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 either |
194 Let $F_*^\ep \sub \bc_*(S^1)$ be the subcomplex where either |
195 (a) the point * is not the left boundary of any blob or |
195 (a) the point * is not on the boundary of any blob or |
196 (b) there are no labeled points to the right of * within distance $\ep$. |
196 (b) there are no labeled points or blob boundaries within distance $\ep$ of *. |
197 Note that all blob diagrams are in $F_*^\ep$ for $\ep$ sufficiently small. |
197 Note that all blob diagrams are in $F_*^\ep$ for $\ep$ sufficiently small. |
198 |
198 Let $b$ be a blob diagram in $F_*^\ep$. |
199 |
199 Define $f(b)$ to be the result of moving any blob boundary points which lie on * |
200 \nn{...} |
200 to distance $\ep$ from *. |
201 |
201 (Move right or left so as to shrink the blob.) |
202 |
202 Extend to get a chain map $f: F_*^\ep \to F_*^\ep$. |
203 |
203 By Lemma \ref{support-shrink}, $f$ is homotopic to the identity. |
204 We want to define a homotopy inverse $s: \bc_*(S^1) \to K_*(C)$ to the inclusion. |
204 Since the image of $f$ is in $J_*$, and since any blob chain is in $F_*^\ep$ |
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205 for $\ep$ sufficiently small, we have that $J_*$ is homotopic to all of $\bc_*(S^1)$. |
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206 |
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207 We now define a homotopy inverse $s: J_* \to K_*(C)$ to the inclusion $i$. |
205 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
208 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
206 * is a labeled point in $y$. |
209 * is a labeled point in $y$. |
207 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
210 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
208 Extending linearly, we get the desired map $s: \bc_*(S^1) \to K_*(C)$. |
211 Extending linearly, we get the desired map $s: \bc_*(S^1) \to K_*(C)$. |
209 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
212 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
213 spanned by blob diagrams |
216 spanned by blob diagrams |
214 where there are no labeled points |
217 where there are no labeled points |
215 in $N_\ep$, except perhaps $*$, and $N_\ep$ is either disjoint from or contained in |
218 in $N_\ep$, except perhaps $*$, and $N_\ep$ is either disjoint from or contained in |
216 every blob in the diagram. |
219 every blob in the diagram. |
217 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
220 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
218 \nn{what if * is on boundary of a blob? need preliminary homotopy to prevent this.} |
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219 |
221 |
220 We define a degree $1$ chain map $j_\ep: L_*^\ep \to L_*^\ep$ as follows. Let $x \in L_*^\ep$ be a blob diagram. |
222 We define a degree $1$ chain map $j_\ep: L_*^\ep \to L_*^\ep$ as follows. Let $x \in L_*^\ep$ be a blob diagram. |
221 \nn{maybe add figures illustrating $j_\ep$?} |
223 \nn{maybe add figures illustrating $j_\ep$?} |
222 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
224 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
223 of $x$ to $N_\ep$. If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
225 of $x$ to $N_\ep$. If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |