# HG changeset patch # User Scott Morrison # Date 1285115294 25200 # Node ID 9dfb5db2acd767e7816323095a4a92a438d22d81 # Parent c9f41c18a96f37ae63b83776549e4899b3481190 remaining changes from tuesday afternoon diff -r c9f41c18a96f -r 9dfb5db2acd7 preamble.tex --- 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} diff -r c9f41c18a96f -r 9dfb5db2acd7 text/basic_properties.tex --- 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. diff -r c9f41c18a96f -r 9dfb5db2acd7 text/blobdef.tex --- 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}; diff -r c9f41c18a96f -r 9dfb5db2acd7 text/evmap.tex --- 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*} diff -r c9f41c18a96f -r 9dfb5db2acd7 text/hochschild.tex --- 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 $H_0$. diff -r c9f41c18a96f -r 9dfb5db2acd7 text/intro.tex --- 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). diff -r c9f41c18a96f -r 9dfb5db2acd7 text/ncat.tex --- 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 \[