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182 is always a labeled point in $K_*(C)$, while in $\bc_*(S^1)$ it may or may not be. |
182 is always a labeled point in $K_*(C)$, while in $\bc_*(S^1)$ it may or may not be. |
183 In particular, there is an inclusion map $i: K_*(C) \to \bc_*(S^1)$. |
183 In particular, there is an inclusion map $i: K_*(C) \to \bc_*(S^1)$. |
184 |
184 |
185 We want to define a homotopy inverse to the above inclusion, but before doing so |
185 We want to define a homotopy inverse to the above inclusion, but before doing so |
186 we must replace $\bc_*(S^1)$ with a homotopy equivalent subcomplex. |
186 we must replace $\bc_*(S^1)$ with a homotopy equivalent subcomplex. |
187 Let $J_* \sub \bc_*(S^1)$ be the subcomplex where * does not lie to the boundary |
187 Let $J_* \sub \bc_*(S^1)$ be the subcomplex where * does not lie on the boundary |
188 of any blob. Note that the image of $i$ is contained in $J_*$. |
188 of any blob. Note that the image of $i$ is contained in $J_*$. |
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 |