332 \begin{proof} |
332 \begin{proof} |
333 Let $c$ be a subset of the blobs of $b$. |
333 Let $c$ be a subset of the blobs of $b$. |
334 There exists $l > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < l$ |
334 There exists $l > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < l$ |
335 and all such $c$. |
335 and all such $c$. |
336 (Here we are using a piecewise smoothness assumption for $\bd c$, and also |
336 (Here we are using a piecewise smoothness assumption for $\bd c$, and also |
337 the fact that $\bd c$ is collared.) |
337 the fact that $\bd c$ is collared. |
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338 We need to consider all such $c$ because all generators appearing in |
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339 iterated boundaries of must be in $G_*^{i,m}$.) |
338 |
340 |
339 Let $r = \deg(b)$ and |
341 Let $r = \deg(b)$ and |
340 \[ |
342 \[ |
341 t = r+n+m+1 . |
343 t = r+n+m+1 = \deg(p\ot b) + m + 1. |
342 \] |
344 \] |
343 |
345 |
344 Choose $k = k_{bmn}$ such that |
346 Choose $k = k_{bmn}$ such that |
345 \[ |
347 \[ |
346 t\ep_k < l |
348 t\ep_k < l |
347 \] |
349 \] |
348 and |
350 and |
349 \[ |
351 \[ |
350 n\cdot ( \phi_t \delta_i) < \ep_k/3 . |
352 n\cdot (2 (\phi_t + 1) \delta_k) < \ep_k . |
351 \] |
353 \] |
352 Let $i \ge k_{bmn}$. |
354 Let $i \ge k_{bmn}$. |
353 Choose $j = j_i$ so that |
355 Choose $j = j_i$ so that |
354 \[ |
356 \[ |
355 t\gamma_j < \ep_i/3 |
357 \gamma_j < \delta_i |
356 \] |
358 \] |
357 and also so that $\gamma_j$ is less than the constant $\eta(X, m, k)$ of Lemma \ref{xxyy5}. |
359 and also so that $\phi_t \gamma_j$ is less than the constant $\rho(M)$ of Lemma \ref{xxzz11}. |
358 |
360 |
359 \nn{...} |
361 Let $j \ge j_i$ and $p\in CD_n(X)$. |
360 |
362 Let $q$ be a generator appearing in $g_j(p)$. |
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363 Note that $|q|$ is contained in a union of $n$ elements of the cover $\cU_j$, |
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364 which implies that $|q|$ is contained in a union of $n$ metric balls of radius $\delta_i$. |
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365 We must show that $q\ot b \in G_*^{i,m}$, which means finding neighborhoods |
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366 $V_0,\ldots,V_m \sub X$ of $|q|\cup |b|$ such that each $V_j$ |
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367 is homeomorphic to a disjoint union of balls and |
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368 \[ |
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369 N_{i,n}(q\ot b) \subeq V_0 \subeq N_{i,n+1}(q\ot b) |
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370 \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,t}(q\ot b) . |
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371 \] |
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372 By repeated applications of Lemma \ref{xx2phi} we can find neighborhoods $U_0,\ldots,U_m$ |
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373 of $|q|$, each homeomorphic to a disjoint union of balls, with |
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374 \[ |
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375 \Nbd_{\phi_{n+l} \delta_i}(|q|) \subeq U_l \subeq \Nbd_{\phi_{n+l+1} \delta_i}(|q|) . |
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376 \] |
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377 The inequalities above \nn{give ref} guarantee that we can find $u_l$ with |
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378 \[ |
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379 (n+l)\ep_i \le u_l \le (n+l+1)\ep_i |
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380 \] |
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381 such that each component of $U_l$ is either disjoint from $\Nbd_{u_l}(|b|)$ or contained in |
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382 $\Nbd_{u_l}(|b|)$. |
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383 This is because there are at most $n$ components of $U_l$, and each component |
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384 has radius $\le (\phi_t + 1) \delta_i$. |
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385 It follows that |
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386 \[ |
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387 V_l \deq \Nbd_{u_l}(|b|) \cup U_l |
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388 \] |
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389 is homeomorphic to a disjoint union of balls and satisfies |
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390 \[ |
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391 N_{i,n+l}(q\ot b) \subeq V_l \subeq N_{i,n+l+1}(q\ot b) . |
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392 \] |
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393 |
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394 The same argument shows that each generator involved in iterated boundaries of $q\ot b$ |
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395 is in $G_*^{i,m}$. |
361 \end{proof} |
396 \end{proof} |
362 |
397 |
363 In the next few lemmas we have made no effort to optimize the various bounds. |
398 In the next few lemmas we have made no effort to optimize the various bounds. |
364 (The bounds are, however, optimal in the sense of minimizing the amount of work |
399 (The bounds are, however, optimal in the sense of minimizing the amount of work |
365 we do. Equivalently, they are the first bounds we thought of.) |
400 we do. Equivalently, they are the first bounds we thought of.) |