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208 which satisfy the operad compatibility conditions, up to coherent homotopy. |
208 which satisfy the operad compatibility conditions, up to coherent homotopy. |
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 |
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213 The ``up to coherent homotopy" in the statement is due to the fact that the isomorphisms of |
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214 \ref{lem:bc-btc} and \ref{thm:gluing} are only defined to up to a contractible set of homotopies. |
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215 |
213 If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ |
216 If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ |
214 to be ``blob cochains", we can summarize the above proposition by saying that the $n$-SC operad acts on |
217 to be ``blob cochains", we can summarize the above proposition by saying that the $n$-SC operad acts on |
215 blob cochains. |
218 blob cochains. |
216 As noted above, the $n$-SC operad contains the little $n{+}1$-balls operad, so this constitutes |
219 As noted above, the $n$-SC operad contains the little $n{+}1$-balls operad, so this constitutes |
217 a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disks operad. |
220 a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disks operad. |