# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1275006956 25200
# Node ID 418919afd07731ce005f8df0f08d1bc52e8c5848
# Parent  79f7b1bd7b1ac68101c0091e42bc0fa9365b9d83
small preliminary changes to Deligne section

diff -r 79f7b1bd7b1a -r 418919afd077 text/deligne.tex
--- a/text/deligne.tex	Tue May 25 16:50:55 2010 -0700
+++ b/text/deligne.tex	Thu May 27 17:35:56 2010 -0700
@@ -11,7 +11,7 @@
 
 We will state this more precisely below as Proposition \ref{prop:deligne}, and just sketch a proof. First, we recall the usual Deligne conjecture, explain how to think of it as a statement about blob complexes, and begin to generalize it.
 
-\def\mapinf{\Maps_\infty}
+%\def\mapinf{\Maps_\infty}
 
 The usual Deligne conjecture \nn{need refs} gives a map
 \[
@@ -25,11 +25,11 @@
 of the blob complex of the interval.
 \nn{need to make sure we prove this above}.
 So the 1-dimensional Deligne conjecture can be restated as
-\begin{eqnarray*}
-	C_*(FG_k)\otimes \mapinf(\bc^C_*(I), \bc^C_*(I))\otimes\cdots
-	\otimes \mapinf(\bc^C_*(I), \bc^C_*(I)) & \\
-	  & \hspace{-5em} \to  \mapinf(\bc^C_*(I), \bc^C_*(I)) .
-\end{eqnarray*}
+\[
+	C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots
+	\otimes \hom(\bc^C_*(I), \bc^C_*(I))
+	  \to  \hom(\bc^C_*(I), \bc^C_*(I)) .
+\]
 See Figure \ref{delfig1}.
 \begin{figure}[!ht]
 $$\mathfig{.9}{deligne/intervals}$$
@@ -39,12 +39,12 @@
 of Figure \ref{delfig1} and ending at the topmost interval.
 The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph.
 We remove the bottom interval of the bigon and replace it with the top interval.
-To map this topological operation to an algebraic one, we need, for each hole, element of
-$\mapinf(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$.
+To map this topological operation to an algebraic one, we need, for each hole, an element of
+$\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$.
 So for each fixed fat graph we have a map
 \[
-	 \mapinf(\bc^C_*(I), \bc^C_*(I))\otimes\cdots
-	\otimes \mapinf(\bc^C_*(I), \bc^C_*(I))  \to  \mapinf(\bc^C_*(I), \bc^C_*(I)) .
+	 \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots
+	\otimes \hom(\bc^C_*(I), \bc^C_*(I))  \to  \hom(\bc^C_*(I), \bc^C_*(I)) .
 \]
 If we deform the fat graph, corresponding to a 1-chain in $C_*(FG_k)$, we get a homotopy
 between the maps associated to the endpoints of the 1-chain.
@@ -65,8 +65,10 @@
 \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 
+\nn{not quite true: this is coarser than components}
+Note that the suboperad where $M_i$, $N_i$ and $R_i\cup M_i$ are all homeomorphic to 
 the $n$-ball is equivalent to the little $n{+}1$-disks operad.
+\nn{what about rotating in the horizontal directions?}
 
 
 If $M$ and $N$ are $n$-manifolds sharing the same boundary, we define
@@ -82,9 +84,9 @@
 \label{prop:deligne}
 There is a collection of maps
 \begin{eqnarray*}
-	C_*(FG^n_{\overline{M}, \overline{N}})\otimes \mapinf(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes 
-\mapinf(\bc_*(M_{k}), \bc_*(N_{k})) & \\
-	& \hspace{-11em}\to  \mapinf(\bc_*(M_0), \bc_*(N_0))
+	C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes 
+\hom(\bc_*(M_{k}), \bc_*(N_{k})) & \\
+	& \hspace{-11em}\to  \hom(\bc_*(M_0), \bc_*(N_0))
 \end{eqnarray*}
 which satisfy an operad type compatibility condition. \nn{spell this out}
 \end{prop}
diff -r 79f7b1bd7b1a -r 418919afd077 text/intro.tex
--- a/text/intro.tex	Tue May 25 16:50:55 2010 -0700
+++ b/text/intro.tex	Thu May 27 17:35:56 2010 -0700
@@ -257,6 +257,7 @@
 \end{property}
 
 Finally, we state two more properties, which we will not prove in this paper.
+\nn{revise this; expect that we will prove these in the paper}
 
 \begin{property}[Mapping spaces]
 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps