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167 Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $k$ |
167 Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $k$ |
168 define |
168 define |
169 \[ |
169 \[ |
170 N_{i,k}(p\ot b) \deq \Nbd_{k\ep_i}(|b|) \cup \Nbd_{4^k\delta_i}(|p|). |
170 N_{i,k}(p\ot b) \deq \Nbd_{k\ep_i}(|b|) \cup \Nbd_{4^k\delta_i}(|p|). |
171 \] |
171 \] |
|
172 \nn{not currently correct; maybe need to split $k$ into two parameters} |
172 In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
173 In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized |
173 by $k$), with $\ep_i$ controlling the size of the buffer around $|b|$ and $\delta_i$ controlling |
174 by $k$), with $\ep_i$ controlling the size of the buffer around $|b|$ and $\delta_i$ controlling |
174 the size of the buffer around $|p|$. |
175 the size of the buffer around $|p|$. |
175 (The $4^k$ comes from Lemma \ref{xxxx}.) |
176 (The $4^k$ comes from Lemma \ref{xxxx}.) |
176 |
177 |
191 |
192 |
192 As sketched above and explained in detail below, |
193 As sketched above and explained in detail below, |
193 $G_*^{i,m}$ is a subcomplex where it is easy to define |
194 $G_*^{i,m}$ is a subcomplex where it is easy to define |
194 the evaluation map. |
195 the evaluation map. |
195 The parameter $m$ controls the number of iterated homotopies we are able to construct |
196 The parameter $m$ controls the number of iterated homotopies we are able to construct |
196 (see Lemma \ref{mhtyLemma}). |
197 (see Lemma \ref{m_order_hty}). |
197 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
198 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of |
198 $CD_*(X)\ot \bc_*(X)$ (see Lemma \ref{xxxlemma}). |
199 $CD_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}). |
199 |
200 |
200 Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$. |
201 Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$. |
201 Let $p\ot b \in G_*^{i,m}$. |
202 Let $p\ot b \in G_*^{i,m}$. |
202 If $\deg(p) = 0$, define |
203 If $\deg(p) = 0$, define |
203 \[ |
204 \[ |
258 This is a standard result in the method of acyclic models. |
259 This is a standard result in the method of acyclic models. |
259 \nn{should we say more here?} |
260 \nn{should we say more here?} |
260 \nn{maybe this lemma should be subsumed into the next lemma. probably it should.} |
261 \nn{maybe this lemma should be subsumed into the next lemma. probably it should.} |
261 \end{proof} |
262 \end{proof} |
262 |
263 |
263 \begin{lemma} |
264 \begin{lemma} \label{m_order_hty} |
264 Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
265 Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with |
265 different choices of $V$ (and hence also different choices of $x'$) at each step. |
266 different choices of $V$ (and hence also different choices of $x'$) at each step. |
266 If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. |
267 If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. |
267 If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic. |
268 If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic. |
268 And so on. |
269 And so on. |
317 The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
318 The next lemma says that for all generators $p\ot b$ we can choose $j$ large enough so that |
318 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
319 $g_j(p)\ot b$ lies in $G_*^{i,m}$, for arbitrary $m$ and sufficiently large $i$ |
319 (depending on $b$, $n = \deg(p)$ and $m$). |
320 (depending on $b$, $n = \deg(p)$ and $m$). |
320 \nn{not the same $n$ as the dimension of the manifolds; fix this} |
321 \nn{not the same $n$ as the dimension of the manifolds; fix this} |
321 |
322 |
322 \begin{lemma} |
323 \begin{lemma} \label{Gim_approx} |
323 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CD_*(X)$. |
324 Fix a blob diagram $b$, a homotopy order $m$ and a degree $n$ for $CD_*(X)$. |
324 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
325 Then there exists a constant $k_{bmn}$ such that for all $i \ge k_{bmn}$ |
325 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CD_n(X)$ |
326 there exists another constant $j_i$ such that for all $j \ge j_i$ and all $p\in CD_n(X)$ |
326 we have $g_j(p)\ot b \in G_*^{i,m}$. |
327 we have $g_j(p)\ot b \in G_*^{i,m}$. |
327 \end{lemma} |
328 \end{lemma} |