--- a/pnas/pnas.tex Sun Nov 14 19:43:47 2010 -0800
+++ b/pnas/pnas.tex Mon Nov 15 09:49:04 2010 -0800
@@ -219,6 +219,9 @@
\nn{In many places we omit details; they can be found in MW.
(Blanket statement in order to avoid too many citations to MW.)}
+\nn{perhaps say something explicit about the relationship of this paper to big blob paper.
+like: in this paper we try to give a clear view of the big picture without getting bogged down in details}
+
\section{Definitions}
\subsection{$n$-categories} \mbox{}
@@ -621,7 +624,7 @@
\xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)}
\end{equation*}
\end{property}
-\nn{maybe should say something about the $A_\infty$ case}
+%\nn{maybe should say something about the $A_\infty$ case}
\begin{proof}(Sketch)
For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram
@@ -630,6 +633,9 @@
$x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$.
\end{proof}
+If $\cC$ is an $A-\infty$ $n$-category then $\bc_*(B^n;\cC)$ is still homotopy equivalent to $\cC(B^n)$,
+but this is no longer concentrated in degree zero.
+
\subsection{Specializations}
\label{sec:specializations}
@@ -808,18 +814,47 @@
\section{Deligne conjecture for $n$-categories}
\label{sec:applications}
+Let $M$ and $N$ be $n$-manifolds with common boundary $E$.
+Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$
+acts on $\bc_*(M)$ and $\bc_*(N)$.
+Let $\hom_A(\bc_*(M), \bc_*(N))$ denote the chain complex of $A_\infty$ module maps
+from $\bc_*(M)$ to $\bc_*(N)$.
+Let $R$ be another $n$-manifold with boundary $-E$.
+There is a chain map
+\[
+ \hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) .
+\]
+We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and
+replaces it with $N$, yielding $N\cup_E R$.
+(This is a more general notion of surgery that usual --- $M$ and $N$ can be any manifolds
+which share a common boundary.)
+In analogy to Hochschild cochains, we will call elements of $\hom_A(\bc_*(M), \bc_*(N))$ ``blob cochains".
+
+Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the
+blob complex.
+An $n$-dimensional surgery cylinder is
+defined to be a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}),
+modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism.
+One can associated to this data an $(n{+}1)$-manifold with a foliation by intervals,
+and the relations we impose correspond to homeomorphisms of the $(n{+}1)$-manifolds
+which preserve the foliation.
+
+Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another.
+
\begin{thm}[Higher dimensional Deligne conjecture]
\label{thm:deligne}
The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains.
-Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad,
-this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball.
\end{thm}
-An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}),
-modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism.
-Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another.
-
-By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module.
+More specifically, let $M_0, N_0, \ldots, M_k, N_k$ be $n$-manifolds and let $SC^n_{\overline{M}, \overline{N}}$
+denote the component of the operad with outer boundary $M_0\cup N_0$ and inner boundaries
+$M_1\cup N_1,\ldots, M_k\cup N_k$.
+Then there is a collection of chain maps
+\begin{multline*}
+ C_*(SC^n_{\overline{M}, \overline{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))
+\end{multline*}
+which satisfy the operad compatibility conditions.
\begin{proof}
We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation},
@@ -828,15 +863,22 @@
This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity.
\end{proof}
-The little disks operad $LD$ is homotopy equivalent to
-\nn{suboperad of}
-the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cochains $Hoch^*(C, C)$.
-The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map
-\[
- C_*(LD_k)\tensor \overbrace{Hoch^*(C, C)\tensor\cdots\tensor Hoch^*(C, C)}^{\text{$k$ copies}}
- \to Hoch^*(C, C),
-\]
-which we now see to be a specialization of Theorem \ref{thm:deligne}.
+Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are 1-balls.
+We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little
+disks operad and $\hom(\bc_*(M_i), \bc_*(N_i))$ is homotopy equivalent to Hochschild cochains.
+This special case is just the usual Deligne conjecture
+(see \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}
+\nn{should check that this is the optimal list of references; what about Gerstenhaber-Voronov?;
+if we revise this list, should propagate change back to main paper}
+).
+
+The general case when $n=1$ goes beyond the original Deligne conjecture, as the $M_i$'s and $N_i$'s
+could be disjoint unions of 1-balls and circles, and the surgery cylinders could be high genus surfaces.
+
+If all of the $M_i$'s and $N_i$'s are $n$-balls, then $SC^n_{\overline{M}, \overline{N}}$
+contains a copy of the little $(n{+}1)$-balls operad.
+Thus the little $(n{+}1)$-balls operad acts on blob cochains of the $n$-ball.
+
%% == end of paper:
@@ -858,7 +900,7 @@
\begin{acknowledgments}
It is a pleasure to acknowledge helpful conversations with
Kevin Costello,
-Mike Freedman,
+Michael Freedman,
Justin Roberts,
and
Peter Teichner.
@@ -952,7 +994,8 @@
\end{figure}
\begin{figure}
-$$\mathfig{.4}{deligne/manifolds}$$
+%$$\mathfig{.4}{deligne/manifolds}$$
+$$\mathfig{.4}{deligne/mapping-cylinders}$$
\caption{An $n$-dimensional surgery cylinder.}\label{delfig2}
\end{figure}