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189 Note also that in $\bc_*(S^1)$ (away from $J_*$) |
189 Note also that in $\bc_*(S^1)$ (away from $J_*$) |
190 a blob diagram could have multiple (nested) blobs whose |
190 a blob diagram could have multiple (nested) blobs whose |
191 boundaries contain *, on both the right and left of *. |
191 boundaries contain *, on both the right and left of *. |
192 |
192 |
193 We claim that $J_*$ is homotopy equivalent to $\bc_*(S^1)$. |
193 We claim that $J_*$ is homotopy equivalent to $\bc_*(S^1)$. |
194 Let $F_*^\ep \sub \bc_*(S^1)$ be the subcomplex where there there are no labeled |
194 Let $F_*^\ep \sub \bc_*(S^1)$ be the subcomplex where either |
195 points within distance $\ep$ of * on the right. |
195 (a) the point * is not the left boundary of any blob or |
196 (This includes * itself.) |
196 (b) there are no labeled points to the right of * within distance $\ep$. |
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197 Note that all blob diagrams are in $F_*^\ep$ for $\ep$ sufficiently small. |
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198 |
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199 |
197 \nn{...} |
200 \nn{...} |
198 |
201 |
199 |
202 |
200 |
203 |
201 We want to define a homotopy inverse $s: \bc_*(S^1) \to K_*(C)$ to the inclusion. |
204 We want to define a homotopy inverse $s: \bc_*(S^1) \to K_*(C)$ to the inclusion. |