
diff -r 7b4a57110e83 -r 9a60488cd2fc pnas/pnas.tex
--- a/pnas/pnas.tex	Tue Oct 26 23:56:33 2010 -0700
+++ b/pnas/pnas.tex	Wed Oct 27 00:08:11 2010 -0700
@@ -597,10 +597,22 @@
 \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 $n$-dimensional surgery cylinder operad,
+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}
-\nn{Explain and sketch}
+
+An $n$-dimensional surgery cylinder is an alternating sequence of mapping cylinders and surgeries, modulo changing the order of distant surgeries, and conjugating the submanifold not modified in a surgery by a homeomorphism. See Figure \ref{delfig2}. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another.
+
+\todo{Explain blob cochains}
+\todo{Sketch proof}
+
+The little disks operad $LD$ is homotopy equivalent to 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 cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923} gives a map
+\[
+	C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}}
+			\to  Hoch^*(C, C),
+\]
+which we now see to be a specialization of Theorem \ref{thm:deligne}.
+
 
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@@ -679,6 +691,11 @@
 \label{partofJfig}
 \end{figure}
 
+\begin{figure}
+$$\mathfig{.4}{deligne/manifolds}$$
+\caption{An $n$-dimensional surgery cylinder}\label{delfig2}
+\end{figure}
+
 
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