# HG changeset patch
# User scott@6e1638ff-ae45-0410-89bd-df963105f760
# Date 1256879022 0
# Node ID 2807257be38201ce99ee3efb784f506cc15b774b
# Parent  db91d0a8ed75aba5824ca0a481af550aa3727366
...

diff -r db91d0a8ed75 -r 2807257be382 blob1.tex
--- a/blob1.tex	Fri Oct 30 04:05:33 2009 +0000
+++ b/blob1.tex	Fri Oct 30 05:03:42 2009 +0000
@@ -71,15 +71,12 @@
 \item medium priority
 \begin{itemize}
 \item $n=2$ examples
-\item dimension $n+1$ (generalized Deligne conjecture?)
 \item should be clear about PL vs Diff; probably PL is better
 (or maybe not)
-\item shuffle product vs gluing product (?)
 \item connection between $A_\infty$ operad and topological $A_\infty$ cat defs
 \end{itemize}
 \item lower priority
 \begin{itemize}
-\item Derive Hochschild standard results from blob point of view?
 \item Kh
 \item Mention somewhere \cite{MR1624157} ``Skein homology''; it's not directly related, but has similar motivations.
 \end{itemize}
@@ -105,6 +102,7 @@
 
 \input{text/comm_alg}
 
+\input{text/deligne}
 
 \appendix
 
diff -r db91d0a8ed75 -r 2807257be382 text/intro.tex
--- a/text/intro.tex	Fri Oct 30 04:05:33 2009 +0000
+++ b/text/intro.tex	Fri Oct 30 05:03:42 2009 +0000
@@ -33,9 +33,6 @@
 \begin{itemize}
 \item explain relation between old and new blob complex definitions
 \item overview of sections
-\item ?? we have resisted the temptation 
-(actually, it was not a temptation) to state things in the greatest
-generality possible
 \item related: we are being unsophisticated from a homotopy theory point of
 view and using chain complexes in many places where we could be by with spaces
 \item ? one of the points we make (far) below is that there is not really much
@@ -250,25 +247,7 @@
 \begin{property}[Higher dimensional Deligne conjecture]
 The singular chains of the $n$-dimensional fat graph operad act on blob cochains.
 \end{property}
-\begin{rem}
-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 $\bc^*(A_i, B_i)$.
-\end{rem}
-
-
-
-
-
+See \S \ref{sec:deligne} for an explanation of the terms appearing here, and (in a later edition of this paper) the proof.
 
 
 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in
@@ -276,4 +255,14 @@
 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.
 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation},
 and Property \ref{property:gluing} in \S \ref{sec:gluing}.
-\nn{need to say where the remaining properties are proved.}
\ No newline at end of file
+\nn{need to say where the remaining properties are proved.}
+
+\subsection{Future directions}
+Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds.
+
+Many results in Hochschild homology can be understood `topologically' via the blob complex. For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ simply corresponds to the gluing operation on $\bc_*(S^1 \times I, A)$, but haven't investigated the details.
+
+Most importantly, however, \nn{applications!} \nn{$n=2$ cases, contact, Kh}
+
+
+\subsection{Thanks and acknowledgements}