equal
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174 \ref{lem:module-blob}, |
174 \ref{lem:module-blob}, |
175 \ref{lem:hochschild-exact}, \ref{lem:hochschild-coinvariants} and |
175 \ref{lem:hochschild-exact}, \ref{lem:hochschild-coinvariants} and |
176 \ref{lem:hochschild-free}. |
176 \ref{lem:hochschild-free}. |
177 \end{proof} |
177 \end{proof} |
178 |
178 |
|
179 \subsection{Technical details} |
179 \begin{proof}[Proof of Lemma \ref{lem:module-blob}] |
180 \begin{proof}[Proof of Lemma \ref{lem:module-blob}] |
180 We show that $K_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
181 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 * |
182 $K_*(C)$ 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. |
183 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 $i: K_*(C) \to \bc_*(S^1)$. |
184 In particular, there is an inclusion map $i: K_*(C) \to \bc_*(S^1)$. |
466 This allows us to construct $x\in K''_1$ such that $\bd x = y$. |
467 This allows us to construct $x\in K''_1$ such that $\bd x = y$. |
467 (The label of $B$ is the restriction of $y$ to $B$.) |
468 (The label of $B$ is the restriction of $y$ to $B$.) |
468 It follows that $H_0(K''_*) \cong C$. |
469 It follows that $H_0(K''_*) \cong C$. |
469 \end{proof} |
470 \end{proof} |
470 |
471 |
471 \medskip |
472 \subsection{An explicit chain map in low degrees} |
472 |
473 |
473 For purposes of illustration, we describe an explicit chain map |
474 For purposes of illustration, we describe an explicit chain map |
474 $\HC_*(M) \to K_*(M)$ |
475 $\HC_*(M) \to K_*(M)$ |
475 between the Hochschild complex and the blob complex (with bimodule point) |
476 between the Hochschild complex and the blob complex (with bimodule point) |
476 for degree $\le 2$. |
477 for degree $\le 2$. |