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210 We claim that $J_*$ is homotopy equivalent to $\bc_*(S^1)$. |
210 We claim that $J_*$ is homotopy equivalent to $\bc_*(S^1)$. |
211 Let $F_*^\ep \sub \bc_*(S^1)$ be the subcomplex where either |
211 Let $F_*^\ep \sub \bc_*(S^1)$ be the subcomplex where either |
212 (a) the point * is not on the boundary of any blob or |
212 (a) the point * is not on the boundary of any blob or |
213 (b) there are no labeled points or blob boundaries within distance $\ep$ of *, |
213 (b) there are no labeled points or blob boundaries within distance $\ep$ of *, |
214 other than blob boundaries at * itself. |
214 other than blob boundaries at * itself. |
215 Note that all blob diagrams are in $F_*^\ep$ for $\ep$ sufficiently small. |
215 Note that all blob diagrams are in some $F_*^\ep$ for $\ep$ sufficiently small. |
216 Let $b$ be a blob diagram in $F_*^\ep$. |
216 Let $b$ be a blob diagram in $F_*^\ep$. |
217 Define $f(b)$ to be the result of moving any blob boundary points which lie on * |
217 Define $f(b)$ to be the result of moving any blob boundary points which lie on * |
218 to distance $\ep$ from *. |
218 to distance $\ep$ from *. |
219 (Move right or left so as to shrink the blob.) |
219 (Move right or left so as to shrink the blob.) |
220 Extend to get a chain map $f: F_*^\ep \to F_*^\ep$. |
220 Extend to get a chain map $f: F_*^\ep \to F_*^\ep$. |
226 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
226 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
227 * is a labeled point in $y$. |
227 * is a labeled point in $y$. |
228 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
228 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
229 Extending linearly, we get the desired map $s: J_* \to K_*(C)$. |
229 Extending linearly, we get the desired map $s: J_* \to K_*(C)$. |
230 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
230 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
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231 What remains is to show that $i \circ s$ is homotopic to the identity. |
231 |
232 |
232 Let $N_\ep$ denote the ball of radius $\ep$ around *. |
233 Let $N_\ep$ denote the ball of radius $\ep$ around *. |
233 Let $L_*^\ep \sub J_*$ be the subcomplex |
234 Let $L_*^\ep \sub J_*$ be the subcomplex |
234 spanned by blob diagrams |
235 spanned by blob diagrams |
235 where there are no labeled points |
236 where there are no labeled points |