197 close to the universal bi-Lipschitz constant for $f$. |
197 close to the universal bi-Lipschitz constant for $f$. |
198 |
198 |
199 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. |
199 Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well. |
200 \end{proof} |
200 \end{proof} |
201 |
201 |
202 |
202 \begin{lemma} |
203 |
203 Let $CM_*(X\to T)$ be the singular chains on the space of continuous maps |
204 |
204 [resp. immersions, diffeomorphisms, PL homeomorphisms, bi-Lipschitz homeomorphisms] |
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205 from $X$ to $T$, and let $G_*\sub CM_*(X\to T)$ denote the chains adapted to an open cover $\cU$ |
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206 of $X$. |
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207 Then $G_*$ is a strong deformation retract of $CM_*(X\to T)$. |
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208 \end{lemma} |
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209 \begin{proof} |
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210 \nn{my current idea is too messy, so I'm going to wait and hopefully think |
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211 of a cleaner proof} |
205 \noop{ |
212 \noop{ |
206 |
213 If suffices to show that |
207 \nn{move this to later:} |
214 ... |
208 |
215 Lemma \ref{basic_adaptation_lemma} |
209 \begin{lemma} \label{extension_lemma_b} |
216 ... |
210 Let $x \in CM_k(X, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
217 } |
211 Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$. |
218 \end{proof} |
212 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
219 |
213 If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with |
220 \medskip |
214 chains of smooth maps or immersions. |
221 |
215 \end{lemma} |
222 \nn{need to clean up references from the main text to the lemmas of this section} |
216 |
223 |
217 \medskip |
224 \medskip |
218 \hrule |
225 |
219 \medskip |
226 \nn{do we want to keep the following?} |
220 |
227 |
221 |
228 The above lemmas remain true if we replace ``adapted" with ``strongly adapted", as defined below. |
222 In this appendix we provide the proof of |
229 The proof of Lemma \ref{basic_adaptation_lemma} is modified by |
223 \nn{should change this to the more general \ref{extension_lemma_b}} |
230 choosing the common refinement $L$ and interpolating maps $\eta$ |
224 |
231 slightly more carefully. |
225 \begin{lem*}[Restatement of Lemma \ref{extension_lemma}] |
232 Since we don't need the stronger result, we omit the details. |
226 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
233 |
227 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
234 Let $X$, $T$ and $\cU$ be as above. |
228 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
235 A $k$-parameter family of maps $f: P \times X \to T$ is |
229 \end{lem*} |
236 {\it strongly adapted to $\cU$} if there is a factorization |
230 |
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231 \nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
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232 later draft} |
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233 |
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234 \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
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235 about boundary later} |
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236 |
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237 } |
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238 |
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239 |
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240 |
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241 |
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242 \medskip |
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243 \hrule |
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244 \medskip |
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245 \nn{the following was removed from earlier section; it should be reincorporated somewhere |
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246 in this section} |
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247 |
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248 Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
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249 A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is |
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250 {\it adapted to $\cU$} if there is a factorization |
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251 \eq{ |
237 \eq{ |
252 P = P_1 \times \cdots \times P_m |
238 P = P_1 \times \cdots \times P_m |
253 } |
239 } |
254 (for some $m \le k$) |
240 (for some $m \le k$) |
255 and families of homeomorphisms |
241 and families of homeomorphisms |
256 \eq{ |
242 \eq{ |
257 f_i : P_i \times X \to X |
243 f_i : P_i \times X \to T |
258 } |
244 } |
259 such that |
245 such that |
260 \begin{itemize} |
246 \begin{itemize} |
261 \item each $f_i$ is supported on some connected $V_i \sub X$; |
247 \item each $f_i$ is supported on some connected $V_i \sub X$; |
262 \item the sets $V_i$ are mutually disjoint; |
248 \item the sets $V_i$ are mutually disjoint; |
263 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
249 \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
264 where $k_i = \dim(P_i)$; and |
250 where $k_i = \dim(P_i)$; and |
265 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
251 \item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
266 for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Homeo(X)$. |
252 for all $p = (p_1, \ldots, p_m)$, for some fixed $gX\to T$. |
267 \end{itemize} |
253 \end{itemize} |
268 A chain $x \in CH_k(X)$ is (by definition) adapted to $\cU$ if it is the sum |
254 |
269 of singular cells, each of which is adapted to $\cU$. |
255 |
270 \medskip |
256 \medskip |
271 \hrule |
257 \hrule |
272 \medskip |
258 \medskip |
273 |
259 |
274 |
260 \nn{do we want to keep this alternative construction?} |
275 |
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276 |
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277 |
261 |
278 \input{text/appendixes/explicit.tex} |
262 \input{text/appendixes/explicit.tex} |
279 |
263 |