--- a/pnas/pnas.tex Sat Nov 13 20:58:40 2010 -0800
+++ b/pnas/pnas.tex Sun Nov 14 15:39:03 2010 -0800
@@ -74,7 +74,6 @@
%\def\s{\sigma}
\input{preamble}
-\input{../text/kw_macros}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Don't type in anything in the following section:
@@ -374,7 +373,7 @@
Product morphisms are compatible with gluing.
Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$
be pinched products with $E = E_1\cup E_2$.
-Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$.
+Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\subset X$.
Then
\[
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) .
@@ -401,7 +400,7 @@
\end{axiom}
To state the next axiom we need the notion of {\it collar maps} on $k$-morphisms.
-Let $X$ be a $k$-ball and $Y\sub\bd X$ be a $(k{-}1)$-ball.
+Let $X$ be a $k$-ball and $Y\subset\bd X$ be a $(k{-}1)$-ball.
Let $J$ be a 1-ball.
Let $Y\times_p J$ denote $Y\times J$ pinched along $(\bd Y)\times J$.
A collar map is an instance of the composition
@@ -434,7 +433,7 @@
\label{axiom:families}
For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes
\[
- C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .
+ C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) .
\]
These action maps are required to be associative up to homotopy,
and also compatible with composition (gluing) in the sense that
@@ -464,7 +463,6 @@
Define product morphisms via product cell decompositions.
-\nn{also do bordism category}
\subsection{The blob complex}
\subsubsection{Decompositions of manifolds}
@@ -497,7 +495,7 @@
a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets
(possibly with additional structure if $k=n$).
Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls,
-and there is a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries
+and there is a subset $\cC(X)\spl \subset \cC(X)$ of morphisms whose boundaries
are splittable along this decomposition.
\begin{defn}
@@ -505,7 +503,7 @@
For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset
\begin{equation*}
%\label{eq:psi-C}
- \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl
+ \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl
\end{equation*}
where the restrictions to the various pieces of shared boundaries amongst the cells
$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). When $k=n$, the `subset' and `product' in the above formula should be interpreted in the appropriate enriching category.
@@ -673,10 +671,10 @@
(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy).
\begin{equation*}
\xymatrix@C+0.3cm{
- \CH{X} \otimes \bc_*(X)
- \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y} &
+ \CH{X} \tensor \bc_*(X)
+ \ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \tensor \gl_Y} &
\bc_*(X) \ar[d]_{\gl_Y} \\
- \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow)
+ \CH{X \bigcup_Y \selfarrow} \tensor \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow)
}
\end{equation*}
\end{enumerate}
@@ -782,7 +780,7 @@
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
\[
- C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}}
+ C_*(LD_k)\tensor \overbrace{Hoch^*(C, C)\tensor\cdots\tensor Hoch^*(C, C)}^{\text{$k$ copies}}
\to Hoch^*(C, C),
\]
which we now see to be a specialization of Theorem \ref{thm:deligne}.