1 %!TEX root = ../blob1.tex

     2 

     3 \section{Higher-dimensional Deligne conjecture}

     4 \label{sec:deligne}

     5 In this section we discuss Property \ref{property:deligne},

     6 \begin{prop}[Higher dimensional Deligne conjecture]

     7 The singular chains of the $n$-dimensional fat graph operad act on blob cochains.

     8 \end{prop}

     9 

    10 The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries

    11 of $n$-manifolds

    12 $R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms

    13 $f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$.

    14 (Note that the suboperad where $A_i$, $B_i$ and $R_i\cup A_i$ are all diffeomorphic to 

    15 the $n$-ball is equivalent to the little $n{+}1$-disks operad.)

    16 

    17 If $A$ and $B$ are $n$-manifolds sharing the same boundary, we define

    18 the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be

    19 $A_\infty$ maps from $\bc_*(A)$ to $\bc_*(B)$, where we think of both

    20 collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$.

    21 The ``holes" in the above 

    22 $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$.