--- a/preamble.tex Tue Sep 21 14:44:17 2010 -0700
+++ b/preamble.tex Tue Sep 21 17:28:14 2010 -0700
@@ -49,7 +49,7 @@
% THEOREMS -------------------------------------------------------
\theoremstyle{plain}
%\newtheorem*{fact}{Fact}
-\newtheorem{prop}{Proposition}[section]
+\newtheorem{prop}{Proposition}[subsection]
\newtheorem{conj}[prop]{Conjecture}
\newtheorem{thm}[prop]{Theorem}
\newtheorem{lem}[prop]{Lemma}
@@ -61,9 +61,9 @@
\newtheorem{defn}[prop]{Definition} % numbered definition
\newtheorem*{defn*}{Definition} % unnumbered definition
\newtheorem{question}{Question}
-\newtheorem{property}{Property}
-\newtheorem{axiom}{Axiom}[section]
-\newtheorem{module-axiom}{Module Axiom}[section]
+\newtheorem{property}[prop]{Property}
+\newtheorem{axiom}[prop]{Axiom}
+\newtheorem{module-axiom}[prop]{Module Axiom}
\newenvironment{rem}{\noindent\textsl{Remark.}}{} % perhaps looks better than rem above?
\newtheorem{rem*}[prop]{Remark}
\newtheorem{remark}[prop]{Remark}
--- a/text/basic_properties.tex Tue Sep 21 14:44:17 2010 -0700
+++ b/text/basic_properties.tex Tue Sep 21 17:28:14 2010 -0700
@@ -1,9 +1,9 @@
%!TEX root = ../blob1.tex
-\section{Basic properties}
+\subsection{Basic properties}
\label{sec:basic-properties}
-In this section we complete the proofs of Properties 2-4.
+In this section we complete the proofs of Properties 2-4. \nn{fix these numbers}
Throughout the paper, where possible, we prove results using Properties 1-4,
rather than the actual definition of blob homology.
This allows the possibility of future improvements on or alternatives to our definition.
--- a/text/blobdef.tex Tue Sep 21 14:44:17 2010 -0700
+++ b/text/blobdef.tex Tue Sep 21 17:28:14 2010 -0700
@@ -1,8 +1,8 @@
%!TEX root = ../blob1.tex
\section{The blob complex}
+\subsection{Definitions}
\label{sec:blob-definition}
-
Let $X$ be an $n$-manifold.
Let $(\cF,U)$ be a fixed system of fields and local relations.
We'll assume it is enriched over \textbf{Vect};
--- a/text/evmap.tex Tue Sep 21 14:44:17 2010 -0700
+++ b/text/evmap.tex Tue Sep 21 17:28:14 2010 -0700
@@ -415,7 +415,7 @@
(For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general
than simplices --- they can be based on any cone-product polyhedron (see Remark \ref{blobsset-remark}).)
-\begin{thm} \label{thm:CH}
+\begin{thm} \label{thm:CH} \label{thm:evaluation}%
For $n$-manifolds $X$ and $Y$ there is a chain map
\eq{
e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) ,
@@ -424,7 +424,7 @@
such that
\begin{enumerate}
\item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of
-$\Homeo(X, Y)$ on $\bc_*(X)$ described in Property (\ref{property:functoriality}), and
+$\Homeo(X, Y)$ on $\bc_*(X)$ described in Property \ref{property:functoriality}, and
\item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$,
the following diagram commutes up to homotopy
\begin{equation*}
--- a/text/hochschild.tex Tue Sep 21 14:44:17 2010 -0700
+++ b/text/hochschild.tex Tue Sep 21 17:28:14 2010 -0700
@@ -455,11 +455,11 @@
($G''_*$ and $G'_*$ depend on $N$, but that is not reflected in the notation.)
Then $G''_*$ and $G'_*$ are both contractible
and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence.
-For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting
+For $G'_*$ the proof is the same as in Lemma \ref{bcontract}, except that the splitting
$G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$.
For $G''_*$ we note that any cycle is supported away from $*$.
Thus any cycle lies in the image of the normal blob complex of a disjoint union
-of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disj-union-contract}).
+of two intervals, which is contractible by Lemma \ref{bcontract} and Corollary \ref{disj-union-contract}.
Finally, it is easy to see that the inclusion
$G''_* \to G'_*$ induces an isomorphism on $H_0$.
--- a/text/intro.tex Tue Sep 21 14:44:17 2010 -0700
+++ b/text/intro.tex Tue Sep 21 17:28:14 2010 -0700
@@ -50,7 +50,7 @@
%and outline anticipated future directions (see \S \ref{sec:future}).
%\nn{recheck this list after done editing intro}
-The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex,
+The first part of the paper (sections \S \ref{sec:fields}--\S \ref{sec:evaluation}) gives the definition of the blob complex,
and establishes some of its properties.
There are many alternative definitions of $n$-categories, and part of the challenge of defining the blob complex is
simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs.
@@ -322,7 +322,6 @@
\newtheorem*{thm:CH}{Theorem \ref{thm:CH}}
\begin{thm:CH}[$C_*(\Homeo(-))$ action]
-\label{thm:evaluation}%
There is a chain map
\begin{equation*}
e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X).
--- a/text/ncat.tex Tue Sep 21 14:44:17 2010 -0700
+++ b/text/ncat.tex Tue Sep 21 17:28:14 2010 -0700
@@ -45,7 +45,7 @@
By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
standard $k$-ball.
We {\it do not} assume that it is equipped with a
-preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below.
+preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. \nn{List the axiom numbers here, mentioning alternate versions, and also the same in the module section.}
Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on
the boundary), we want a corresponding
@@ -465,7 +465,7 @@
same (traditional) $i$-morphism as the corresponding codimension $i$ cell $c$.
-\addtocounter{axiom}{-1}
+%\addtocounter{axiom}{-1}
\begin{axiom}[Product (identity) morphisms]
For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$),
there is a map $\pi^*:\cC(X)\to \cC(E)$.
@@ -592,7 +592,7 @@
The revised axiom is
-\addtocounter{axiom}{-1}
+%\addtocounter{axiom}{-1}
\begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.]
\label{axiom:extended-isotopies}
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
@@ -610,7 +610,7 @@
$C_*(\Homeo_\bd(X))$ denote the singular chains on this space.
-\addtocounter{axiom}{-1}
+%\addtocounter{axiom}{-1}
\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]
For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes
\[
@@ -1434,7 +1434,7 @@
For $A_\infty$ modules we require
-\addtocounter{module-axiom}{-1}
+%\addtocounter{module-axiom}{-1}
\begin{module-axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act]
For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes
\[