217 |
217 |
218 We now define a homotopy inverse $s: J_* \to K_*(C)$ to the inclusion $i$. |
218 We now define a homotopy inverse $s: J_* \to K_*(C)$ to the inclusion $i$. |
219 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
219 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
220 * is a labeled point in $y$. |
220 * is a labeled point in $y$. |
221 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
221 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
222 Extending linearly, we get the desired map $s: \bc_*(S^1) \to K_*(C)$. |
222 Extending linearly, we get the desired map $s: J_* \to K_*(C)$. |
223 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
223 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
224 |
224 |
225 Let $N_\ep$ denote the ball of radius $\ep$ around *. |
225 Let $N_\ep$ denote the ball of radius $\ep$ around *. |
226 Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex |
226 Let $L_*^\ep \sub J_*$ be the subcomplex |
227 spanned by blob diagrams |
227 spanned by blob diagrams |
228 where there are no labeled points |
228 where there are no labeled points |
229 in $N_\ep$, except perhaps $*$, and $N_\ep$ is either disjoint from or contained in |
229 in $N_\ep$, except perhaps $*$, and $N_\ep$ is either disjoint from or contained in |
230 every blob in the diagram. |
230 every blob in the diagram. |
231 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
231 Note that for any chain $x \in J_*$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
232 |
232 |
233 We define a degree $1$ map $j_\ep: L_*^\ep \to L_*^\ep$ as follows. |
233 We define a degree $1$ map $j_\ep: L_*^\ep \to L_*^\ep$ as follows. |
234 Let $x \in L_*^\ep$ be a blob diagram. |
234 Let $x \in L_*^\ep$ be a blob diagram. |
235 \nn{maybe add figures illustrating $j_\ep$?} |
235 %\nn{maybe add figures illustrating $j_\ep$?} |
236 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding |
236 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding |
237 $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
237 $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
238 of $x$ to $N_\ep$. |
238 of $x$ to $N_\ep$. |
239 If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
239 If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
240 \nn{SM: I don't think we need to consider sums here} |
240 %\nn{SM: I don't think we need to consider sums here} |
241 \nn{KW: It depends on whether we allow linear combinations of fields outside of twig blobs} |
241 %\nn{KW: It depends on whether we allow linear combinations of fields outside of twig blobs} |
242 write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let |
242 write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let |
243 $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$, |
243 $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$, |
244 and have an additional blob $N_\ep$ with label $y_i - s(y_i)$. |
244 and have an additional blob $N_\ep$ with label $y_i - s(y_i)$. |
245 Define $j_\ep(x) = \sum x_i$. |
245 Define $j_\ep(x) = \sum x_i$. |
246 |
246 |
248 \[ |
248 \[ |
249 \bd j_\ep + j_\ep \bd = \id - i \circ s . |
249 \bd j_\ep + j_\ep \bd = \id - i \circ s . |
250 \] |
250 \] |
251 (To get the signs correct here, we add $N_\ep$ as the first blob.) |
251 (To get the signs correct here, we add $N_\ep$ as the first blob.) |
252 Since for $\ep$ small enough $L_*^\ep$ captures all of the |
252 Since for $\ep$ small enough $L_*^\ep$ captures all of the |
253 homology of $\bc_*(S^1)$, |
253 homology of $J_*$, |
254 it follows that the mapping cone of $i \circ s$ is acyclic and therefore (using the fact that |
254 it follows that the mapping cone of $i \circ s$ is acyclic and therefore (using the fact that |
255 these complexes are free) $i \circ s$ is homotopic to the identity. |
255 these complexes are free) $i \circ s$ is homotopic to the identity. |
256 \end{proof} |
256 \end{proof} |
257 |
257 |
258 \begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
258 \begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
469 x - \bd h(x) - h(\bd x) \in K''_* . |
469 x - \bd h(x) - h(\bd x) \in K''_* . |
470 } |
470 } |
471 Since $K'_0 = K''_0$, we can take $h_0 = 0$. |
471 Since $K'_0 = K''_0$, we can take $h_0 = 0$. |
472 Let $x \in K'_1$, with single blob $B \sub S^1$. |
472 Let $x \in K'_1$, with single blob $B \sub S^1$. |
473 If $* \notin B$, then $x \in K''_1$ and we define $h_1(x) = 0$. |
473 If $* \notin B$, then $x \in K''_1$ and we define $h_1(x) = 0$. |
474 If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$). |
474 If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with $B$ playing the role of $N$ above). |
475 Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$. |
475 Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$. |
476 Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$. |
476 Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$. |
477 Define $h_1(x) = y$. |
477 Define $h_1(x) = y$. |
478 The general case is similar, except that we have to take lower order homotopies into account. |
478 The general case is similar, except that we have to take lower order homotopies into account. |
479 Let $x \in K'_k$. |
479 Let $x \in K'_k$. |
484 So $x' \in G'_l$ for some $l \le k$. |
484 So $x' \in G'_l$ for some $l \le k$. |
485 Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$. |
485 Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$. |
486 Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$. |
486 Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$. |
487 Define $h_k(x) = y \bullet p$. |
487 Define $h_k(x) = y \bullet p$. |
488 This completes the proof that $i: K''_* \to K'_*$ is a homotopy equivalence. |
488 This completes the proof that $i: K''_* \to K'_*$ is a homotopy equivalence. |
489 \nn{need to say above more clearly and settle on notation/terminology} |
489 %\nn{need to say above more clearly and settle on notation/terminology} |
490 |
490 |
491 Finally, we show that $K''_*$ is contractible with $H_0\cong C$. |
491 Finally, we show that $K''_*$ is contractible with $H_0\cong C$. |
492 This is similar to the proof of Proposition \ref{bcontract}, but a bit more |
492 This is similar to the proof of Proposition \ref{bcontract}, but a bit more |
493 complicated since there is no single blob which contains the support of all blob diagrams |
493 complicated since there is no single blob which contains the support of all blob diagrams |
494 in $K''_*$. |
494 in $K''_*$. |