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\section{Higher-dimensional Deligne conjecture}
\label{sec:deligne}
In this section we discuss Property \ref{property:deligne},
\begin{prop}[Higher dimensional Deligne conjecture]
The singular chains of the $n$-dimensional fat graph operad act on blob cochains.
\end{prop}

The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries
of $n$-manifolds
$R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms
$f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$.
(Note that the suboperad where $A_i$, $B_i$ and $R_i\cup A_i$ are all diffeomorphic to 
the $n$-ball is equivalent to the little $n{+}1$-disks operad.)

If $A$ and $B$ are $n$-manifolds sharing the same boundary, we define
the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be
$A_\infty$ maps from $\bc_*(A)$ to $\bc_*(B)$, where we think of both
collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$.
The ``holes" in the above 
$n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$.