%!TEX root = ../blob1.tex
\section{Higher-dimensional Deligne conjecture}
\label{sec:deligne}
In this section we prove a higher dimensional version of the Deligne conjecture
about the action of the little disks operad on Hochschild cochains.
The first several paragraphs lead up to a precise statement of the result
(Theorem \ref{thm:deligne} below).
Then we give the proof.
%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, MR1328534, 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.
We now reinterpret $C_*(LD_k)$ and $Hoch^*(C, C)$ in such a way as to make the generalization to
higher dimensions clear.
The little disks operad is homotopy equivalent to configurations of little bigons inside a big bigon,
as shown in Figure \ref{delfig1}.
We can think of such a configuration as encoding a sequence of surgeries, starting at the bottommost interval
of Figure \ref{delfig1} and ending at the topmost interval.
\begin{figure}[t]
$$\mathfig{.9}{deligne/intervals}$$
\caption{Little bigons, thought of as encoding surgeries}\label{delfig1}\end{figure}
The surgeries correspond to the $k$ bigon-shaped ``holes".
We remove the bottom interval of each little 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
$\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$, which is homotopy equivalent to $Hoch^*(C, C)$.
So for each fixed configuration 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 configuration, corresponding to a 1-chain in $C_*(LD_k)$, we get a homotopy
between the maps associated to the endpoints of the 1-chain.
Similarly, higher-dimensional chains in $C_*(LD_k)$ give rise to higher homotopies.
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
\S\ref{ss:module-morphisms}.
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.
So we will define, below, the operad of $n$-dimensional surgery cylinders, analogous to mapping
cylinders of homeomorphisms (Figure \ref{delfig2}).
\begin{figure}[t]
$$\mathfig{.9}{deligne/manifolds}$$
\caption{An $n$-dimensional surgery cylinder}\label{delfig2}
\end{figure}
(Note that $n$ is the dimension of the manifolds we are doing surgery on; the surgery cylinders
are $n{+}1$-dimensional.)
An $n$-dimensional surgery cylinder ($n$-SC for short) consists of:
\begin{itemize}
\item ``Lower" $n$-manifolds $M_0,\ldots,M_k$ and ``upper" $n$-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 \ref{xdfig2}).
\begin{figure}[t]
$$\mathfig{.9}{deligne/mapping-cylinders}$$
\caption{An $n$-dimensional surgery cylinder constructed from mapping cylinders}\label{xdfig2}
\end{figure}
%The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part.
We regard two such surgery cylinders 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 which restricts to the identity on
$\bd R_i = \bd R'_i = E_0\cup \bd M_i$, we can replace
\begin{eqnarray*}
(\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& (\ldots, R_{i-1}, R'_i, R_{i+1}, \ldots) \\
(\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots),
\end{eqnarray*}
leaving the $M_i$ and $N_i$ fixed.
(Keep in mind the case $R'_i = R_i$.)
(See Figure \ref{xdfig3}.)
\begin{figure}[t]
$$\mathfig{.4}{deligne/dfig3a} \to \mathfig{.4}{deligne/dfig3b} $$
\caption{Conjugating by a homeomorphism.}
\label{xdfig3}
\end{figure}
\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 \ref{xdfig1}.)
\begin{figure}[t]
$$\mathfig{.3}{deligne/dfig1a} \leftarrow \mathfig{.3}{deligne/dfig1b} \rightarrow \mathfig{.3}{deligne/dfig1c}$$
\caption{Changing the order of a surgery.}\label{xdfig1}
\end{figure}
\end{itemize}
Note that the second equivalence increases the number of holes (or arity) by 1.
We can make a similar identification with the roles 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 a colored operad structure on $n$-dimensional surgery cylinders, given by gluing the outer boundary
of one cylinder into one of the inner boundaries of another cylinder.
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
$SC^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional surgery cylinders as above.
(Note that in different parts of $SC^n_{\ol{M}\ol{N}}$ the $M_i$'s and $N_i$'s
are ordered differently.)
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 $SC^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$.
\medskip
%The little $n{+}1$-balls operad injects into the $n$-SC operad.
The $n$-SC operad contains the little $n{+}1$-balls operad.
Roughly speaking, given a configuration of $k$ little $n{+}1$-balls in the standard
$n{+}1$-ball, we fiber the complement of the balls by vertical intervals
and let $M_i$ [$N_i$] be the southern [northern] hemisphere of the $i$-th ball.
More precisely, let $x_1,\ldots,x_{n+1}$ be the coordinates of $\r^{n+1}$.
Let $z$ be a point of the $k$-th space of the little $n{+}1$-balls operad, with
little balls $D_1,\ldots,D_k$ inside the standard $n{+}1$-ball.
We assume the $D_i$'s are ordered according to the $x_{n+1}$ coordinate of their centers.
Let $\pi:\r^{n+1}\to \r^n$ be the projection corresponding to $x_{n+1}$.
Let $B\sub\r^n$ be the standard $n$-ball.
Let $M_i$ and $N_i$ be $B$ for all $i$.
Identify $\pi(D_i)$ with $B$ (a.k.a.\ $M_i$ or $N_i$) via translations and dilations (no rotations).
Let $R_i = B\setmin \pi(D_i)$.
Let $f_i = \rm{id}$ for all $i$.
We have now defined a map from the little $n{+}1$-balls operad to the $n$-SC operad,
with contractible fibers.
(The fibers correspond to moving the $D_i$'s in the $x_{n+1}$
direction while keeping them disjoint.)
%\nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s.
%does this need more explanation?}
Another familiar subspace of the $n$-SC operad is $\Homeo(M_0\to N_0)$, which corresponds to
case $k=0$ (no holes).
In this case the surgery cylinder is just a single mapping cylinder.
\medskip
Let $\ol{f} \in SC^n_{\ol{M}\ol{N}}$.
As usual, fix a system of field $\cF$ and let $\bc_*$ denote the blob complex construction based on $\cF$.
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 $\bc_*(E_i)$ (see \S\ref{ss:module-morphisms}).
We will 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
$p(\ol{f})(\alpha_1\ot\cdots\ot\alpha_k)$ to be the composition
\[
\bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1)
\stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1)
\stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to}
\cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k)
\stackrel{f_k}{\to} \bc_*(N_0)
\]
(Recall that the maps $\id\ot\alpha_i$ were defined in \S\ref{ss:module-morphisms}.)
It is easy to check that the above definition is compatible with the equivalence relations
and also the operad structure.
We can reinterpret the above as a chain map
\[
p: C_0(SC^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k))
\to \hom(\bc_*(M_0), \bc_*(N_0)) .
\]
The main result of this section is that this chain map extends to the full singular
chain complex $C_*(SC^n_{\ol{M}\ol{N}})$.
\begin{thm}
\label{thm:deligne}
There is a collection of chain maps
\[
C_*(SC^n_{\ol{M}\ol{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes
\hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0))
\]
which satisfy the operad compatibility conditions, up to coherent homotopy.
On $C_0(SC^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above.
When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of \S\ref{sec:evaluation}.
\end{thm}
The ``up to coherent homotopy" in the statement is due to the fact that the isomorphisms of
\ref{lem:bc-btc} and \ref{thm:gluing} are only defined up to a contractible set of homotopies.
If, in analogy to Hochschild cochains, we define elements of $\hom(\bc_*(M), \bc_*(N))$
to be ``blob cochains", we can summarize the above proposition by saying that the $n$-SC operad acts on
blob cochains.
As noted above, the $n$-SC operad contains the little $n{+}1$-balls operad, so this constitutes
a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disks operad.
\begin{proof}
As described above, $SC^n_{\ol{M}\ol{N}}$ is equal to the disjoint
union of products of homeomorphism spaces, modulo some relations.
By Theorem \ref{thm:CH} and the Eilenberg-Zilber theorem, we have for each such product $P$
a chain map
\[
C_*(P)\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes
\hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) .
\]
It suffices to show that the above maps are compatible with the relations whereby
$SC^n_{\ol{M}\ol{N}}$ is constructed from the various $P$'s.
This in turn follows easily from the fact that
the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative
(up to coherent homotopy).
%\nn{should add some detail to above}
\end{proof}
We note that even when $n=1$, the above theorem goes beyond an action of the little disks operad.
$M_i$ could be a disjoint union of intervals, and $N_i$ could connect the end points of the intervals
in a different pattern from $M_i$.
The genus of the surface associated to the surgery cylinder could be greater than zero.