
diff -r b66138a1cde8 -r e448415ad80a pnas/pnas.tex
--- a/pnas/pnas.tex	Sun Nov 14 16:15:17 2010 -0800
+++ b/pnas/pnas.tex	Sun Nov 14 16:33:36 2010 -0800
@@ -761,6 +761,17 @@
 \end{itemize}
 \end{thm}
 
+\begin{proof} (Sketch.)
+The $A_\infty$ action of $\bc_*(Y)$ follows from the naturality of the blob complex with respect to gluing
+and the $C_*(\Homeo(-))$ action of Theorem \ref{thm:evaluation}.
+
+Let $T_*$ denote the self tensor product of $\bc_*(X)$, which is a homotopy colimit.
+Let $X_{\mathrm gl}$ denote $X$ glued to itself along $Y$.
+There is a tautological map from the 0-simplices of $T_*$ to $\bc_*(X_{\mathrm gl})$,
+and this map can be extended to a chain map on all of $T_*$ by sending the higher simplices to zero.
+Constructing a homotopy inverse to this natural map invloves making various choices, but one can show that the
+choices form contractible subcomplexes and apply the acyclic models theorem.
+\end{proof}
 
 We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above.
 
@@ -777,7 +788,11 @@
 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps
 (see \cite[\S7.1]{1009.5025}).
 
-\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.}
+\begin{proof} (Sketch.)
+
+\end{proof}
+
+%\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.}
 
 \section{Higher Deligne conjecture}
 \label{sec:applications}




