174 Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ |
174 Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ |
175 define |
175 define |
176 \[ |
176 \[ |
177 N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|). |
177 N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|). |
178 \] |
178 \] |
179 In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
179 In other words, for each $i$ |
|
180 we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
180 by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling |
181 by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling |
181 the size of the buffers around $|p|$. |
182 the size of the buffers around $|p|$. |
182 |
183 |
183 Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$. |
184 Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$. |
184 Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
185 Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
223 V^j \subeq N_{i,k}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V . |
224 V^j \subeq N_{i,k}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V . |
224 \] |
225 \] |
225 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
226 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
226 We therefore have splittings |
227 We therefore have splittings |
227 \[ |
228 \[ |
228 p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet b'' , |
229 p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' , |
229 \] |
230 \] |
230 where $p' \in CH_*(V)$, $p'' \in CH_*(X\setmin V)$, |
231 where $p' \in CH_*(V)$, $p'' \in CH_*(X\setmin V)$, |
231 $b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, |
232 $b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, |
232 $f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$. |
233 $f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$. |
233 (Note that since the family of homeomorphisms $p$ is constant (independent of parameters) |
234 (Note that since the family of homeomorphisms $p$ is constant (independent of parameters) |
313 |
314 |
314 Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
315 Next we show how to homotope chains in $CH_*(X)\ot \bc_*(X)$ to one of the |
315 $G_*^{i,m}$. |
316 $G_*^{i,m}$. |
316 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
317 Choose a monotone decreasing sequence of real numbers $\gamma_j$ converging to zero. |
317 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
318 Let $\cU_j$ denote the open cover of $X$ by balls of radius $\gamma_j$. |
318 Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}. |
319 Let $h_j: CH_*(X)\to CH_*(X)$ be a chain map homotopic to the identity whose image is spanned by families of homeomorphisms with support compatible with $\cU_j$, as described in Lemma \ref{xxxxx}. |
319 Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
320 Recall that $h_j$ and also the homotopy connecting it to the identity do not increase |
320 supports. |
321 supports. |
321 Define |
322 Define |
322 \[ |
323 \[ |
323 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
324 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
324 \] |
325 \] |
325 The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
326 The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
326 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
327 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
327 (depending on $b$, $n = \deg(p)$ and $m$). |
328 (depending on $b$, $\deg(p)$ and $m$). |
328 (Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.) |
329 %(Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.) |
329 |
330 |
330 \begin{lemma} \label{Gim_approx} |
331 \begin{lemma} \label{Gim_approx} |
331 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$. |
332 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CH_*(X)$. |
332 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
333 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
333 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CH_n(X)$ |
334 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CH_n(X)$ |
339 There exists $l > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < l$ |
340 There exists $l > 0$ such that $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < l$ |
340 and all such $c$. |
341 and all such $c$. |
341 (Here we are using a piecewise smoothness assumption for $\bd c$, and also |
342 (Here we are using a piecewise smoothness assumption for $\bd c$, and also |
342 the fact that $\bd c$ is collared. |
343 the fact that $\bd c$ is collared. |
343 We need to consider all such $c$ because all generators appearing in |
344 We need to consider all such $c$ because all generators appearing in |
344 iterated boundaries of must be in $G_*^{i,m}$.) |
345 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.) |
345 |
346 |
346 Let $r = \deg(b)$ and |
347 Let $r = \deg(b)$ and |
347 \[ |
348 \[ |
348 t = r+n+m+1 = \deg(p\ot b) + m + 1. |
349 t = r+n+m+1 = \deg(p\ot b) + m + 1. |
349 \] |
350 \] |