--- a/blob1.tex Wed Jun 02 22:09:52 2010 -0700
+++ b/blob1.tex Wed Jun 02 22:28:04 2010 -0700
@@ -1,10 +1,14 @@
\documentclass[11pt,leqno]{article}
+%\usepackage{amsthm}
+
\newcommand{\pathtotrunk}{./}
\input{text/article_preamble}
\input{text/top_matter}
\input{text/kw_macros}
+
+%\title{Blob Homology}
\title{The Blob Complex}
\begin{document}
@@ -16,10 +20,12 @@
\maketitle
-[revision $\ge$ 320; $\ge$ 2 June 2010]
+
+[revision $\ge$ 276; $\ge$ 30 May 2010]
\textbf{Draft version, read with caution.}
+
\nn{maybe to do: add appendix on various versions of acyclic models}
\paragraph{To do list}
@@ -54,6 +60,7 @@
\input{text/intro}
+%\input{text/definitions}
\input{text/tqftreview}
\input{text/blobdef}
@@ -68,8 +75,6 @@
\input{text/a_inf_blob}
-\input{text/comm_alg}
-
\input{text/deligne}
\appendix
@@ -80,9 +85,15 @@
\input{text/appendixes/comparing_defs}
+\input{text/comm_alg}
+
+%\input{text/obsolete}
+
% ----------------------------------------------------------------
%\newcommand{\urlprefix}{}
\bibliographystyle{plain}
+%Included for winedt:
+%input "bibliography/bibliography.bib"
\bibliography{bibliography/bibliography}
% ----------------------------------------------------------------
@@ -90,6 +101,8 @@
\url{http://tqft.net/blobs},
and at \url{http://canyon23.net/math/}.
+% A GTART necessity:
+% \Addresses
% ----------------------------------------------------------------
\end{document}
% ----------------------------------------------------------------
--- 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}
-
--- a/text/appendixes/famodiff.tex Wed Jun 02 22:09:52 2010 -0700
+++ b/text/appendixes/famodiff.tex Wed Jun 02 22:28:04 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 22:09:52 2010 -0700
+++ b/text/comm_alg.tex Wed Jun 02 22:28:04 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.}