# HG changeset patch
# User Scott Morrison <scott@tqft.net>
# Date 1288525313 -32400
# Node ID 99611dfed1f38104deacad83e411d3d8d51ec6bb
# Parent  0bd4aca0546b8247d1f7e6b5a21ed2b5a3e14e1f
k-blobs for small k, and blob cochains

diff -r 0bd4aca0546b -r 99611dfed1f3 pnas/pnas.tex
--- a/pnas/pnas.tex	Sun Oct 31 20:15:44 2010 +0900
+++ b/pnas/pnas.tex	Sun Oct 31 20:41:53 2010 +0900
@@ -381,16 +381,19 @@
 
 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of
 \begin{itemize}
-\item a permissible collection of $k$ embedded balls (called `blobs') in $W$,
+\item a permissible collection of $k$ embedded balls,
 \item an ordering of the balls, and
 \item for each resulting piece of $W$, a field,
 \end{itemize}
-such that for any innermost blob $B$, the field on $B$ goes to zero under the composition map from $\cC$.
+such that for any innermost blob $B$, the field on $B$ goes to zero under the composition map from $\cC$. We call such a field a `null field on $B$'.
 
 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with signs given by the ordering.
 
 \todo{Say why this really is the homotopy colimit}
-\todo{Spell out $k=0, 1, 2$}
+
+We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$. For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field that that ball is a null field. The differential simply forgets the ball. Thus we see that $H_0$ of the blob complex is the quotient of fields by null fields.
+
+For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. This ensures that the differential is well-defined.
 
 \section{Properties of the blob complex}
 \subsection{Formal properties}
@@ -590,7 +593,7 @@
 
 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. 
 Note that there is no restriction on the connectivity of $T$ as there is for the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}.
-
+\todo{sketch proof}
 
 \begin{thm}[Higher dimensional Deligne conjecture]
 \label{thm:deligne}
@@ -599,14 +602,13 @@
 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 an alternating sequence of mapping cylinders and surgeries, modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. See Figure \ref{delfig2}. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another.
-
+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.
 
-\todo{Explain blob cochains}
 \todo{Sketch proof}
 
-The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923} gives a map
+The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map
 \[
 	C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}}
 			\to  Hoch^*(C, C),