%!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 ``Upper" $n$-manifolds $M_0,\ldots,M_k$ and ``lower" $n$-manifolds $N_0,\ldots,N_k$,

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

We call $M_0$ and $N_0$ the outer boundary and the remaining $M_i$'s and $N_i$'s the inner

boundaries.

\item Additional manifolds $R_1,\ldots,R_{k}$, with $\bd R_i = E_0\cup \bd M_i = E_0\cup \bd N_i$.

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

\item Homeomorphisms 

\begin{eqnarray*}

	f_0: M_0 &\to& R_1\cup M_1 \\

	f_i: R_i\cup N_i &\to& R_{i+1}\cup M_{i+1}\;\; \mbox{for}\, 1\le i \le k-1 \\

	f_k: R_k\cup N_k &\to& N_0 .

\end{eqnarray*}

Each $f_i$ should be the identity restricted to $E_0$.

\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).

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

\[

	(\ldots, f_{i-1}, f_i, \ldots) \to (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots),

\]

leaving the $M_i$, $N_i$ and $R_i$ fixed.

(See Figure xxxx.)

\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.

There is an operad structure on $n$-dimensional fat graphs, given by gluing the outer boundary

of one graph into one of the inner boundaries of another graph.

We leave it to the reader to work out a more precise statement in terms of $M_i$'s, $f_i$'s etc.

For fixed $\ol{M} = (M_0,\ldots,M_k)$ and $\ol{N} = (N_0,\ldots,N_k)$, we let

$FG^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional fat graphs as above.

The topology comes from the spaces

\[

	\Homeo(M_0\to R_1\cup M_1)\times \Homeo(R_1\cup N_1\to R_2\cup M_2)\times

			\cdots\times \Homeo(R_k\cup N_k\to N_0)

\]

and the above equivalence relations.

We will denote the typical element of $FG^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$.

\medskip

Let $\ol{f} \in FG^n_{\ol{M}\ol{N}}$.

Let $\hom(\bc_*(M_i), \bc_*(N_i))$ denote the morphisms from $\bc_*(M_i)$ to $\bc_*(N_i)$,

as modules of the $A_\infty$ 1-category $\bc_*(E_i)$.

We define a map

\[

	p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k))

				\to \hom(\bc_*(M_0), \bc_*(N_0)) .

\]

Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define $p(\ol{f}$ to be the composition

\[

	\bc_*(M_0)  \stackrel{f_0}{\to} \bc_*(R_1\cup M_1)

				 \stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1)

				 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \to

				 \cdots  \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k)

				 \stackrel{f_k}{\to} \bc_*(N_0)

\]

(Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.)

It is easy to check that the above definition is compatible with the equivalence relations

and also the operad structure.

\nn{little m-disks operad; }

\nn{*** resume revising here}

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

\hrule\medskip