212 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
212 If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
213 * is a labeled point in $y$. |
213 * is a labeled point in $y$. |
214 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
214 Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
215 Let $x \in \bc_*(S^1)$. |
215 Let $x \in \bc_*(S^1)$. |
216 Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
216 Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
217 $x$ with $y$. |
217 $x$ with $s(y)$. |
218 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
218 It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
219 |
219 |
220 Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
220 Let $L_*^\ep \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
221 in a neighborhood $B_\ep$ of *, except perhaps *. |
221 in a neighborhood $B_\ep$ of $*$, except perhaps $*$, and $B_\ep$ is either disjoint from or contained in every blob. |
222 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
222 Note that for any chain $x \in \bc_*(S^1)$, $x \in L_*^\ep$ for sufficiently small $\ep$. |
223 \nn{rest of argument goes similarly to above} |
223 \nn{rest of argument goes similarly to above} |
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224 |
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225 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. |
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226 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding $B_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
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227 of $x$ to $B_\ep$. If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
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228 write $y_i$ for the restriction of $z_i$ to $B_\ep$, and let |
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229 $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin B_\ep$, |
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230 and have an additional blob $B_\ep$ with label $y_i - s(y_i)$. |
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231 Define $j_\ep(x) = \sum x_i$. |
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232 \todo{need to check signs coming from blob complex differential} |
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233 \todo{finish this} |
224 \end{proof} |
234 \end{proof} |
225 \begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
235 \begin{proof}[Proof of Lemma \ref{lem:hochschild-exact}] |
226 \todo{p. 1478 of scott's notes} |
236 We now prove that $K_*$ is an exact functor. |
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237 |
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238 %\todo{p. 1478 of scott's notes} |
227 Essentially, this comes down to the unsurprising fact that the functor on $C$-$C$ bimodules |
239 Essentially, this comes down to the unsurprising fact that the functor on $C$-$C$ bimodules |
228 \begin{equation*} |
240 \begin{equation*} |
229 M \mapsto \ker(C \tensor M \tensor C \xrightarrow{c_1 \tensor m \tensor c_2 \mapsto c_1 m c_2} M) |
241 M \mapsto \ker(C \tensor M \tensor C \xrightarrow{c_1 \tensor m \tensor c_2 \mapsto c_1 m c_2} M) |
230 \end{equation*} |
242 \end{equation*} |
231 is exact. For completeness we'll explain this below. |
243 is exact. For completeness we'll explain this below. |
246 & = q - 0 |
258 & = q - 0 |
247 \end{align*} |
259 \end{align*} |
248 (here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$). |
260 (here we used that $g$ is a map of $C$-$C$ bimodules, and that $\sum_i a_i q_i b_i = 0$). |
249 |
261 |
250 Identical arguments show that the functors |
262 Identical arguments show that the functors |
251 \begin{equation*} |
263 \begin{equation} |
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264 \label{eq:ker-functor}% |
252 M \mapsto \ker(C^{\tensor k} \tensor M \tensor C^{\tensor l} \to M) |
265 M \mapsto \ker(C^{\tensor k} \tensor M \tensor C^{\tensor l} \to M) |
253 \end{equation*} |
266 \end{equation} |
254 are all exact too. |
267 are all exact too. Moreover, tensor products of such functors with each |
255 |
268 other and with $C$ (e.g., producing the functor $M \mapsto \ker(M \tensor C \to M) |
256 Finally, then \todo{explain why this is all we need.} |
269 \tensor C \tensor \ker(C \tensor M \to M)$) are all still exact. |
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270 |
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271 Finally, then we see that the functor $K_*$ is simply an (infinite) |
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272 direct sum of this sort of functor. The direct sum is indexed by |
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273 configurations of nested blobs and positions of labels; for each such configuration, we have one of the above tensor product functors, |
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274 with the labels of twig blobs corresponding to tensor factors as in \eqref{eq:ker-functor}, and all other labelled points corresponding |
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275 to tensor factors of $C$. |
257 \end{proof} |
276 \end{proof} |
258 \begin{proof}[Proof of Lemma \ref{lem:hochschild-coinvariants}] |
277 \begin{proof}[Proof of Lemma \ref{lem:hochschild-coinvariants}] |
259 \todo{} |
278 \todo{} |
260 \end{proof} |
279 \end{proof} |
261 \begin{proof}[Proof of Lemma \ref{lem:hochschild-free}] |
280 \begin{proof}[Proof of Lemma \ref{lem:hochschild-free}] |
262 We show that $K_*(C\otimes C)$ is |
281 We show that $K_*(C\otimes C)$ is |
263 quasi-isomorphic to the 0-step complex $C$. |
282 quasi-isomorphic to the 0-step complex $C$. We'll do this in steps, establishing quasi-isomorphisms and homotopy equivalences |
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283 $$K_*(C \tensor C) \quismto K'_* \htpyto K''_* \quismto C.$$ |
264 |
284 |
265 Let $K'_* \sub K_*(C\otimes C)$ be the subcomplex where the label of |
285 Let $K'_* \sub K_*(C\otimes C)$ be the subcomplex where the label of |
266 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
286 the point $*$ is $1 \otimes 1 \in C\otimes C$. |
267 We will show that the inclusion $i: K'_* \to K_*(C\otimes C)$ is a quasi-isomorphism. |
287 We will show that the inclusion $i: K'_* \to K_*(C\otimes C)$ is a quasi-isomorphism. |
268 |
288 |