103 |
103 |
104 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
104 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
105 of small collar maps, plus a shrunken version of $b$. |
105 of small collar maps, plus a shrunken version of $b$. |
106 The composition of all the collar maps shrinks $B$ to a sufficiently small ball. |
106 The composition of all the collar maps shrinks $B$ to a sufficiently small ball. |
107 |
107 |
108 Let $\cV_1$ be an auxiliary open cover of $X$, satisfying conditions specified below. |
108 Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
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109 also satisfying conditions specified below. |
109 Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. |
110 Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. |
110 Choose a sequence of collar maps $f_j:\bc_0(B)\to\bc_0(B)$ such that each has support |
111 Choose a sequence of collar maps $f_j:\bc_0(B)\to\bc_0(B)$ such that each has support |
111 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
112 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
112 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
113 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
113 \nn{need to say this better; maybe give fig} |
114 \nn{need to say this better; maybe give fig} |
140 $s(b)$ consists of a series of 2-blob diagrams implementing a series |
141 $s(b)$ consists of a series of 2-blob diagrams implementing a series |
141 of small collar maps, plus a shrunken version of $b$. |
142 of small collar maps, plus a shrunken version of $b$. |
142 The composition of all the collar maps shrinks $B$ to a sufficiently small |
143 The composition of all the collar maps shrinks $B$ to a sufficiently small |
143 disjoint union of balls. |
144 disjoint union of balls. |
144 |
145 |
145 Let $\cV_2$ be an auxiliary open cover of $X$, satisfying conditions specified below. |
146 Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and |
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147 also satisfying conditions specified below. |
146 As before, choose a sequence of collar maps $f_j$ |
148 As before, choose a sequence of collar maps $f_j$ |
147 such that each has support |
149 such that each has support |
148 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
150 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
149 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
151 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
150 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
152 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
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153 |
151 Fix $j$. |
154 Fix $j$. |
152 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 $\bd(d_j) = g_j(s(\bd b)) - g_{j-1}(s(\bd b))$. |
153 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 |
154 appearing in the boundaries of the $e_k$. |
157 appearing in the boundaries of the $e_k$. |
155 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 |
156 $\bd q_m = g_j(p_m) = g_{j-1}(p_m)$. |
159 $\bd q_m = f_j(p_m) = p_m$. |
157 Furthermore, we can arrange that all of the $q_m$ have the same support, and that this support |
160 Furthermore, we can arrange that all of the $q_m$ have the same support, and that this support |
158 is contained in a open set of $\cV_1$. |
161 is contained in a open set of $\cV_1$. |
159 (This is possible since there are only finitely many $p_m$.) |
162 (This is possible since there are only finitely many $p_m$.) |
160 Now consider |
163 If $x$ is a sum of $p_m$'s, we denote the corresponding sum of $q_m$'s by $q(x)$. |
161 |
164 |
162 |
165 Now consider, for each $k$, $e_k + q(\bd e_k)$. |
163 |
166 This is a 1-chain whose boundary is $f_j(\bd e_k)$. |
164 |
167 The support of $e_k$ is $g_{j-1}(V)$ for some $V\in \cV_1$, and |
165 |
168 the support of $q(\bd e_k)$ is contained in $V'$ for some $V'\in \cV_1$. |
166 \nn{...} |
169 We now reveal the mysterious condition (mentioned above) which $\cV_1$ satisfies: |
167 |
170 the union of $g_{j-1}(V)$ and $V'$, for all of the finitely many instances |
168 |
171 arising in the construction of $h_2$, lies inside a disjoint union of balls $U$ |
169 |
172 such that each individual ball lies in an open set of $\cV_2$. |
170 |
173 (In this case there are either one or two balls in the disjoint union.) |
171 |
174 For any fixed open cover $\cV_2$ this condition can be satisfied by choosing $\cV_1$ small enough. |
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175 It follows from \ref{disj-union-contract} |
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176 that we can choose $x_k \in \bc_2(X)$ with $\bd x_k = f_j(e_k) - e_k - q(\bd e_k)$ |
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177 and with $\supp(x_k) = U$. |
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178 We can now take $d_j \deq \sum x_k$. |
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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. |
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180 \nn{should maybe have figure} |
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181 |
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182 We now define |
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183 \[ |
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184 s(b) = \sum d_j + g(b), |
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185 \] |
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186 where $g$ is the composition of all the $f_j$'s. |
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187 It is easy to verify that $s(b) \in \sbc_2$, $\supp(s(b)) = \supp(b)$, and |
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188 $\bd(s(b)) = s(\bd b)$. |
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189 If follows that we can choose $h_2(b)\in \bc_2(X)$ such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$. |
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190 This completes the definition of $h_2$. |
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191 |
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192 The general case $h_l$ is similar. |
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193 When constructing the analogue of $x_k$ above, we will need to find a disjoint union of balls $U$ |
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194 which contains finitely many open sets from $\cV_{l-1}$ |
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195 such that each ball is contained in some open set of $\cV_l$. |
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196 For sufficiently fine $\cV_{l-1}$ this will be possible. |
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197 |
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198 \nn{should probably be more specific at the end} |
172 \end{proof} |
199 \end{proof} |
173 |
200 |
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201 |
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202 \medskip |
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203 |
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204 Next we define the sort-of-simplicial space version of the blob complex, $\btc_*(X)$. |
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205 First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$. |
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206 We give $\BD_k$ the finest topology such that |
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207 \begin{itemize} |
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208 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
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209 \item \nn{something about blob labels and vector space structure} |
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210 \item \nn{maybe also something about gluing} |
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211 \end{itemize} |
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212 |
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213 Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |
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214 whose $(i,j)$ entry is $C_i(\BD_j)$, the singular $i$-chains on the space of $j$-blob diagrams. |
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215 The horizontal boundary of the double complex, |
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216 denoted $\bd_t$, is the singular boundary, and the vertical boundary, denoted $\bd_b$, is |
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217 the blob boundary. |
174 |
218 |
175 |
219 |
176 |
220 |
177 |
221 |
178 \subsection{Action of \texorpdfstring{$\CH{X}$}{C_*(Homeo(M))}} |
222 \subsection{Action of \texorpdfstring{$\CH{X}$}{C_*(Homeo(M))}} |