162 Choose a metric on $X$. |
162 Choose a metric on $X$. |
163 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
163 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero |
164 (e.g.\ $\ep_i = 2^{-i}$). |
164 (e.g.\ $\ep_i = 2^{-i}$). |
165 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
165 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$ |
166 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
166 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$). |
167 Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $k$ |
167 Let $\phi_l$ be an increasing sequence of positive numbers |
|
168 satisfying the inequalities of Lemma \ref{xx2phi} (e.g. $\phi_l = 2^{3^{l-1}}$). |
|
169 Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$ |
168 define |
170 define |
169 \[ |
171 \[ |
170 N_{i,k}(p\ot b) \deq \Nbd_{k\ep_i}(|b|) \cup \Nbd_{4^k\delta_i}(|p|). |
172 N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|). |
171 \] |
173 \] |
172 \nn{not currently correct; maybe need to split $k$ into two parameters} |
|
173 In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
174 In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
174 by $k$), with $\ep_i$ controlling the size of the buffer around $|b|$ and $\delta_i$ controlling |
175 by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling |
175 the size of the buffer around $|p|$. |
176 the size of the buffers around $|p|$. |
176 (The $4^k$ comes from Lemma \ref{xxxx}.) |
|
177 |
177 |
178 Next we define subcomplexes $G_*^{i,m} \sub CD_*(X)\otimes \bc_*(X)$. |
178 Next we define subcomplexes $G_*^{i,m} \sub CD_*(X)\otimes \bc_*(X)$. |
179 Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
179 Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) |
180 = \deg(p) + \deg(b)$. |
180 = \deg(p) + \deg(b)$. |
181 $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) |
181 $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) |
213 \] |
213 \] |
214 Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V^j$ be the choice of neighborhood |
214 Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V^j$ be the choice of neighborhood |
215 of $|p_j|\cup |b_j|$ made at the preceding stage of the induction. |
215 of $|p_j|\cup |b_j|$ made at the preceding stage of the induction. |
216 For all $j$, |
216 For all $j$, |
217 \[ |
217 \[ |
218 V^j \subeq N_{i,(k-1)+1}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V . |
218 V^j \subeq N_{i,k}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V . |
219 \] |
219 \] |
220 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
220 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) |
221 We therefore have splittings |
221 We therefore have splittings |
222 \[ |
222 \[ |
223 p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet b'' , |
223 p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet b'' , |
229 near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
229 near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are |
230 unambiguous.) |
230 unambiguous.) |
231 We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
231 We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$. |
232 %We also have that $\deg(b'') = 0 = \deg(p'')$. |
232 %We also have that $\deg(b'') = 0 = \deg(p'')$. |
233 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
233 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. |
234 This is possible by \nn{...}. |
234 This is possible by \ref{bcontract}, \ref{disjunion} and \nn{prop. 2 of local relations (isotopy)}. |
235 Finally, define |
235 Finally, define |
236 \[ |
236 \[ |
237 e(p\ot b) \deq x' \bullet p''(b'') . |
237 e(p\ot b) \deq x' \bullet p''(b'') . |
238 \] |
238 \] |
239 |
239 |
241 For each generator $p\ot b$ we specify the acyclic (in positive degrees) |
241 For each generator $p\ot b$ we specify the acyclic (in positive degrees) |
242 target complex $\bc_*(p(V)) \bullet p''(b'')$. |
242 target complex $\bc_*(p(V)) \bullet p''(b'')$. |
243 |
243 |
244 The definition of $e: G_*^{i,m} \to \bc_*(X)$ depends on two sets of choices: |
244 The definition of $e: G_*^{i,m} \to \bc_*(X)$ depends on two sets of choices: |
245 The choice of neighborhoods $V$ and the choice of inverse boundaries $x'$. |
245 The choice of neighborhoods $V$ and the choice of inverse boundaries $x'$. |
246 The next two lemmas show that up to (iterated) homotopy $e$ is independent |
246 The next lemma shows that up to (iterated) homotopy $e$ is independent |
247 of these choices. |
247 of these choices. |
248 |
248 (Note that independence of choices of $x'$ (for fixed choices of $V$) |
249 \begin{lemma} |
249 is a standard result in the method of acyclic models.) |
250 Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
250 |
251 different choices of $x'$ at each step. |
251 %\begin{lemma} |
252 (Same choice of $V$ at each step.) |
252 %Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
253 Then $e$ and $\tilde{e}$ are homotopic via a homotopy in $\bc_*(p(V)) \bullet p''(b'')$. |
253 %different choices of $x'$ at each step. |
254 Any two choices of such a first-order homotopy are second-order homotopic, and so on, |
254 %(Same choice of $V$ at each step.) |
255 to arbitrary order. |
255 %Then $e$ and $\tilde{e}$ are homotopic via a homotopy in $\bc_*(p(V)) \bullet p''(b'')$. |
256 \end{lemma} |
256 %Any two choices of such a first-order homotopy are second-order homotopic, and so on, |
257 |
257 %to arbitrary order. |
258 \begin{proof} |
258 %\end{lemma} |
259 This is a standard result in the method of acyclic models. |
259 |
260 \nn{should we say more here?} |
260 %\begin{proof} |
261 \nn{maybe this lemma should be subsumed into the next lemma. probably it should.} |
261 %This is a standard result in the method of acyclic models. |
262 \end{proof} |
262 %\nn{should we say more here?} |
|
263 %\nn{maybe this lemma should be subsumed into the next lemma. probably it should.} |
|
264 %\end{proof} |
263 |
265 |
264 \begin{lemma} \label{m_order_hty} |
266 \begin{lemma} \label{m_order_hty} |
265 Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
267 Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
266 different choices of $V$ (and hence also different choices of $x'$) at each step. |
268 different choices of $V$ (and hence also different choices of $x'$) at each step. |
267 If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. |
269 If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. |
286 \;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 , |
288 \;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 , |
287 \] |
289 \] |
288 where $p'_1 \in CD_*(V_1)$, $p''_1 \in CD_*(X\setmin V_1)$, |
290 where $p'_1 \in CD_*(V_1)$, $p''_1 \in CD_*(X\setmin V_1)$, |
289 $b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$, |
291 $b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$, |
290 $f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$. |
292 $f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$. |
291 Inductively, $\bd f'_1 = 0$. |
293 Inductively, $\bd f'_1 = 0$ and $f_1'' = p_1''(b_1'')$. |
292 Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$. |
294 Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$. |
293 Define |
295 Define |
294 \[ |
296 \[ |
295 h(p\ot b) \deq x'_1 \bullet p''_1(b''_1) . |
297 h(p\ot b) \deq x'_1 \bullet p''_1(b''_1) . |
296 \] |
298 \] |
316 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
318 g_j \deq h_j\circ h_{j-1} \circ \cdots \circ h_1 . |
317 \] |
319 \] |
318 The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
320 The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
319 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
321 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
320 (depending on $b$, $n = \deg(p)$ and $m$). |
322 (depending on $b$, $n = \deg(p)$ and $m$). |
321 \nn{not the same $n$ as the dimension of the manifolds; fix this} |
323 (Note: Don't confuse this $n$ with the top dimension $n$ used elsewhere in this paper.) |
322 |
324 |
323 \begin{lemma} \label{Gim_approx} |
325 \begin{lemma} \label{Gim_approx} |
324 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CD_*(X)$. |
326 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CD_*(X)$. |
325 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
327 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
326 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CD_n(X)$ |
328 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CD_n(X)$ |