105 Suppose that there exists $V \sub X$ such that |
105 Suppose that there exists $V \sub X$ such that |
106 \begin{enumerate} |
106 \begin{enumerate} |
107 \item $V$ is homeomorphic to a disjoint union of balls, and |
107 \item $V$ is homeomorphic to a disjoint union of balls, and |
108 \item $\supp(p) \cup \supp(b) \sub V$. |
108 \item $\supp(p) \cup \supp(b) \sub V$. |
109 \end{enumerate} |
109 \end{enumerate} |
110 Let $W = X \setmin V$, and let $V' = p(V)$ and $W' = p(W)$. |
110 Let $W = X \setmin V$, $W' = p(W)$ and $V' = X\setmin W'$. |
111 We then have a factorization |
111 We then have a factorization |
112 \[ |
112 \[ |
113 p = \gl(q, r), |
113 p = \gl(q, r), |
114 \] |
114 \] |
115 where $q \in CD_k(V, V')$ and $r' \in CD_0(W, W')$. |
115 where $q \in CD_k(V, V')$ and $r \in CD_0(W, W')$. |
116 We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$. |
116 We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$. |
117 According to the commutative diagram of the proposition, we must have |
117 According to the commutative diagram of the proposition, we must have |
118 \[ |
118 \[ |
119 e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = |
119 e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = |
120 gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) . |
120 gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) . |
462 If these balls are disjoint, let $U$ be their union. |
462 If these balls are disjoint, let $U$ be their union. |
463 Otherwise, assume WLOG that $S_{k-1}$ and $S_k$ are distance less than $2\phi_{k-1}r$ apart. |
463 Otherwise, assume WLOG that $S_{k-1}$ and $S_k$ are distance less than $2\phi_{k-1}r$ apart. |
464 Let $R_i = \Nbd_{\phi_{k-1} r}(S_i)$ for $i = 1,\ldots,k-2$ |
464 Let $R_i = \Nbd_{\phi_{k-1} r}(S_i)$ for $i = 1,\ldots,k-2$ |
465 and $R_{k-1} = \Nbd_{\phi_{k-1} r}(S_{k-1})\cup \Nbd_{\phi_{k-1} r}(S_k)$. |
465 and $R_{k-1} = \Nbd_{\phi_{k-1} r}(S_{k-1})\cup \Nbd_{\phi_{k-1} r}(S_k)$. |
466 Each $R_i$ is contained in a metric ball of radius $r' \deq (2\phi_{k-1}+2)r$. |
466 Each $R_i$ is contained in a metric ball of radius $r' \deq (2\phi_{k-1}+2)r$. |
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467 Note that the defining inequality of the $\phi_i$ guarantees that |
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468 \[ |
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469 \phi_{k-1}r' = \phi_{k-1}(2\phi_{k-1}+2)r \le \phi_k r \le \rho(M) . |
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470 \] |
467 By induction, there is a neighborhood $U$ of $R \deq \bigcup_i R_i$, |
471 By induction, there is a neighborhood $U$ of $R \deq \bigcup_i R_i$, |
468 homeomorphic to a disjoint union |
472 homeomorphic to a disjoint union |
469 of balls, and such that |
473 of balls, and such that |
470 \[ |
474 \[ |
471 U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , |
475 U \subeq \Nbd_{\phi_{k-1}r'}(R) = \Nbd_{t}(S) \subeq \Nbd_{\phi_k r}(S) , |