%!TEX root = ../blob1.tex

\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 $\bc^*(A_i, B_i)$.