203 There is a collection of chain maps

   203 There is a collection of chain maps

   204 \[

   204 \[

   205 	C_*(SC^n_{\ol{M}\ol{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes 

   205 	C_*(SC^n_{\ol{M}\ol{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes 

   206 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to  \hom(\bc_*(M_0), \bc_*(N_0))

   206 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to  \hom(\bc_*(M_0), \bc_*(N_0))

   207 \]

   207 \]

   209 On $C_0(SC^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above.

   209 On $C_0(SC^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above.

   210 When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of \S\ref{sec:evaluation}.

   210 When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of \S\ref{sec:evaluation}.

   211 \end{thm}

   211 \end{thm}

   212 

   212 

   213 If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$

   213 If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$

   226 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to  \hom(\bc_*(M_0), \bc_*(N_0)) .

   226 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to  \hom(\bc_*(M_0), \bc_*(N_0)) .

   227 \]

   227 \]

   228 It suffices to show that the above maps are compatible with the relations whereby

   228 It suffices to show that the above maps are compatible with the relations whereby

   229 $SC^n_{\ol{M}\ol{N}}$ is constructed from the various $P$'s.

   229 $SC^n_{\ol{M}\ol{N}}$ is constructed from the various $P$'s.

   230 This in turn follows easily from the fact that

   230 This in turn follows easily from the fact that

   232 %\nn{should add some detail to above}

   233 %\nn{should add some detail to above}

   233 \end{proof}

   234 \end{proof}

   234 

   235 

   235 We note that even when $n=1$, the above theorem goes beyond an action of the little disks operad.

   236 We note that even when $n=1$, the above theorem goes beyond an action of the little disks operad.

   236 $M_i$ could be a disjoint union of intervals, and $N_i$ could connect the end points of the intervals

   237 $M_i$ could be a disjoint union of intervals, and $N_i$ could connect the end points of the intervals