235 near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
235 near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
236 unambiguous.) |
236 unambiguous.) |
237 We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
237 We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
238 %We also have that $\deg(b'') = 0 = \deg(p'')$. |
238 %We also have that $\deg(b'') = 0 = \deg(p'')$. |
239 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
239 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
240 This is possible by \ref{bcontract}, \ref{disjunion} and \nn{property 2 of local relations (isotopy)}. |
240 This is possible by \ref{bcontract}, \ref{disjunion} and the fact that isotopic fields |
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241 differ by a local relation \nn{give reference?}. |
241 Finally, define |
242 Finally, define |
242 \[ |
243 \[ |
243 e(p\ot b) \deq x' \bullet p''(b'') . |
244 e(p\ot b) \deq x' \bullet p''(b'') . |
244 \] |
245 \] |
245 |
246 |
335 we have $g_j(p)\ot b \in G_*^{i,m}$. |
336 we have $g_j(p)\ot b \in G_*^{i,m}$. |
336 \end{lemma} |
337 \end{lemma} |
337 |
338 |
338 \begin{proof} |
339 \begin{proof} |
339 Let $c$ be a subset of the blobs of $b$. |
340 Let $c$ be a subset of the blobs of $b$. |
340 There exists $l > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < l$ |
341 There exists $\lambda > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ |
341 and all such $c$. |
342 and all such $c$. |
342 (Here we are using a piecewise smoothness assumption for $\bd c$, and also |
343 (Here we are using a piecewise smoothness assumption for $\bd c$, and also |
343 the fact that $\bd c$ is collared. |
344 the fact that $\bd c$ is collared. |
344 We need to consider all such $c$ because all generators appearing in |
345 We need to consider all such $c$ because all generators appearing in |
345 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
346 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
373 is homeomorphic to a disjoint union of balls and |
374 is homeomorphic to a disjoint union of balls and |
374 \[ |
375 \[ |
375 N_{i,n}(q\ot b) \subeq V_0 \subeq N_{i,n+1}(q\ot b) |
376 N_{i,n}(q\ot b) \subeq V_0 \subeq N_{i,n+1}(q\ot b) |
376 \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,t}(q\ot b) . |
377 \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,t}(q\ot b) . |
377 \] |
378 \] |
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379 Recall that |
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380 \[ |
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381 N_{i,a}(q\ot b) \deq \Nbd_{a\ep_i}(|b|) \cup \Nbd_{\phi_a\delta_i}(|q|). |
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382 \] |
378 By repeated applications of Lemma \ref{xx2phi} we can find neighborhoods $U_0,\ldots,U_m$ |
383 By repeated applications of Lemma \ref{xx2phi} we can find neighborhoods $U_0,\ldots,U_m$ |
379 of $|q|$, each homeomorphic to a disjoint union of balls, with |
384 of $|q|$, each homeomorphic to a disjoint union of balls, with |
380 \[ |
385 \[ |
381 \Nbd_{\phi_{n+l} \delta_i}(|q|) \subeq U_l \subeq \Nbd_{\phi_{n+l+1} \delta_i}(|q|) . |
386 \Nbd_{\phi_{n+l} \delta_i}(|q|) \subeq U_l \subeq \Nbd_{\phi_{n+l+1} \delta_i}(|q|) . |
382 \] |
387 \] |
383 The inequalities above \nn{give ref} guarantee that we can find $u_l$ with |
388 The inequalities above guarantee that |
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389 for each $0\le l\le m$ we can find $u_l$ with |
384 \[ |
390 \[ |
385 (n+l)\ep_i \le u_l \le (n+l+1)\ep_i |
391 (n+l)\ep_i \le u_l \le (n+l+1)\ep_i |
386 \] |
392 \] |
387 such that each component of $U_l$ is either disjoint from $\Nbd_{u_l}(|b|)$ or contained in |
393 such that each component of $U_l$ is either disjoint from $\Nbd_{u_l}(|b|)$ or contained in |
388 $\Nbd_{u_l}(|b|)$. |
394 $\Nbd_{u_l}(|b|)$. |
450 Let $S \sub M$ be contained in a union (not necessarily disjoint) |
456 Let $S \sub M$ be contained in a union (not necessarily disjoint) |
451 of $k$ metric balls of radius $r$. |
457 of $k$ metric balls of radius $r$. |
452 Let $\phi_1, \phi_2, \ldots$ be an increasing sequence of real numbers satisfying |
458 Let $\phi_1, \phi_2, \ldots$ be an increasing sequence of real numbers satisfying |
453 $\phi_1 \ge 2$ and $\phi_{i+1} \ge \phi_i(2\phi_i + 2) + \phi_i$. |
459 $\phi_1 \ge 2$ and $\phi_{i+1} \ge \phi_i(2\phi_i + 2) + \phi_i$. |
454 For convenience, let $\phi_0 = 0$. |
460 For convenience, let $\phi_0 = 0$. |
455 Assume also that $\phi_k r \le \rho(M)$. |
461 Assume also that $\phi_k r \le \rho(M)$, |
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462 where $\rho(M)$ is as in Lemma \ref{xxzz11}. |
456 Then there exists a neighborhood $U$ of $S$, |
463 Then there exists a neighborhood $U$ of $S$, |
457 homeomorphic to a disjoint union of balls, such that |
464 homeomorphic to a disjoint union of balls, such that |
458 \[ |
465 \[ |
459 \Nbd_{\phi_{k-1} r}(S) \subeq U \subeq \Nbd_{\phi_k r}(S) . |
466 \Nbd_{\phi_{k-1} r}(S) \subeq U \subeq \Nbd_{\phi_k r}(S) . |
460 \] |
467 \] |