251 The sense of ``small" we mean will be explained below. |
253 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. |
254 |
253 |
255 We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$. |
254 We may assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU^N$. |
256 |
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 |
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256 the $U_\alpha^N$'s. Define $V_\beta^i \sup V_\beta^N$ to be the corresponding union of $k-1$ |
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257 of the $U_\alpha^i$'s. |
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258 |
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259 Define |
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260 \[ |
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261 W_j^i = U_1^i \cup U_2^i \cup \cdots \cup U_j^i . |
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262 \] |
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263 |
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264 We will construct a sequence of maps $f_i : P\to \Homeo(X)$, for $i = 0, 1, \ldots, N$, with the following properties: |
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265 \begin{itemize} |
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266 \item[(A)] $f_0 = f$; |
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267 \item[(B)] $f_i = g$ on $W_i^i$; |
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268 \item[(C)] $f_i$ restricted to $Q_\beta$ has support contained in $V_\beta^{N-i}$; and |
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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 |
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270 support of $F_i$ restricted to $Q_\beta\times I$ is contained in $U_i^i\cup V_\beta^{N-i}$. |
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271 \nn{check this when done writing} |
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272 \end{itemize} |
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273 |
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274 Once we have the $F_i$'s as in (D), we can finish the argument as follows. |
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275 Assemble the $F_i$'s into a map $F: P\times [0,N] \to \Homeo(X)$. |
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276 View $F$ as a homotopy rel boundary from $F$ restricted to $P\times\{0\}$ (which is just $f$ by (A)) |
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277 to $F$ restricted to $\bd P \times [0,N] \cup P\times\{N\}$. |
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278 $F$ restricted to $\bd P \times [0,N]$ is adapted to $\cU$ by (D). |
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279 $F$ restricted to $P\times\{N\}$ is constant on $W_N^N = X$ by (B), and therefore is, {\it a fortiori}, also adapted |
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280 to $\cU$. |
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281 |
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282 \nn{resume revising here} |
257 |
283 |
258 Theorem 5.1 of \cite{MR0283802}, applied $N$ times, allows us |
284 Theorem 5.1 of \cite{MR0283802}, applied $N$ times, allows us |
259 to choose a homotopy $h:P\times [0,N] \to \Homeo(X)$ with the following properties: |
285 to choose a homotopy $h:P\times [0,N] \to \Homeo(X)$ with the following properties: |
260 \begin{itemize} |
286 \begin{itemize} |
261 \item $h(p, 0) = f(p)$ for all $p\in P$. |
287 \item $h(p, 0) = f(p)$ for all $p\in P$. |