180 We show that $K_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
180 We show that $K_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
181 $K_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * |
181 $K_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * |
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 define a left inverse $s: \bc_*(S^1) \to K_*(C)$ to the inclusion as follows. |
185 We want to define a homotopy inverse to the above inclusion, but before doing so |
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186 we must replace $\bc_*(S^1)$ with a homotopy equivalent subcomplex. |
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187 Let $J_* \sub \bc_*(S^1)$ be the subcomplex where * does not lie to the boundary |
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188 of any blob. Note that the image of $i$ is contained in $J_*$. |
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189 Note also that in $\bc_*(S^1)$ (away from $J_*$) |
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190 a blob diagram could have multiple (nested) blobs whose |
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191 boundaries contain *, on both the right and left of *. |
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192 |
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193 We claim that $J_*$ is homotopy equivalent to $\bc_*(S^1)$. |
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194 Let $F_*^\ep \sub \bc_*(S^1)$ be the subcomplex where there there are no labeled |
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195 points within distance $\ep$ of * on the right. |
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196 (This includes * itself.) |
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197 \nn{...} |
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198 |
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199 |
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200 |
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201 We want to define a homotopy inverse $s: \bc_*(S^1) \to K_*(C)$ to the inclusion. |
186 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
202 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
187 * is a labeled point in $y$. |
203 * is a labeled point in $y$. |
188 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
204 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
189 Extending linearly, we get the desired map $s: \bc_*(S^1) \to K_*(C)$. |
205 Extending linearly, we get the desired map $s: \bc_*(S^1) \to K_*(C)$. |
190 %Let $x \in \bc_*(S^1)$. |
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191 %Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
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192 %$x$ with $s(y)$. |
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193 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
206 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
194 |
207 |
195 Let $N_\ep$ denote the ball of radius $\ep$ around *. |
208 Let $N_\ep$ denote the ball of radius $\ep$ around *. |
196 Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex |
209 Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex |
197 spanned by blob diagrams |
210 spanned by blob diagrams |