--- a/blob1.tex Wed Jun 02 17:45:13 2010 -0700
+++ b/blob1.tex Thu Jun 03 09:47:18 2010 -0700
@@ -68,8 +68,6 @@
\input{text/a_inf_blob}
-\input{text/comm_alg}
-
\input{text/deligne}
\appendix
@@ -80,6 +78,8 @@
\input{text/appendixes/comparing_defs}
+\input{text/comm_alg}
+
% ----------------------------------------------------------------
%\newcommand{\urlprefix}{}
\bibliographystyle{plain}
@@ -94,4 +94,3 @@
\end{document}
% ----------------------------------------------------------------
-
--- a/preamble.tex Wed Jun 02 17:45:13 2010 -0700
+++ b/preamble.tex Thu Jun 03 09:47:18 2010 -0700
@@ -190,6 +190,8 @@
\newcommand{\CD}[1]{C_*(\Diff(#1))}
\newcommand{\CH}[1]{C_*(\Homeo(#1))}
+\newcommand{\cl}[1]{\underrightarrow{#1}}
+
\newcommand{\directSumStack}[2]{{\begin{matrix}#1 \\ \DirectSum \\#2\end{matrix}}}
\newcommand{\directSumStackThree}[3]{{\begin{matrix}#1 \\ \DirectSum \\#2 \\ \DirectSum \\#3\end{matrix}}}
--- a/text/a_inf_blob.tex Wed Jun 02 17:45:13 2010 -0700
+++ b/text/a_inf_blob.tex Thu Jun 03 09:47:18 2010 -0700
@@ -217,7 +217,7 @@
This concludes the proof of Theorem \ref{product_thm}.
\end{proof}
-\nn{need to say something about dim $< n$ above}
+\nn{need to prove a version where $E$ above has dimension $m<n$; result is an $n{-}m$-category}
\medskip
@@ -235,11 +235,11 @@
\[
F \to E \to Y .
\]
-We outline two approaches.
+We outline one approach here and a second in Subsection xxxx.
We can generalize the definition of a $k$-category by replacing the categories
-of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$.
-\nn{need citation to other work that does this; Stolz and Teichner?}
+of $j$-balls ($j\le k$) with categories of $j$-balls $D$ equipped with a map $p:D\to Y$
+(c.f. \cite{MR2079378}).
Call this a $k$-category over $Y$.
A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$:
assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$.
@@ -254,6 +254,7 @@
+\nn{put this later}
\nn{The second approach: Choose a decomposition $Y = \cup X_i$
such that the restriction of $E$ to $X_i$ is a product $F\times X_i$.
@@ -275,28 +276,56 @@
Let $X$ be a closed $k$-manifold with a splitting $X = X'_1\cup_Y X'_2$.
We will need an explicit collar on $Y$, so rewrite this as
$X = X_1\cup (Y\times J) \cup X_2$.
-\nn{need figure}
-Given this data we have: \nn{need refs to above for these}
+Given this data we have:
\begin{itemize}
\item An $A_\infty$ $n{-}k$-category $\bc(X)$, which assigns to an $m$-ball
$D$ fields on $D\times X$ (for $m+k < n$) or the blob complex $\bc_*(D\times X; c)$
-(for $m+k = n$). \nn{need to explain $c$}.
+(for $m+k = n$).
+(See Example \ref{ex:blob-complexes-of-balls}.)
+%\nn{need to explain $c$}.
\item An $A_\infty$ $n{-}k{+}1$-category $\bc(Y)$, defined similarly.
\item Two $\bc(Y)$ modules $\bc(X_1)$ and $\bc(X_2)$, which assign to a marked
$m$-ball $(D, H)$ either fields on $(D\times Y) \cup (H\times X_i)$ (if $m+k < n$)
or the blob complex $\bc_*((D\times Y) \cup (H\times X_i))$ (if $m+k = n$).
+(See Example \nn{need example for this}.)
\end{itemize}
\begin{thm}
\label{thm:gluing}
-$\bc(X) \cong \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
+$\bc(X) \simeq \bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
\end{thm}
\begin{proof}
+\nn{for now, just prove $k=0$ case.}
The proof is similar to that of Theorem \ref{product_thm}.
-\nn{need to say something about dimensions less than $n$,
-but for now concentrate on top dimension.}
+We give a short sketch with emphasis on the differences from
+the proof of Theorem \ref{product_thm}.
+
+Let $\cT$ denote the chain complex $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
+Recall that this is a homotopy colimit based on decompositions of the interval $J$.
+
+We define a map $\psi:\cT\to \bc_*(X)$. On filtration degree zero summands it is given
+by gluing the pieces together to get a blob diagram on $X$.
+On filtration degree 1 and greater $\psi$ is zero.
+The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split
+over some decomposition of $J$.
+It follows from Proposition \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to
+a subcomplex of $G_*$.
+
+Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models.
+As in the proof of Theorem \ref{product_thm}, we assign to a generator $a$ of $G_*$
+an acyclic subcomplex which is (roughly) $\psi\inv(a)$.
+The proof of acyclicity is easier in this case since any pair of decompositions of $J$ have
+a common refinement.
+
+The proof that these two maps are inverse to each other is the same as in
+Theorem \ref{product_thm}.
+\end{proof}
+
+This establishes Property \ref{property:gluing}.
+
+\noop{
Let $\cT$ denote the $n{-}k$-category $\bc(X_1) \otimes_{\bc(Y), J} \bc(X_2)$.
Let $D$ be an $n{-}k$-ball.
There is an obvious map from $\cT(D)$ to $\bc_*(D\times X)$.
@@ -307,9 +336,8 @@
decomposition of $D\times X$.
The proof that these two maps are inverse to each other is the same as in
Theorem \ref{product_thm}.
-\end{proof}
+}
-This establishes Property \ref{property:gluing}.
\medskip
@@ -329,12 +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, $j \ge 0$.
@@ -364,42 +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
@@ -407,7 +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:
@@ -418,21 +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}
-
--- a/text/appendixes/famodiff.tex Wed Jun 02 17:45:13 2010 -0700
+++ b/text/appendixes/famodiff.tex Thu Jun 03 09:47:18 2010 -0700
@@ -207,7 +207,7 @@
Since PL homeomorphisms are bi-Lipschitz, we have established this last remaining case of claim 4 of the lemma as well.
\end{proof}
-\begin{lemma}
+\begin{lemma} \label{extension_lemma_c}
Let $\cX_*$ be any of $C_*(\Maps(X \to T))$ or singular chains on the subspace of $\Maps(X\to T)$ consisting of immersions, diffeomorphisms, bi-Lipschitz homeomorphisms or PL homeomorphisms.
Let $G_* \subset \cX_*$ denote the chains adapted to an open cover $\cU$
of $X$.
--- a/text/comm_alg.tex Wed Jun 02 17:45:13 2010 -0700
+++ b/text/comm_alg.tex Thu Jun 03 09:47:18 2010 -0700
@@ -3,8 +3,7 @@
\section{Commutative algebras as $n$-categories}
\label{sec:comm_alg}
-\nn{this should probably not be a section by itself. i'm just trying to write down the outline
-while it's still fresh in my mind.}
+\nn{should consider leaving this out; for now, make it an appendix.}
\nn{also, this section needs a little updating to be compatible with the rest of the paper.}
--- a/text/ncat.tex Wed Jun 02 17:45:13 2010 -0700
+++ b/text/ncat.tex Thu Jun 03 09:47:18 2010 -0700
@@ -1044,7 +1044,7 @@
Suppose $S$ is a topological space, with a subspace $T$. We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ for $k<n$ the set $\pi_{\leq n}(S,T)(B,N)$ consists of all continuous maps of pairs $(B,N) \to (S,T)$ and on each marked $n$-ball $(B,N)$ it consists of all such maps modulo homotopies fixed on $\bdy B \setminus N$. This is a module over the fundamental $n$-category $\pi_{\leq n}(S)$ of $S$, from Example \ref{ex:maps-to-a-space}. Modifications corresponding to Examples \ref{ex:maps-to-a-space-with-a-fiber} and \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains.
\end{example}
-\subsection{Modules as boundary labels}
+\subsection{Modules as boundary labels (colimits for decorated manifolds)}
\label{moddecss}
Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. Let $W$ be a $k$-manifold ($k\le n$),