
diff -r 75f5c197a0d4 -r cfab8c2189a7 text/deligne.tex
--- a/text/deligne.tex	Sun Nov 01 20:29:41 2009 +0000
+++ b/text/deligne.tex	Sun Nov 01 21:40:23 2009 +0000
@@ -55,38 +55,41 @@
 involved were 1-dimensional.
 Thus we can define a $n$-dimensional fat graph to sequence of general surgeries
 on an $n$-manifold.
-More specifically, \nn{...}
+More specifically,
+the $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries
+$R_i \cup M_i \leadsto R_i \cup N_i$ together with mapping cylinders of diffeomorphisms
+$f_i: R_i\cup N_i \to R_{i+1}\cup M_{i+1}$.
+(See Figure \ref{delfig2}.)
+\begin{figure}[!ht]
+$$\mathfig{.9}{tempkw/delfig2}$$
+\caption{A fat graph}\label{delfig2}\end{figure}
+The components of the $n$-dimensional fat graph operad are indexed by tuples
+$(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$.
+Note that the suboperad where $M_i$, $N_i$ and $R_i\cup M_i$ are all diffeomorphic to 
+the $n$-ball is equivalent to the little $n{+}1$-disks operad.
+
+
+If $M$ and $N$ are $n$-manifolds sharing the same boundary, we define
+the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be
+$A_\infty$ maps from $\bc_*(M)$ to $\bc_*(N)$, where we think of both
+collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$.
+The ``holes" in the above 
+$n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$.
+\nn{need to make up my mind which notation I'm using for the module maps}
+
+Putting this together we get a collection of maps
+\begin{eqnarray*}
+	C_*(FG^n_{\overline{M}, \overline{N}})\otimes \mapinf(\bc_*(M_0), \bc_*(N_0))\otimes\cdots\otimes 
+\mapinf(\bc_*(M_{k-1}), \bc_*(N_{k-1})) & \\
+	& \hspace{-11em}\to  \mapinf(\bc_*(M_k), \bc_*(N_k))
+\end{eqnarray*}
+which satisfy an operad type compatibility condition.
+
+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
 
-
-Figure \ref{delfig2}
-\begin{figure}[!ht]
-$$\mathfig{.9}{tempkw/delfig2}$$
-\caption{A fat graph}\label{delfig2}\end{figure}
-
-
-\begin{eqnarray*}
-	C_*(FG^n_{\overline{M}, \overline{N}})\otimes \mapinf(\bc_*(M_0), \bc_*(N_0))\otimes\cdots\otimes 
-\mapinf(\bc_*(M_{k-1}), \bc_*(N_{k-1})) & \\
-	& \hspace{-5em}\to  \mapinf(\bc_*(M_k), \bc_*(N_k))
-\end{eqnarray*}
-
-\medskip
-\hrule\medskip
-
-The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries
-of $n$-manifolds
-$R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms
-$f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$.
-(Note that the suboperad where $A_i$, $B_i$ and $R_i\cup A_i$ are all diffeomorphic to 
-the $n$-ball is equivalent to the little $n{+}1$-disks operad.)
-
-If $A$ and $B$ are $n$-manifolds sharing the same boundary, we define
-the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be
-$A_\infty$ maps from $\bc_*(A)$ to $\bc_*(B)$, where we think of both
-collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$.
-The ``holes" in the above 
-$n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$.




