
diff -r a20e2318cbb0 -r 0bfcb02658ce text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Wed Jun 02 22:09:52 2010 -0700
+++ b/text/a_inf_blob.tex	Wed Jun 02 22:28:04 2010 -0700
@@ -357,12 +357,15 @@
 \end{thm}
 \begin{rem}
 \nn{This just isn't true, Lurie doesn't do this! I just heard this from Ricardo...}
+\nn{KW: Are you sure about that?}
 Lurie has shown in \cite{0911.0018} that the topological chiral homology of an $n$-manifold $M$ with coefficients in \nn{a certain $E_n$ algebra constructed from $T$} recovers  the same space of singular chains on maps from $M$ to $T$, with the additional hypothesis that $T$ is $n-1$-connected. This extra hypothesis is not surprising, in view of the idea that an $E_n$ algebra is roughly equivalent data as an $A_\infty$ $n$-category which is trivial at all but the topmost level.
 \end{rem}
 
+\nn{proof is again similar to that of Theorem \ref{product_thm}.  should probably say that explicitly}
+
 \begin{proof}
 We begin by constructing chain map $g: \cB^\cT(M) \to C_*(\Maps(M\to T))$.
-We then use \ref{extension_lemma_b} to show that $g$ induces isomorphisms on homology.
+We then use Lemma \ref{extension_lemma_c} to show that $g$ induces isomorphisms on homology.
 
 Recall that the homotopy colimit $\cB^\cT(M)$ is constructed out of a series of
 $j$-fold mapping cylinders, $j \ge 0$.
@@ -392,42 +395,12 @@
 It is not hard to see that this defines a chain map from 
 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$.
 
-
-%%%%%%%%%%%%%%%%%
-\noop{
-Next we show that $g$ induces a surjection on homology.
-Fix $k > 0$ and choose an open cover $\cU$ of $M$ fine enough so that the union 
-of any $k$ open sets of $\cU$ is contained in a disjoint union of balls in $M$.
-\nn{maybe should refer to elsewhere in this paper where we made a very similar argument}
-Let $S_*$ be the subcomplex of $C_*(\Maps(M\to T))$ generated by chains adapted to $\cU$.
-It follows from Lemma \ref{extension_lemma_b} that $C_*(\Maps(M\to T))$
-retracts onto $S_*$.
-
-Let $S_{\le k}$ denote the chains of $S_*$ of degree less than or equal to $k$.
-We claim that $S_{\le k}$ lies in the image of $g$.
-Let $c$ be a generator of $S_{\le k}$ --- that is, a $j$-parameter family of maps $M\to T$,
-$j \le k$.
-We chose $\cU$ fine enough so that the support of $c$ is contained in a disjoint union of balls
-in $M$.
-It follow that we can choose a decomposition $K$ of $M$ so that the support of $c$ is 
-disjoint from the $n{-}1$-skeleton of $K$.
-It is now easy to see that $c$ is in the image of $g$.
-
-Next we show that $g$ is injective on homology.
-}
-
-
-
 \nn{...}
 
-
-
 \end{proof}
 
 \nn{maybe should also mention version where we enrich over
-spaces rather than chain complexes; should comment on Lurie's \cite{0911.0018} (and others') similar result
-for the $E_\infty$ case, and mention that our version does not require 
-any connectivity assumptions}
+spaces rather than chain complexes;}
 
 \medskip
 \hrule
@@ -435,7 +408,7 @@
 
 \nn{to be continued...}
 \medskip
-\nn{still to do: fiber bundles, general maps}
+\nn{still to do: general maps}
 
 \todo{}
 Various citations we might want to make:
@@ -446,21 +419,4 @@
 \item \cite{MR1256989} definition of framed little-discs operad
 \end{itemize}
 
-We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction
-\begin{itemize}
-%\mbox{}% <-- gets the indenting right
-\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is
-naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below.
 
-\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an
-$A_\infty$ module for $\bc_*(Y \times I)$.
-
-\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension
-$0$-submanifold of its boundary, the blob homology of $X'$, obtained from
-$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of
-$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule.
-\begin{equation*}
-\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]}
-\end{equation*}
-\end{itemize}
-




