# HG changeset patch
# User kevin@6e1638ff-ae45-0410-89bd-df963105f760
# Date 1257101500 0
# Node ID 0993acb4f314326db0a06976217deb477adc6403
# Parent  cb70a71710a5201e4b369935010fb8b715e10313
...

diff -r cb70a71710a5 -r 0993acb4f314 diagrams/pdf/tempkw/delfig1.pdf
Binary file diagrams/pdf/tempkw/delfig1.pdf has changed
diff -r cb70a71710a5 -r 0993acb4f314 diagrams/pdf/tempkw/delfig2.pdf
Binary file diagrams/pdf/tempkw/delfig2.pdf has changed
diff -r cb70a71710a5 -r 0993acb4f314 text/comm_alg.tex
--- a/text/comm_alg.tex	Sun Nov 01 17:02:10 2009 +0000
+++ b/text/comm_alg.tex	Sun Nov 01 18:51:40 2009 +0000
@@ -12,12 +12,16 @@
 (Thomas Tradler's idea.)
 Should prove (or at least conjecture) that here.}
 
+\nn{also, this section needs a little updating to be compatible with the rest of the paper.}
+
 If $C$ is a commutative algebra it
-can (and will) also be thought of as an $n$-category whose $j$-morphisms are trivial for
+can also be thought of as an $n$-category whose $j$-morphisms are trivial for
 $j<n$ and whose $n$-morphisms are $C$. 
 The goal of this \nn{subsection?} is to compute
 $\bc_*(M^n, C)$ for various commutative algebras $C$.
 
+\medskip
+
 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$.
 
 Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$
@@ -45,6 +49,7 @@
 
 \begin{proof}
 \nn{easy, but should probably write the details eventually}
+\nn{this is just the standard ``method of acyclic models" set up, so we should just give a reference for that}
 \end{proof}
 
 Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$
@@ -161,9 +166,10 @@
 
 \medskip
 
-Next we consider the case $C = k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$.
-Define $\Delta_l \sub \Sigma^\infty(M)$ to be those configurations of points with $l$ or
-more points coinciding.
+Next we consider the case $C$ is the truncated polynomial
+algebra $k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$.
+Define $\Delta_l \sub \Sigma^\infty(M)$ to be configurations of points in $M$ with $l$ or
+more of the points coinciding.
 
 \begin{prop}
 $\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$
@@ -174,5 +180,14 @@
 \nn{...}
 \end{proof}
 
-\nn{...}
+\medskip
+\hrule
+\medskip
 
+Still to do:
+\begin{itemize}
+\item compare the topological computation for truncated polynomial algebra with [Loday]
+\item multivariable truncated polynomial algebras (at least mention them)
+\item ideally, say something more about higher hochschild homology (maybe sketch idea for proof of equivalence)
+\end{itemize}
+
diff -r cb70a71710a5 -r 0993acb4f314 text/deligne.tex
--- a/text/deligne.tex	Sun Nov 01 17:02:10 2009 +0000
+++ b/text/deligne.tex	Sun Nov 01 18:51:40 2009 +0000
@@ -7,6 +7,76 @@
 The singular chains of the $n$-dimensional fat graph operad act on blob cochains.
 \end{prop}
 
+We will give a more precise statement of the proposition below.
+
+\nn{for now, we just sketch the proof.}
+
+\def\mapinf{\Maps_\infty}
+
+The usual Deligne conjecture \nn{need refs} gives a map
+\[
+	C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}}
+			\to  Hoch^*(C, C) .
+\]
+Here $LD_k$ is the $k$-th space of the little disks operad, and $Hoch^*(C, C)$ denotes Hochschild
+cochains.
+The little disks operad is homotopy equivalent to the fat graph operad
+\nn{need ref; and need to restrict which fat graphs}, and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms
+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*}
+See Figure \ref{delfig1}.
+\begin{figure}[!ht]
+$$\mathfig{.9}{tempkw/delfig1}$$
+\caption{A fat graph}\label{delfig1}\end{figure}
+
+We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval
+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
+$\mapinf(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$.
+So for each fixed fat graph we have a map
+\[
+	 \mapinf(\bc^C_*(I), \bc^C_*(I))\otimes\cdots
+	\otimes \mapinf(\bc^C_*(I), \bc^C_*(I))  \to  \mapinf(\bc^C_*(I), \bc^C_*(I)) .
+\]
+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.
+Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies.
+
+It should now be clear how to generalize this to higher dimensions.
+In the sequence-of-surgeries description above, we never used the fact that the manifolds
+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{...}
+
+
+\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
diff -r cb70a71710a5 -r 0993acb4f314 text/intro.tex
--- a/text/intro.tex	Sun Nov 01 17:02:10 2009 +0000
+++ b/text/intro.tex	Sun Nov 01 18:51:40 2009 +0000
@@ -275,6 +275,7 @@
 \end{property}
 
 \begin{property}[Higher dimensional Deligne conjecture]
+\label{property:deligne}
 The singular chains of the $n$-dimensional fat graph operad act on blob cochains.
 \end{property}
 See \S \ref{sec:deligne} for an explanation of the terms appearing here, and (in a later edition of this paper) the proof.