337 the fact that $\bd c$ is collared.)

   337 the fact that $\bd c$ is collared.

   338 We need to consider all such $c$ because all generators appearing in

   339 iterated boundaries of must be in $G_*^{i,m}$.)

   341 	t = r+n+m+1 .

   343 	t = r+n+m+1 = \deg(p\ot b) + m + 1.

   350 	n\cdot ( \phi_t \delta_i) < \ep_k/3 .

   352 	n\cdot (2 (\phi_t + 1) \delta_k) < \ep_k .

   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 $g_j(p)$.

   363 Note that $|q|$ is contained in a union of $n$ elements of the cover $\cU_j$,

   364 which implies that $|q|$ is contained in a union of $n$ metric balls of radius $\delta_i$.

   365 We must show that $q\ot b \in G_*^{i,m}$, which means finding neighborhoods

   366 $V_0,\ldots,V_m \sub X$ of $|q|\cup |b|$ such that each $V_j$

   367 is homeomorphic to a disjoint union of balls and

   368 \[

   369 	N_{i,n}(q\ot b) \subeq V_0 \subeq N_{i,n+1}(q\ot b)

   370 			\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,t}(q\ot b) .

   371 \]

   372 By repeated applications of Lemma \ref{xx2phi} we can find neighborhoods $U_0,\ldots,U_m$

   373 of $|q|$, each homeomorphic to a disjoint union of balls, with

   374 \[

   375 	\Nbd_{\phi_{n+l} \delta_i}(|q|) \subeq U_l \subeq \Nbd_{\phi_{n+l+1} \delta_i}(|q|) .

   376 \]

   377 The inequalities above \nn{give ref} guarantee that we can find $u_l$ with 

   378 \[

   379 	(n+l)\ep_i \le u_l \le (n+l+1)\ep_i

   380 \]

   381 such that each component of $U_l$ is either disjoint from $\Nbd_{u_l}(|b|)$ or contained in 

   382 $\Nbd_{u_l}(|b|)$.

   383 This is because there are at most $n$ components of $U_l$, and each component

   384 has radius $\le (\phi_t + 1) \delta_i$.

   385 It follows that

   386 \[

   387 	V_l \deq \Nbd_{u_l}(|b|) \cup U_l

   388 \]

   389 is homeomorphic to a disjoint union of balls and satisfies

   390 \[

   391 	N_{i,n+l}(q\ot b) \subeq V_l \subeq N_{i,n+l+1}(q\ot b) .

   392 \]

   393 

   394 The same argument shows that each generator involved in iterated boundaries of $q\ot b$

   395 is in $G_*^{i,m}$.