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 $f: F_*^\ep \to F_*^\ep$. |
221 By Lemma \ref{support-shrink}, $f$ is homotopic to the identity. |
221 By Corollary \ref{disj-union-contract}, |
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222 $f$ is homotopic to the identity. |
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223 (Use the facts that $f$ factors though a map from a disjoint union of balls |
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224 into $S^1$, and that $f$ is the identity in degree 0.) |
222 Since the image of $f$ is in $J_*$, and since any blob chain is in $F_*^\ep$ |
225 Since the image of $f$ is in $J_*$, and since any blob chain is in $F_*^\ep$ |
223 for $\ep$ sufficiently small, we have that $J_*$ is homotopic to all of $\bc_*(S^1)$. |
226 for $\ep$ sufficiently small, we have that $J_*$ is homotopic to all of $\bc_*(S^1)$. |
224 |
227 |
225 We now define a homotopy inverse $s: J_* \to K_*(C)$ to the inclusion $i$. |
228 We now define a homotopy inverse $s: J_* \to K_*(C)$ to the inclusion $i$. |
226 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
229 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |