%!TEX root = ../blob1.tex

\section{Higher-dimensional Deligne conjecture}

\label{sec:deligne}

In this section we 

sketch

\nn{revisit ``sketch" after proof is done} 

the proof of a higher dimensional version of the Deligne conjecture

about the action of the little disks operad on Hochschild cohomology.

The first several paragraphs lead up to a precise statement of the result

(Proposition \ref{prop:deligne} below).

Then we sketch the proof.

\nn{Does this generalisation encompass Kontsevich's proposed generalisation from \cite{MR1718044}, that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S}

%from http://www.ams.org/mathscinet-getitem?mr=1805894

%Different versions of the geometric counterpart of Deligne's conjecture have been proven by Tamarkin [``Formality of chain operad of small squares'', preprint, http://arXiv.org/abs/math.QA/9809164], the reviewer [in Confˇrence Moshˇ Flato 1999, Vol. II (Dijon), 307--331, Kluwer Acad. Publ., Dordrecht, 2000; MR1805923 (2002d:55009)], and J. E. McClure and J. H. Smith [``A solution of Deligne's conjecture'', preprint, http://arXiv.org/abs/math.QA/9910126] (see also a later simplified version [J. E. McClure and J. H. Smith, ``Multivariable cochain operations and little $n$-cubes'', preprint, http://arXiv.org/abs/math.QA/0106024]). The paper under review gives another proof of Deligne's conjecture, which, as the authors indicate, may be generalized to a proof of a higher-dimensional generalization of Deligne's conjecture, suggested in [M. Kontsevich, Lett. Math. Phys. 48 (1999), no. 1, 35--72; MR1718044 (2000j:53119)]. 

The usual Deligne conjecture (proved variously in \cite{MR1805894, MR2064592, hep-th/9403055, MR1805923} 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.

The little disks operad is homotopy equivalent to the 

(transversely orient) fat graph operad

\nn{need ref, or say more precisely what we mean}, 

and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms

of the blob complex of the interval, thought of as a bimodule for itself.

\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 emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module

for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the 

morphisms of such modules as defined in 

Subsection \ref{ss:module-morphisms}.

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.

To convert this topological operation to an algebraic one, we need, for each hole, an element of

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

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.

Thus we can define an $n$-dimensional fat graph to be a sequence of general surgeries

on an $n$-manifold (Figure \ref{delfig2}).

\begin{figure}[!ht]

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

\caption{An  $n$-dimensional fat graph}\label{delfig2}

\end{figure}

More specifically, an $n$-dimensional fat graph consists of:

\begin{itemize}

\item ``Incoming" $n$-manifolds $M_1,\ldots,M_k$ and ``outgoing" $n$-manifolds $N_1,\ldots,N_k$,

with $\bd M_i = \bd N_i$ for all $i$.

\item An ``outer boundary" $n{-}1$-manifold $E$.

\item Additional manifolds $R_0,\ldots,R_{k+1}$, with $\bd R_i = E\cup \bd M_i = E\cup \bd N_i$.

(By convention, $M_i = N_i = \emptyset$ if $i <1$ or $i>k$.)

We call $R_0$ the outer incoming manifold and $R_{k+1}$ the outer outgoing manifold

\item Homeomorphisms $f_i : R_i\cup N_i\to R_{i+1}\cup M_{i+1}$, $0\le i \le k$.

\end{itemize}

We can think of the above data as encoding the union of the mapping cylinders $C(f_0),\ldots,C(f_k)$,

with $C(f_i)$ glued to $C(f_{i+1})$ along $R_{i+1}$

(see Figure xxxx).

\nn{also need to revise outer labels of older fig}

The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part.

We regard two such fat graphs as the same if there is a homeomorphism between them which is the 

identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping

cylinders.

More specifically, we impose the following two equivalence relations:

\begin{itemize}

\item If $g:R_i\to R_i$ is a homeomorphism, we can replace

\[

\]

(See Figure xxx.)

\item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a

compatible disjoint union of $\bd M = \bd N$), we can replace

\begin{eqnarray*}

	(\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\

	(\ldots, N_{i-1}, N_i, N_{i+1}, \ldots) &\to& (\ldots, N_{i-1}, N'_i, N''_i, N_{i+1}, \ldots) \\

	(\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& 

						(\ldots, R_{i-1}, R_i\cup M''_i, R_i\cup N'_i, R_{i+1}, \ldots) \\

	(\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, f_{i-1}, \rm{id}, f_i, \ldots) .

\end{eqnarray*}

(See Figure xxxx.)

\end{itemize}

Note that the second equivalence increases the number of holes (or arity) by 1.

We can make a similar identification with the rolls of $M'_i$ and $M''_i$ reversed.

In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries

do not overlap, we can perform them in reverse order or simultaneously.

\nn{operad structure (need to ntro mroe terminology above}

\nn{*** resume revising here}

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{...}

\medskip

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