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 $X$, satisfying conditions specified below. |
109 Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. |
109 Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. |
110 Choose a series of collar maps $f_j:\bc_0(B)\to\bc_0(B)$ such that each has support |
110 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 |
111 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 $\cV_1$. |
112 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} |
113 \nn{need to say this better; maybe give fig} |
114 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
114 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
115 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
115 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
141 of small collar maps, plus a shrunken version of $b$. |
141 of small collar maps, plus a shrunken version of $b$. |
142 The composition of all the collar maps shrinks $B$ to a sufficiently small |
142 The composition of all the collar maps shrinks $B$ to a sufficiently small |
143 disjoint union of balls. |
143 disjoint union of balls. |
144 |
144 |
145 Let $\cV_2$ be an auxiliary open cover of $X$, satisfying conditions specified below. |
145 Let $\cV_2$ be an auxiliary open cover of $X$, satisfying conditions specified below. |
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146 As before, choose a sequence of collar maps $f_j$ |
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147 such that each has support |
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148 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
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149 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
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150 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
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151 Fix $j$. |
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152 We will construct a 2-chain $d_j$ such that $\bd(d_j) = g_j(s(\bd b)) - g_{j-1}(s(\bd b))$. |
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153 Let $g_{j-1}(s(\bd b)) = \sum e_k$, and let $\{p_m\}$ be the 0-blob diagrams |
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154 appearing in the boundaries of the $e_k$. |
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155 As in the construction of $h_1$, we can choose 1-blob diagrams $q_m$ such that |
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156 $\bd q_m = g_j(p_m) = g_{j-1}(p_m)$. |
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157 Furthermore, we can arrange that all of the $q_m$ have the same support, and that this support |
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158 is contained in a open set of $\cV_1$. |
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159 (This is possible since there are only finitely many $p_m$.) |
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160 Now consider |
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161 |
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162 |
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163 |
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164 |
146 |
165 |
147 \nn{...} |
166 \nn{...} |
148 |
167 |
149 |
168 |
150 |
169 |
151 |
170 |
152 |
171 |
153 \end{proof} |
172 \end{proof} |
154 |
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155 |
173 |
156 |
174 |
157 |
175 |
158 |
176 |
159 |
177 |