%!TEX root = ../blob1.tex

\section{Higher-dimensional Deligne conjecture}

\label{sec:deligne}

The usual Deligne conjecture \nn{need refs} gives a map

\[

	C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}}

			\to  Hoch^*(C, C) .

\]

Here $LD_k$ is the $k$-th space of the little disks operad, and $Hoch^*(C, C)$ denotes Hochschild

cochains.

\nn{need to make sure we prove this above}.

So the 1-dimensional Deligne conjecture can be restated as

\[

	C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots

	\otimes \hom(\bc^C_*(I), \bc^C_*(I))

	  \to  \hom(\bc^C_*(I), \bc^C_*(I)) .

\]

See Figure \ref{delfig1}.

\begin{figure}[!ht]

$$\mathfig{.9}{deligne/intervals}$$

\caption{A fat graph}\label{delfig1}\end{figure}

We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval

of Figure \ref{delfig1} and ending at the topmost interval.

The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph.

We remove the bottom interval of the bigon and replace it with the top interval.

$\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$.

So for each fixed fat graph we have a map

\[

	 \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots

	\otimes \hom(\bc^C_*(I), \bc^C_*(I))  \to  \hom(\bc^C_*(I), \bc^C_*(I)) .

\]

If we deform the fat graph, corresponding to a 1-chain in $C_*(FG_k)$, we get a homotopy

between the maps associated to the endpoints of the 1-chain.

Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies.

It should now be clear how to generalize this to higher dimensions.

In the sequence-of-surgeries description above, we never used the fact that the manifolds

involved were 1-dimensional.

on an $n$-manifold.

More specifically,

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

$R_i \cup M_i \leadsto R_i \cup N_i$ together with mapping cylinders of diffeomorphisms

$f_i: R_i\cup N_i \to R_{i+1}\cup M_{i+1}$.

(See Figure \ref{delfig2}.)

\begin{figure}[!ht]

$$\mathfig{.9}{deligne/manifolds}$$

The components of the $n$-dimensional fat graph operad are indexed by tuples

$(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$.

\nn{not quite true: this is coarser than components}

Note that the suboperad where $M_i$, $N_i$ and $R_i\cup M_i$ are all homeomorphic to 

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

\nn{what about rotating in the horizontal directions?}

If $M$ and $N$ 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 $\bc_*(M)$ to $\bc_*(N)$, 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 $\bc^*(A_i, B_i)$.

\nn{need to make up my mind which notation I'm using for the module maps}

Putting this together we get 

\begin{prop}(Precise statement of Property \ref{property:deligne})

\label{prop:deligne}

There is a collection of maps

\begin{eqnarray*}

	C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes 

\hom(\bc_*(M_{k}), \bc_*(N_{k})) & \\

	& \hspace{-11em}\to  \hom(\bc_*(M_0), \bc_*(N_0))

\end{eqnarray*}

which satisfy an operad type compatibility condition. \nn{spell this out}

\end{prop}

Note that if $k=0$ then this is just the action of chains of diffeomorphisms from Section \ref{sec:evaluation}.

And indeed, the proof is very similar \nn{...}

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