154 Fix $j$. |
154 Fix $j$. |
155 We will construct a 2-chain $d_j$ such that $\bd(d_j) = g_j(s(\bd b)) - g_{j-1}(s(\bd b))$. |
155 We will construct a 2-chain $d_j$ such that $\bd(d_j) = g_j(s(\bd b)) - g_{j-1}(s(\bd b))$. |
156 Let $g_{j-1}(s(\bd b)) = \sum e_k$, and let $\{p_m\}$ be the 0-blob diagrams |
156 Let $g_{j-1}(s(\bd b)) = \sum e_k$, and let $\{p_m\}$ be the 0-blob diagrams |
157 appearing in the boundaries of the $e_k$. |
157 appearing in the boundaries of the $e_k$. |
158 As in the construction of $h_1$, we can choose 1-blob diagrams $q_m$ such that |
158 As in the construction of $h_1$, we can choose 1-blob diagrams $q_m$ such that |
159 $\bd q_m = f_j(p_m) = p_m$. |
159 $\bd q_m = f_j(p_m) - p_m$ and $\supp(q_m)$ is contained in an open set of $\cV_1$. |
160 Furthermore, we can arrange that all of the $q_m$ have the same support, and that this support |
160 %%% \nn{better not to do this, to make things more parallel with general case (?)} |
161 is contained in a open set of $\cV_1$. |
161 %Furthermore, we can arrange that all of the $q_m$ have the same support, and that this support |
162 (This is possible since there are only finitely many $p_m$.) |
162 %is contained in a open set of $\cV_1$. |
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163 %(This is possible since there are only finitely many $p_m$.) |
163 If $x$ is a sum of $p_m$'s, we denote the corresponding sum of $q_m$'s by $q(x)$. |
164 If $x$ is a sum of $p_m$'s, we denote the corresponding sum of $q_m$'s by $q(x)$. |
164 |
165 |
165 Now consider, for each $k$, $e_k + q(\bd e_k)$. |
166 Now consider, for each $k$, $e_k + q(\bd e_k)$. |
166 This is a 1-chain whose boundary is $f_j(\bd e_k)$. |
167 This is a 1-chain whose boundary is $f_j(\bd e_k)$. |
167 The support of $e_k$ is $g_{j-1}(V)$ for some $V\in \cV_1$, and |
168 The support of $e_k$ is $g_{j-1}(V)$ for some $V\in \cV_1$, and |
168 the support of $q(\bd e_k)$ is contained in $V'$ for some $V'\in \cV_1$. |
169 the support of $q(\bd e_k)$ is contained in a union $V'$ of finitely many open sets |
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170 of $\cV_1$, all of which contain the support of $f_j$. |
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171 %the support of $q(\bd e_k)$ is contained in $V'$ for some $V'\in \cV_1$. |
169 We now reveal the mysterious condition (mentioned above) which $\cV_1$ satisfies: |
172 We now reveal the mysterious condition (mentioned above) which $\cV_1$ satisfies: |
170 the union of $g_{j-1}(V)$ and $V'$, for all of the finitely many instances |
173 the union of $g_{j-1}(V)$ and $V'$, for all of the finitely many instances |
171 arising in the construction of $h_2$, lies inside a disjoint union of balls $U$ |
174 arising in the construction of $h_2$, lies inside a disjoint union of balls $U$ |
172 such that each individual ball lies in an open set of $\cV_2$. |
175 such that each individual ball lies in an open set of $\cV_2$. |
173 (In this case there are either one or two balls in the disjoint union.) |
176 (In this case there are either one or two balls in the disjoint union.) |
174 For any fixed open cover $\cV_2$ this condition can be satisfied by choosing $\cV_1$ small enough. |
177 For any fixed open cover $\cV_2$ this condition can be satisfied by choosing $\cV_1$ |
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178 to be a sufficiently fine cover. |
175 It follows from \ref{disj-union-contract} |
179 It follows from \ref{disj-union-contract} |
176 that we can choose $x_k \in \bc_2(X)$ with $\bd x_k = f_j(e_k) - e_k - q(\bd e_k)$ |
180 that we can choose $x_k \in \bc_2(X)$ with $\bd x_k = f_j(e_k) - e_k - q(\bd e_k)$ |
177 and with $\supp(x_k) = U$. |
181 and with $\supp(x_k) = U$. |
178 We can now take $d_j \deq \sum x_k$. |
182 We can now take $d_j \deq \sum x_k$. |
179 It is clear that $\bd d_j = \sum (f_j(e_k) - e_k) = g_j(s(\bd b)) - g_{j-1}(s(\bd b))$, as desired. |
183 It is clear that $\bd d_j = \sum (f_j(e_k) - e_k) = g_j(s(\bd b)) - g_{j-1}(s(\bd b))$, as desired. |
192 The general case $h_l$ is similar. |
196 The general case $h_l$ is similar. |
193 When constructing the analogue of $x_k$ above, we will need to find a disjoint union of balls $U$ |
197 When constructing the analogue of $x_k$ above, we will need to find a disjoint union of balls $U$ |
194 which contains finitely many open sets from $\cV_{l-1}$ |
198 which contains finitely many open sets from $\cV_{l-1}$ |
195 such that each ball is contained in some open set of $\cV_l$. |
199 such that each ball is contained in some open set of $\cV_l$. |
196 For sufficiently fine $\cV_{l-1}$ this will be possible. |
200 For sufficiently fine $\cV_{l-1}$ this will be possible. |
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201 Since $C_*$ is finite, the process terminates after finitely many, say $r$, steps. |
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202 We take $\cV_r = \cU$. |
197 |
203 |
198 \nn{should probably be more specific at the end} |
204 \nn{should probably be more specific at the end} |
199 \end{proof} |
205 \end{proof} |
200 |
206 |
201 |
207 |
214 whose $(i,j)$ entry is $C_i(\BD_j)$, the singular $i$-chains on the space of $j$-blob diagrams. |
220 whose $(i,j)$ entry is $C_i(\BD_j)$, the singular $i$-chains on the space of $j$-blob diagrams. |
215 The horizontal boundary of the double complex, |
221 The horizontal boundary of the double complex, |
216 denoted $\bd_t$, is the singular boundary, and the vertical boundary, denoted $\bd_b$, is |
222 denoted $\bd_t$, is the singular boundary, and the vertical boundary, denoted $\bd_b$, is |
217 the blob boundary. |
223 the blob boundary. |
218 |
224 |
219 |
225 We will regard $\bc_*(X)$ as the subcomplex $\btc_{0*}(X) \sub \btc_{**}(X)$. |
220 |
226 The main result of this subsection is |
221 |
227 |
222 \subsection{Action of \texorpdfstring{$\CH{X}$}{C_*(Homeo(M))}} |
228 \begin{lemma} \label{lem:bt-btc} |
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229 The inclusion $\bc_*(X) \sub \btc_*(X)$ is a homotopy equivalence |
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230 \end{lemma} |
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231 |
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232 Before giving the proof we need a few preliminary results. |
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233 |
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234 |
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235 |
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236 |
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237 |
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238 \subsection{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}} |
223 \label{ss:emap-def} |
239 \label{ss:emap-def} |
224 |
240 |
225 |
241 |
226 |
242 |
227 \subsection{[older version still hanging around]} |
243 \subsection{[older version still hanging around]} |