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 \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 

   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 $\le 2$.