45 If $f$ is smooth, Lipschitz or PL, as appropriate, and $f(p, \cdot):X\to T$ is in $\cX$ for all $p \in P$ |
45 If $f$ is smooth, Lipschitz or PL, as appropriate, and $f(p, \cdot):X\to T$ is in $\cX$ for all $p \in P$ |
46 then $F(t, p, \cdot)$ is also in $\cX$ for all $t\in I$ and $p\in P$. |
46 then $F(t, p, \cdot)$ is also in $\cX$ for all $t\in I$ and $p\in P$. |
47 \end{enumerate} |
47 \end{enumerate} |
48 \end{lemma} |
48 \end{lemma} |
49 |
49 |
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50 Note: We will prove a version of item 4 of Lemma \ref{basic_adaptation_lemma} for topological |
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51 homeomorphisms in Lemma \ref{basic_adaptation_lemma_2} below. |
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52 Since the proof is rather different we segregate it to a separate lemma. |
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53 |
50 \begin{proof} |
54 \begin{proof} |
51 Our homotopy will have the form |
55 Our homotopy will have the form |
52 \eqar{ |
56 \eqar{ |
53 F: I \times P \times X &\to& X \\ |
57 F: I \times P \times X &\to& X \\ |
54 (t, p, x) &\mapsto& f(u(t, p, x), x) |
58 (t, p, x) &\mapsto& f(u(t, p, x), x) |
210 close to the universal bi-Lipschitz constant for $f$. |
214 close to the universal bi-Lipschitz constant for $f$. |
211 |
215 |
212 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. |
216 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. |
213 \end{proof} |
217 \end{proof} |
214 |
218 |
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219 |
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220 % Edwards-Kirby: MR0283802 |
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221 |
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222 The above proof doesn't work for homeomorphisms which are merely continuous. |
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223 The $k=1$ case for plain, continuous homeomorphisms |
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224 is more or less equivalent to Corollary 1.3 of \cite{MR0283802}. |
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225 The proof found in \cite{MR0283802} of that corollary can be adapted to many-parameter families of |
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226 homeomorphisms: |
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227 |
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228 \begin{lemma} \label{basic_adaptation_lemma_2} |
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229 Lemma \ref{basic_adaptation_lemma} holds for continuous homeomorphisms |
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230 in item 4. |
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231 \end{lemma} |
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232 |
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233 \begin{proof} |
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234 We will imitate the proof of Corollary 1.3 of \cite{MR0283802}. |
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235 |
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236 Let $P$ be some $k$-dimensional polyhedron and $f:P\to \Homeo(X)$. |
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237 After subdividing $P$, we may assume that there exists $g\in \Homeo(X)$ |
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238 such that $g\circ f(P)$ is a small neighborhood of the |
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239 identity in $\Homeo(X)$. |
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240 The sense of ``small" we mean will be explained below. |
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241 It depends only on $\cU$ and some auxiliary covers. |
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242 |
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243 We assume, inductively, that the restriction of $f$ to $\bd P$ is adapted to $\cU$. |
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244 |
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245 |
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246 |
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247 \nn{...} |
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248 |
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249 \end{proof} |
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250 |
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251 |
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252 |
215 \begin{lemma} \label{extension_lemma_c} |
253 \begin{lemma} \label{extension_lemma_c} |
216 Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the |
254 Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the |
217 subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, |
255 subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, |
218 bi-Lipschitz homeomorphisms or PL homeomorphisms. |
256 bi-Lipschitz homeomorphisms, PL homeomorphisms or plain old continuous homeomorphisms. |
219 Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$ |
257 Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$ |
220 of $X$. |
258 of $X$. |
221 Then $G_*$ is a strong deformation retract of $\cX_*$. |
259 Then $G_*$ is a strong deformation retract of $\cX_*$. |
222 \end{lemma} |
260 \end{lemma} |
223 \begin{proof} |
261 \begin{proof} |
224 It suffices to show that given a generator $f:P\times X\to T$ of $\cX_k$ with |
262 It suffices to show that given a generator $f:P\times X\to T$ of $\cX_k$ with |
225 $\bd f \in G_{k-1}$ there exists $h\in \cX_{k+1}$ with $\bd h = f + g$ and $g \in G_k$. |
263 $\bd f \in G_{k-1}$ there exists $h\in \cX_{k+1}$ with $\bd h = f + g$ and $g \in G_k$. |
226 This is exactly what Lemma \ref{basic_adaptation_lemma} |
264 This is exactly what Lemma \ref{basic_adaptation_lemma} (or \ref{basic_adaptation_lemma_2}) |
227 gives us. |
265 gives us. |
228 More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$. |
266 More specifically, let $\bd P = \sum Q_i$, with each $Q_i\in G_{k-1}$. |
229 Let $F: I\times P\times X\to T$ be the homotopy constructed in Lemma \ref{basic_adaptation_lemma}. |
267 Let $F: I\times P\times X\to T$ be the homotopy constructed in Lemma \ref{basic_adaptation_lemma}. |
230 Then $\bd F$ is equal to $f$ plus $F(1, \cdot, \cdot)$ plus the restrictions of $F$ to $I\times Q_i$. |
268 Then $\bd F$ is equal to $f$ plus $F(1, \cdot, \cdot)$ plus the restrictions of $F$ to $I\times Q_i$. |
231 Part 2 of Lemma \ref{basic_adaptation_lemma} says that $F(1, \cdot, \cdot)\in G_k$, |
269 Part 2 of Lemma \ref{basic_adaptation_lemma} says that $F(1, \cdot, \cdot)\in G_k$, |
232 while part 3 of Lemma \ref{basic_adaptation_lemma} says that the restrictions to $I\times Q_i$ are in $G_k$. |
270 while part 3 of Lemma \ref{basic_adaptation_lemma} says that the restrictions to $I\times Q_i$ are in $G_k$. |
233 \end{proof} |
271 \end{proof} |
234 |
272 |
235 \medskip |
273 \medskip |
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274 |
236 |
275 |
237 %%%%%% Lo, \noop{...} |
276 %%%%%% Lo, \noop{...} |
238 \noop{ |
277 \noop{ |
239 |
278 |
240 \medskip |
279 \medskip |