Binary file pnas/diagrams/deligne/mapping-cylinders.pdf has changed
--- a/pnas/pnas.tex Sun Nov 14 22:54:29 2010 -0800
+++ b/pnas/pnas.tex Sun Nov 14 23:13:40 2010 -0800
@@ -823,11 +823,20 @@
\]
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.)
-Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the
+Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the
blob complex.
-\nn{...}
+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.
+\nn{more to do...}
\begin{thm}[Higher dimensional Deligne conjecture]
\label{thm:deligne}
@@ -836,10 +845,6 @@
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.
\begin{proof}
@@ -976,7 +981,8 @@
\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}