--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/blob1.tex Sat Feb 09 01:01:03 2008 +0000
@@ -0,0 +1,1103 @@
+\documentclass[11pt,leqno]{article}
+
+\usepackage{amsmath,amssymb,amsthm}
+
+\usepackage[all]{xy}
+
+
+%%%%% excerpts from my include file of standard macros
+
+\def\bc{{\cal B}}
+
+\def\z{\mathbb{Z}}
+\def\r{\mathbb{R}}
+\def\c{\mathbb{C}}
+\def\t{\mathbb{T}}
+
+\def\du{\sqcup}
+\def\bd{\partial}
+\def\sub{\subset}
+\def\sup{\supset}
+%\def\setmin{\smallsetminus}
+\def\setmin{\setminus}
+\def\ep{\epsilon}
+\def\sgl{_\mathrm{gl}}
+\def\deq{\stackrel{\mathrm{def}}{=}}
+\def\pd#1#2{\frac{\partial #1}{\partial #2}}
+
+\def\nn#1{{{\it \small [#1]}}}
+
+
+% equations
+\newcommand{\eq}[1]{\begin{displaymath}#1\end{displaymath}}
+\newcommand{\eqar}[1]{\begin{eqnarray*}#1\end{eqnarray*}}
+\newcommand{\eqspl}[1]{\begin{displaymath}\begin{split}#1\end{split}\end{displaymath}}
+
+% tricky way to iterate macros over a list
+\def\semicolon{;}
+\def\applytolist#1{
+ \expandafter\def\csname multi#1\endcsname##1{
+ \def\multiack{##1}\ifx\multiack\semicolon
+ \def\next{\relax}
+ \else
+ \csname #1\endcsname{##1}
+ \def\next{\csname multi#1\endcsname}
+ \fi
+ \next}
+ \csname multi#1\endcsname}
+
+% \def\cA{{\cal A}} for A..Z
+\def\calc#1{\expandafter\def\csname c#1\endcsname{{\cal #1}}}
+\applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM;
+
+% \DeclareMathOperator{\pr}{pr} etc.
+\def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}
+\applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{End}{Hom}{Mat}{Tet}{cat}{Diff}{sign};
+
+
+
+%%%%%% end excerpt
+
+
+
+
+
+\title{Blob Homology}
+
+\begin{document}
+
+
+
+\makeatletter
+\@addtoreset{equation}{section}
+\gdef\theequation{\thesection.\arabic{equation}}
+\makeatother
+\newtheorem{thm}[equation]{Theorem}
+\newtheorem{prop}[equation]{Proposition}
+\newtheorem{lemma}[equation]{Lemma}
+\newtheorem{cor}[equation]{Corollary}
+\newtheorem{defn}[equation]{Definition}
+
+
+
+\maketitle
+
+\section{Introduction}
+
+(motivation, summary/outline, etc.)
+
+(motivation:
+(1) restore exactness in pictures-mod-relations;
+(1') add relations-amongst-relations etc. to pictures-mod-relations;
+(2) want answer independent of handle decomp (i.e. don't
+just go from coend to derived coend (e.g. Hochschild homology));
+(3) ...
+)
+
+\section{Definitions}
+
+\subsection{Fields}
+
+Fix a top dimension $n$.
+
+A {\it system of fields}
+\nn{maybe should look for better name; but this is the name I use elsewhere}
+is a collection of functors $\cC$ from manifolds of dimension $n$ or less
+to sets.
+These functors must satisfy various properties (see KW TQFT notes for details).
+For example:
+there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$;
+there is a restriction map $\cC(X) \to \cC(\bd X)$;
+gluing manifolds corresponds to fibered products of fields;
+given a field $c \in \cC(Y)$ there is a ``product field"
+$c\times I \in \cC(Y\times I)$; ...
+\nn{should eventually include full details of definition of fields.}
+
+\nn{note: probably will suppress from notation the distinction
+between fields and their (orientation-reversal) duals}
+
+\nn{remark that if top dimensional fields are not already linear
+then we will soon linearize them(?)}
+
+The definition of a system of fields is intended to generalize
+the relevant properties of the following two examples of fields.
+
+The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$
+is a manifold of dimension $n$ or less) to be the set of
+all maps from $X$ to $B$.
+
+The second example will take longer to explain.
+Given an $n$-category $C$ with the right sort of duality
+(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category),
+we can construct a system of fields as follows.
+Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$
+with codimension $i$ cells labeled by $i$-morphisms of $C$.
+We'll spell this out for $n=1,2$ and then describe the general case.
+
+If $X$ has boundary, we require that the cell decompositions are in general
+position with respect to the boundary --- the boundary intersects each cell
+transversely, so cells meeting the boundary are mere half-cells.
+
+Put another way, the cell decompositions we consider are dual to standard cell
+decompositions of $X$.
+
+We will always assume that our $n$-categories have linear $n$-morphisms.
+
+For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with
+an object (0-morphism) of the 1-category $C$.
+A field on a 1-manifold $S$ consists of
+\begin{itemize}
+ \item A cell decomposition of $S$ (equivalently, a finite collection
+of points in the interior of $S$);
+ \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$)
+by an object (0-morphism) of $C$;
+ \item a transverse orientation of each 0-cell, thought of as a choice of
+``domain" and ``range" for the two adjacent 1-cells; and
+ \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with
+domain and range determined by the transverse orientation and the labelings of the 1-cells.
+\end{itemize}
+
+If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels
+of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the
+interior of $S$, each transversely oriented and each labeled by an element (1-morphism)
+of the algebra.
+
+For $n=2$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with
+an object of the 2-category $C$.
+A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$.
+A field on a 2-manifold $Y$ consists of
+\begin{itemize}
+ \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such
+that each component of the complement is homeomorphic to a disk);
+ \item a labeling of each 2-cell (and each half 2-cell adjacent to $\bd Y$)
+by a 0-morphism of $C$;
+ \item a transverse orientation of each 1-cell, thought of as a choice of
+``domain" and ``range" for the two adjacent 2-cells;
+ \item a labeling of each 1-cell by a 1-morphism of $C$, with
+domain and range determined by the transverse orientation of the 1-cell
+and the labelings of the 2-cells;
+ \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood
+of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped
+to $\pm 1 \in S^1$; and
+ \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range
+determined by the labelings of the 1-cells and the parameterizations of the previous
+bullet.
+\end{itemize}
+\nn{need to say this better; don't try to fit everything into the bulleted list}
+
+For general $n$, a field on a $k$-manifold $X^k$ consists of
+\begin{itemize}
+ \item A cell decomposition of $X$;
+ \item an explicit general position homeomorphism from the link of each $j$-cell
+to the boundary of the standard $(k-j)$-dimensional bihedron; and
+ \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with
+domain and range determined by the labelings of the link of $j$-cell.
+\end{itemize}
+
+\nn{next definition might need some work; I think linearity relations should
+be treated differently (segregated) from other local relations, but I'm not sure
+the next definition is the best way to do it}
+
+For top dimensional ($n$-dimensional) manifolds, we're actually interested
+in the linearized space of fields.
+By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is
+the vector space of finite
+linear combinations of fields on $X$.
+If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$.
+Thus the restriction (to boundary) maps are well defined because we never
+take linear combinations of fields with differing boundary conditions.
+
+In some cases we don't linearize the default way; instead we take the
+spaces $\cC_l(X; a)$ to be part of the data for the system of fields.
+In particular, for fields based on linear $n$-category pictures we linearize as follows.
+Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
+obvious relations on 0-cell labels.
+More specifically, let $L$ be a cell decomposition of $X$
+and let $p$ be a 0-cell of $L$.
+Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
+$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
+Then the subspace $K$ is generated by things of the form
+$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader
+to infer the meaning of $\alpha_{\lambda c + d}$.
+Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.
+
+\nn{Maybe comment further: if there's a natural basis of morphisms, then no need;
+will do something similar below; in general, whenever a label lives in a linear
+space we do something like this; ? say something about tensor
+product of all the linear label spaces? Yes:}
+
+For top dimensional ($n$-dimensional) manifolds, we linearize as follows.
+Define an ``almost-field" to be a field without labels on the 0-cells.
+(Recall that 0-cells are labeled by $n$-morphisms.)
+To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
+space determined by the labeling of the link of the 0-cell.
+(If the 0-cell were labeled, the label would live in this space.)
+We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
+We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the
+above tensor products.
+
+
+
+\subsection{Local relations}
+
+Let $B^n$ denote the standard $n$-ball.
+A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$
+(for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties.
+
+\nn{implies (extended?) isotopy; stable under gluing; open covers?; ...}
+
+For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$,
+where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
+
+For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map
+$\cC_l(B^n; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
+domain and range.
+
+\nn{maybe examples of local relations before general def?}
+
+Note that the $Y$ is an $n$-manifold which is merely homeomorphic to the standard $B^n$,
+then any homeomorphism $B^n \to Y$ induces the same local subspaces for $Y$.
+We'll denote these by $U(Y; c) \sub \cC_l(Y; c)$, $c \in \cC(\bd Y)$.
+
+Given a system of fields and local relations, we define the skein space
+$A(Y^n; c)$ to be the space of all finite linear combinations of fields on
+the $n$-manifold $Y$ modulo local relations.
+The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations
+is defined to be the dual of $A(Y; c)$.
+(See KW TQFT notes or xxxx for details.)
+
+The blob complex is in some sense the derived version of $A(Y; c)$.
+
+
+
+\subsection{The blob complex}
+
+Let $X$ be an $n$-manifold.
+Assume a fixed system of fields.
+In this section we will usually suppress boundary conditions on $X$ from the notation
+(e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$).
+
+We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
+submanifold of $X$, then $X \setmin Y$ implicitly means the closure
+$\overline{X \setmin Y}$.
+
+We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case.
+
+Define $\bc_0(X) = \cC_l(X)$.
+(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cC_l(X; c)$.
+We'll omit this sort of detail in the rest of this section.)
+In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$.
+
+$\bc_1(X)$ is the space of all local relations that can be imposed on $\bc_0(X)$.
+More specifically, define a 1-blob diagram to consist of
+\begin{itemize}
+\item An embedded closed ball (``blob") $B \sub X$.
+%\nn{Does $B$ need a homeo to the standard $B^n$? I don't think so.
+%(See note in previous subsection.)}
+%\item A field (boundary condition) $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$.
+\item A field $r \in \cC(X \setmin B; c)$
+(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$).
+\item A local relation field $u \in U(B; c)$
+(same $c$ as previous bullet).
+\end{itemize}
+%(Note that the the field $c$ is determined (implicitly) as the boundary of $u$ and/or $r$,
+%so we will omit $c$ from the notation.)
+Define $\bc_1(X)$ to be the space of all finite linear combinations of
+1-blob diagrams, modulo the simple relations relating labels of 0-cells and
+also the label ($u$ above) of the blob.
+\nn{maybe spell this out in more detail}
+(See xxxx above.)
+\nn{maybe restate this in terms of direct sums of tensor products.}
+
+There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear
+combination of fields on $X$ obtained by gluing $r$ to $u$.
+In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by
+just erasing the blob from the picture
+(but keeping the blob label $u$).
+
+Note that the skein module $A(X)$
+is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
+
+$\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$.
+More specifically, $\bc_2(X)$ is the space of all finite linear combinations of
+2-blob diagrams (defined below), modulo the usual linear label relations.
+\nn{and also modulo blob reordering relations?}
+
+\nn{maybe include longer discussion to motivate the two sorts of 2-blob diagrams}
+
+There are two types of 2-blob diagram: disjoint and nested.
+A disjoint 2-blob diagram consists of
+\begin{itemize}
+\item A pair of disjoint closed balls (blobs) $B_0, B_1 \sub X$.
+%\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$.
+\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$
+(where $c_i \in \cC(\bd B_i)$).
+\item Local relation fields $u_i \in U(B_i; c_i)$.
+\end{itemize}
+Define $\bd(B_0, B_1, r, u_0, u_1) = (B_1, ru_0, u_1) - (B_0, ru_1, u_0) \in \bc_1(X)$.
+In other words, the boundary of a disjoint 2-blob diagram
+is the sum (with alternating signs)
+of the two ways of erasing one of the blobs.
+It's easy to check that $\bd^2 = 0$.
+
+A nested 2-blob diagram consists of
+\begin{itemize}
+\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$.
+\item A field $r \in \cC(X \setmin B_0; c_0)$
+(for some $c_0 \in \cC(\bd B_0)$).
+Let $r = r_1 \cup r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
+(for some $c_1 \in \cC(B_1)$) and
+$r' \in \cC(X \setmin B_1; c_1)$.
+\item A local relation field $u_0 \in U(B_0; c_0)$.
+\end{itemize}
+Define $\bd(B_0, B_1, r, u_0) = (B_1, r', r_1u_0) - (B_0, r, u_0)$.
+Note that xxxx above guarantees that $r_1u_0 \in U(B_1)$.
+As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating
+sum of the two ways of erasing one of the blobs.
+If we erase the inner blob, the outer blob inherits the label $r_1u_0$.
+
+Now for the general case.
+A $k$-blob diagram consists of
+\begin{itemize}
+\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$.
+For each $i$ and $j$, we require that either $B_i \cap B_j$ is empty or
+$B_i \sub B_j$ or $B_j \sub B_i$.
+(The case $B_i = B_j$ is allowed.
+If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.)
+If a blob has no other blobs strictly contained in it, we call it a twig blob.
+%\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$.
+%(These are implied by the data in the next bullets, so we usually
+%suppress them from the notation.)
+%$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$
+%if the latter space is not empty.
+\item A field $r \in \cC(X \setmin B^t; c^t)$,
+where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$.
+\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$,
+where $c_j$ is the restriction of $c^t$ to $\bd B_j$.
+If $B_i = B_j$ then $u_i = u_j$.
+\end{itemize}
+
+We define $\bc_k(X)$ to be the vector space of all finite linear combinations
+of $k$-blob diagrams, modulo the linear label relations and
+blob reordering relations defined in the remainder of this paragraph.
+Let $x$ be a blob diagram with one undetermined $n$-morphism label.
+The unlabeled entity is either a blob or a 0-cell outside of the twig blobs.
+Let $a$ and $b$ be two possible $n$-morphism labels for
+the unlabeled blob or 0-cell.
+Let $c = \lambda a + b$.
+Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly.
+Then we impose the relation
+\eq{
+ x_c = \lambda x_a + x_b .
+}
+\nn{should do this in terms of direct sums of tensor products}
+Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$
+of their blob labelings.
+Then we impose the relation
+\eq{
+ x = \sign(\pi) x' .
+}
+
+(Alert readers will have noticed that for $k=2$ our definition
+of $\bc_k(X)$ is slightly different from the previous definition
+of $\bc_2(X)$.
+The general definition takes precedence;
+the earlier definition was simplified for purposes of exposition.)
+
+The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows.
+Let $b = (\{B_i\}, r, \{u_j\})$ be a $k$-blob diagram.
+Let $E_j(b)$ denote the result of erasing the $j$-th blob.
+If $B_j$ is not a twig blob, this involves only decrementing
+the indices of blobs $B_{j+1},\ldots,B_{k-1}$.
+If $B_j$ is a twig blob, we have to assign new local relation labels
+if removing $B_j$ creates new twig blobs.
+If $B_l$ becomes a twig after removing $B_j$, then set $u_l = r_lu_j$,
+where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$.
+Finally, define
+\eq{
+ \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b).
+}
+The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel.
+Thus we have a chain complex.
+
+\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)}
+
+
+\nn{TO DO: ((?)) allow $n$-morphisms to be chain complex instead of just
+a vector space; relations to Chas-Sullivan string stuff}
+
+
+
+\section{Basic properties of the blob complex}
+
+\begin{prop} \label{disjunion}
+There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.
+\end{prop}
+\begin{proof}
+Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them
+(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a
+blob diagram $(b_1, b_2)$ on $X \du Y$.
+Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way.
+In the other direction, any blob diagram on $X\du Y$ is equal (up to sign)
+to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines
+a pair of blob diagrams on $X$ and $Y$.
+These two maps are compatible with our sign conventions \nn{say more about this?} and
+with the linear label relations.
+The two maps are inverses of each other.
+\nn{should probably say something about sign conventions for the differential
+in a tensor product of chain complexes; ask Scott}
+\end{proof}
+
+For the next proposition we will temporarily restore $n$-manifold boundary
+conditions to the notation.
+
+Suppose that for all $c \in \cC(\bd B^n)$
+we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$
+of the quotient map
+$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$.
+\nn{always the case if we're working over $\c$}.
+Then
+\begin{prop} \label{bcontract}
+For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$
+is a chain homotopy equivalence
+with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$.
+Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0.
+\end{prop}
+\begin{proof}
+By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map
+$h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$.
+For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding
+an $(i{+}1)$-st blob equal to all of $B^n$.
+In other words, add a new outermost blob which encloses all of the others.
+Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to
+the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$.
+\nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$}
+\end{proof}
+
+(Note that for the above proof to work, we need the linear label relations
+for blob labels.
+Also we need to blob reordering relations (?).)
+
+(Note also that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy
+equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.)
+
+(For fields based on $n$-cats, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$.)
+
+\medskip
+
+As we noted above,
+\begin{prop}
+There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$.
+\qed
+\end{prop}
+
+
+\begin{prop}
+For fixed fields ($n$-cat), $\bc_*$ is a functor from the category
+of $n$-manifolds and diffeomorphisms to the category of chain complexes and
+(chain map) isomorphisms.
+\qed
+\end{prop}
+
+
+In particular,
+\begin{prop} \label{diff0prop}
+There is an action of $\Diff(X)$ on $\bc_*(X)$.
+\qed
+\end{prop}
+
+The above will be greatly strengthened in Section \ref{diffsect}.
+
+\medskip
+
+For the next proposition we will temporarily restore $n$-manifold boundary
+conditions to the notation.
+
+Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$.
+Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$
+with boundary $Z\sgl$.
+Given compatible fields (pictures, boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$,
+we have the blob complex $\bc_*(X; a, b, c)$.
+If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on
+$X$ to get blob diagrams on $X\sgl$:
+
+\begin{prop}
+There is a natural chain map
+\eq{
+ \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl).
+}
+The sum is over all fields $a$ on $Y$ compatible at their
+($n{-}2$-dimensional) boundaries with $c$.
+`Natural' means natural with respect to the actions of diffeomorphisms.
+\qed
+\end{prop}
+
+The above map is very far from being an isomorphism, even on homology.
+This will be fixed in Section \ref{gluesect} below.
+
+An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$
+and $X\sgl = X_1 \cup_Y X_2$.
+(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.)
+For $x_i \in \bc_*(X_i)$, we introduce the notation
+\eq{
+ x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
+}
+Note that we have resumed our habit of omitting boundary labels from the notation.
+
+
+\bigskip
+
+\nn{what else?}
+
+
+
+
+\section{$n=1$ and Hochschild homology}
+
+In this section we analyze the blob complex in dimension $n=1$
+and find that for $S^1$ the homology of the blob complex is the
+Hochschild homology of the category (algebroid) that we started with.
+
+Notation: $HB_i(X) = H_i(\bc_*(X))$.
+
+Let us first note that there is no loss of generality in assuming that our system of
+fields comes from a category.
+(Or maybe (???) there {\it is} a loss of generality.
+Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be
+thought of as the morphisms of a 1-category $C$.
+More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$
+are $A(I; a, b)$, and composition is given by gluing.
+If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change
+and neither does $A(I; a, b) = HB_0(I; a, b)$.
+But what about $HB_i(I; a, b)$ for $i > 0$?
+Might these higher blob homology groups be different?
+Seems unlikely, but I don't feel like trying to prove it at the moment.
+In any case, we'll concentrate on the case of fields based on 1-category
+pictures for the rest of this section.)
+
+(Another question: $\bc_*(I)$ is an $A_\infty$-category.
+How general of an $A_\infty$-category is it?
+Given an arbitrary $A_\infty$-category can one find fields and local relations so
+that $\bc_*(I)$ is in some sense equivalent to the original $A_\infty$-category?
+Probably not, unless we generalize to the case where $n$-morphisms are complexes.)
+
+Continuing...
+
+Let $C$ be a *-1-category.
+Then specializing the definitions from above to the case $n=1$ we have:
+\begin{itemize}
+\item $\cC(pt) = \ob(C)$ .
+\item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$.
+Then an element of $\cC(R; c)$ is a collection of (transversely oriented)
+points in the interior
+of $R$, each labeled by a morphism of $C$.
+The intervals between the points are labeled by objects of $C$, consistent with
+the boundary condition $c$ and the domains and ranges of the point labels.
+\item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by
+composing the morphism labels of the points.
+\item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single
+point (at some standard location) labeled by $x$.
+Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the
+form $y - \chi(e(y))$.
+Thus we can, if we choose, restrict the blob twig labels to things of this form.
+\end{itemize}
+
+We want to show that $HB_*(S^1)$ is naturally isomorphic to the
+Hochschild homology of $C$.
+\nn{Or better that the complexes are homotopic
+or quasi-isomorphic.}
+In order to prove this we will need to extend the blob complex to allow points to also
+be labeled by elements of $C$-$C$-bimodules.
+%Given an interval (1-ball) so labeled, there is an evaluation map to some tensor product
+%(over $C$) of $C$-$C$-bimodules.
+%Define the local relations $U(I; a, b)$ to be the direct sum of the kernels of these maps.
+%Now we can define the blob complex for $S^1$.
+%This complex is the sum of complexes with a fixed cyclic tuple of bimodules present.
+%If $M$ is a $C$-$C$-bimodule, let $G_*(M)$ denote the summand of $\bc_*(S^1)$ corresponding
+%to the cyclic 1-tuple $(M)$.
+%In other words, $G_*(M)$ is a blob-like complex where exactly one point is labeled
+%by an element of $M$ and the remaining points are labeled by morphisms of $C$.
+%It's clear that $G_*(C)$ is isomorphic to the original bimodule-less
+%blob complex for $S^1$.
+%\nn{Is it really so clear? Should say more.}
+
+%\nn{alternative to the above paragraph:}
+Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$.
+We define a blob-like complex $F_*(S^1, (p_i), (M_i))$.
+The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling
+other points.
+The blob twig labels lie in kernels of evaluation maps.
+(The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.)
+Let $F_*(M) = F_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point.
+In other words, fields for $F_*(M)$ have an element of $M$ at the fixed point $*$
+and elements of $C$ at variable other points.
+
+We claim that the homology of $F_*(M)$ is isomorphic to the Hochschild
+homology of $M$.
+\nn{Or maybe we should claim that $M \to F_*(M)$ is the/a derived coend.
+Or maybe that $F_*(M)$ is quasi-isomorphic (or perhaps homotopic) to the Hochschild
+complex of $M$.}
+This follows from the following lemmas:
+\begin{itemize}
+\item $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$.
+\item An exact sequence $0 \to M_1 \to M_2 \to M_3 \to 0$
+gives rise to an exact sequence $0 \to F_*(M_1) \to F_*(M_2) \to F_*(M_3) \to 0$.
+(See below for proof.)
+\item $F_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is
+homotopic to the 0-step complex $C$.
+(See below for proof.)
+\item $F_*(C)$ (here $C$ is wearing its $C$-$C$-bimodule hat) is homotopic to $\bc_*(S^1)$.
+(See below for proof.)
+\end{itemize}
+
+First we show that $F_*(C\otimes C)$ is
+homotopic to the 0-step complex $C$.
+
+Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of
+the point $*$ is $1 \otimes 1 \in C\otimes C$.
+We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism.
+
+Fix a small $\ep > 0$.
+Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$.
+Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex where $B_\ep$ is either disjoint from
+or contained in all blobs, and the two boundary points of $B_\ep$ are not labeled points.
+For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$
+labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$.
+(See Figure xxxx.)
+\nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.}
+
+Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows.
+Let $x \in F^\ep_*$ be a blob diagram.
+If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to
+$x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$.
+If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows.
+Let $y_i$ be the restriction of $z_i$ to $*$.
+Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$,
+and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$.
+Define $j_\ep(x) = \sum x_i$.
+
+Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also.
+
+The key property of $j_\ep$ is
+\eq{
+ \bd j_\ep + j_\ep \bd = \id - \sigma_\ep ,
+}
+where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction of each field
+mentioned in $x \in F^\ep_*$ (call the restriction $y$) with $s_\ep(y)$.
+Note that $\sigma_\ep(x) \in F'$.
+
+If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$
+is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$.
+One strategy would be to try to stitch together various $j_\ep$ for progressively smaller
+$\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$.
+Instead, we'll be less ambitious and just show that
+$F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
+
+If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$
+$x \in F_*^\ep$.
+(This is true for any chain in $F_*(C\otimes C)$, since chains are sums of
+finitely many blob diagrams.)
+Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map
+is surjective on homology.
+If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$
+and
+\eq{
+ \bd x = \bd (\sigma_\ep(y) + j_\ep(x)) .
+}
+Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology.
+This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$.
+
+\medskip
+
+Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob.
+We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence.
+
+First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with
+$S^1$ replaced some (any) neighborhood of $* \in S^1$.
+Then $G''_*$ and $G'_*$ are both contractible.
+For $G'_*$ the proof is the same as in (\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 \nn{need to establish terminology for this; maybe
+in ``basic properties" section above} 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{disjunion}).
+Actually, we need the further (easy) result that the inclusion
+$G''_* \to G'_*$ induces an isomorphism on $H_0$.
+
+Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that
+for all $x \in F'_*$ we have
+\eq{
+ x - \bd h(x) - h(\bd x) \in F''_* .
+}
+Since $F'_0 = F''_0$, we can take $h_0 = 0$.
+Let $x \in F'_1$, with single blob $B \sub S^1$.
+If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$.
+If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$).
+Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$.
+Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$.
+Define $h_1(x) = y$.
+The general case is similar, except that we have to take lower order homotopies into account.
+Let $x \in F'_k$.
+If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$.
+Otherwise, let $B$ be the outermost blob of $x$ containing $*$.
+By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$.
+So $x' \in G'_l$ for some $l \le k$.
+Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$.
+Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$.
+Define $h_k(x) = y \bullet p$.
+This completes the proof that $i: F''_* \to F'_*$ is a homotopy equivalence.
+\nn{need to say above more clearly and settle on notation/terminology}
+
+Finally, we show that $F''_*$ is contractible.
+\nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now}
+Let $x$ be a cycle in $F''_*$.
+The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a
+ball $B \subset S^1$ containing the union of the supports and not containing $*$.
+Adding $B$ as a blob to $x$ gives a contraction.
+\nn{need to say something else in degree zero}
+
+This completes the proof that $F_*(C\otimes C)$ is
+homotopic to the 0-step complex $C$.
+
+\medskip
+
+Next we show that $F_*(C)$ is homotopic \nn{q-isom?} to $\bc_*(S^1)$
+\nn{...}
+
+\bigskip
+
+\nn{still need to prove exactness claim}
+
+\nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex?
+Do we need a map from hoch to blob?
+Does the above exactness and contractibility guarantee such a map without writing it
+down explicitly?
+Probably it's worth writing down an explicit map even if we don't need to.}
+
+
+
+\section{Action of $C_*(\Diff(X))$} \label{diffsect}
+
+Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of
+the space of diffeomorphisms
+of the $n$-manifold $X$ (fixed on $\bd X$).
+For convenience, we will permit the singular cells generating $CD_*(X)$ to be more general
+than simplices --- they can be based on any linear polyhedron.
+\nn{be more restrictive here? does more need to be said?}
+
+\begin{prop} \label{CDprop}
+For each $n$-manifold $X$ there is a chain map
+\eq{
+ e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) .
+}
+On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$
+(Proposition (\ref{diff0prop})).
+For any splitting $X = X_1 \cup X_2$, the following diagram commutes
+\eq{ \xymatrix{
+ CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\
+ CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2)
+ \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} &
+ \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl}
+} }
+Any other map satisfying the above two properties is homotopic to $e_X$.
+\end{prop}
+
+The proof will occupy the remainder of this section.
+
+\medskip
+
+Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$.
+We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all
+$x \notin S$ and $p, q \in P$.
+Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$.
+
+Let $\cU = \{U_\alpha\}$ be an open cover of $X$.
+A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is
+{\it adapted to $\cU$} if there is a factorization
+\eq{
+ P = P_1 \times \cdots \times P_m
+}
+(for some $m \le k$)
+and families of diffeomorphisms
+\eq{
+ f_i : P_i \times X \to X
+}
+such that
+\begin{itemize}
+\item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$;
+\item the $V_i$'s are mutually disjoint;
+\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s,
+where $k_i = \dim(P_i)$; and
+\item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$
+for all $p = (p_1, \ldots, p_m)$.
+\end{itemize}
+A chain $x \in C_k(\Diff(M))$ is (by definition) adapted to $\cU$ if is is the sum
+of singular cells, each of which is adapted to $\cU$.
+
+\begin{lemma} \label{extension_lemma}
+Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$.
+Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
+\end{lemma}
+
+The proof will be given in Section \ref{fam_diff_sect}.
+
+\medskip
+
+Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$
+(e.g.~the support of a blob diagram).
+We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if
+$f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$
+either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$.
+A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells,
+each of which is compatible.
+(Note that we could strengthen the definition of compatibility to incorporate
+a factorization condition, similar to the definition of ``adapted to" above.
+The weaker definition given here will suffice for our needs below.)
+
+\begin{cor} \label{extension_lemma_2}
+Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is compatible with $\{B_j\}$.
+Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$.
+\end{cor}
+\begin{proof}
+This will follow from Lemma \ref{extension_lemma} for
+appropriate choice of cover $\cU = \{U_\alpha\}$.
+Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let
+$V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$.
+Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$
+either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$.
+
+Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$,
+with the (compatible) support of $f_i$ in place of $X$.
+This insures that the resulting homotopy $h_i$ is compatible.
+Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$.
+\nn{actually, need to start with the 0-skeleton of $\bd x$, then 1-skeleton, etc.; fix this}
+\end{proof}
+
+
+
+
+\section{Families of Diffeomorphisms} \label{fam_diff_sect}
+
+
+Lo, the proof of Lemma (\ref{extension_lemma}):
+
+\nn{should this be an appendix instead?}
+
+\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in
+later draft}
+
+\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry
+about boundary later}
+
+Recall that we are given
+an open cover $\cU = \{U_\alpha\}$ and an
+$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$.
+We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
+
+Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$.
+
+As a first approximation to the argument we will eventually make, let's replace $x$
+with a single singular cell
+\eq{
+ f: P \times X \to X .
+}
+Also, we'll ignore for now issues around $\bd P$.
+
+Our homotopy will have the form
+\eqar{
+ F: I \times P \times X &\to& X \\
+ (t, p, x) &\mapsto& f(u(t, p, x), x)
+}
+for some function
+\eq{
+ u : I \times P \times X \to P .
+}
+First we describe $u$, then we argue that it does what we want it to do.
+
+For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$.
+The various $K_\alpha$ should be in general position with respect to each other.
+We will see below that the $K_\alpha$'s need to be sufficiently fine in order
+to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not
+merely a homotopy through maps $X\to X$.
+
+Let $L$ be the union of all the $K_\alpha$'s.
+$L$ is itself a cell decomposition of $P$.
+\nn{next two sentences not needed?}
+To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$,
+where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$.
+Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$.
+
+Let $J$ denote the handle decomposition of $P$ corresponding to $L$.
+Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and,
+more importantly, a $k{-}i$-dimensional normal coordinate.
+
+For each $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$.
+Let $D$ be a $k$-handle of $J$, and let $d$ also denote the corresponding
+$k$-cell of $L$.
+To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s
+which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$.
+
+For $p \in D$ we define
+\eq{
+ u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} .
+}
+(Recall that $P$ is a single linear cell, so the weighted average of points of $P$
+makes sense.)
+
+So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$.
+For handles of $J$ of index less than $k$, we will define $u$ to
+interpolate between the values on $k$-handles defined above.
+
+If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate
+of $E$.
+In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$
+with a $k$-handle.
+Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell
+corresponding to $E$.
+Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$
+adjacent to the $k{-}1$-cell corresponding to $E$.
+For $p \in E$, define
+\eq{
+ u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha}
+ + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) .
+}
+
+In general, for $E$ a $k{-}j$-handle, there is a normal coordinate
+$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron.
+The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$.
+If we triangulate $R$ (without introducing new vertices), we can linearly extend
+a map from the the vertices of $R$ into $P$ to a map of all of $R$ into $P$.
+Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets
+the $k{-}j$-cell corresponding to $E$.
+For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells.
+Now define, for $p \in E$,
+\eq{
+ u(t, p, x) = (1-t)p + t \left(
+ \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha}
+ + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right)
+ \right) .
+}
+Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension
+mentioned above.
+
+This completes the definition of $u: I \times P \times X \to P$.
+
+\medskip
+
+Next we verify that $u$ has the desired properties.
+
+Since $u(0, p, x) = p$ for all $p\in P$ and $x\in X$, $F(0, p, x) = f(p, x)$ for all $p$ and $x$.
+Therefore $F$ is a homotopy from $f$ to something.
+
+Next we show that the the $K_\alpha$'s are sufficiently fine cell decompositions,
+then $F$ is a homotopy through diffeomorphisms.
+We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$.
+We have
+\eq{
+% \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) .
+ \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} .
+}
+Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and
+\nn{bounded away from zero, or something like that}.
+(Recall that $X$ and $P$ are compact.)
+Also, $\pd{f}{p}$ is bounded.
+So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done.
+It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$
+and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s.
+These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine.
+This completes the proof that $F$ is a homotopy through diffeomorphisms.
+
+\medskip
+
+Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$
+is a singular cell adapted to $\cU$.
+This will complete the proof of the lemma.
+\nn{except for boundary issues and the `$P$ is a cell' assumption}
+
+Let $j$ be the codimension of $D$.
+(Or rather, the codimension of its corresponding cell. From now on we will not make a distinction
+between handle and corresponding cell.)
+Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$,
+where the $j_i$'s are the codimensions of the $K_\alpha$
+cells of codimension greater than 0 which intersect to form $D$.
+We will show that
+if the relevant $U_\alpha$'s are disjoint, then
+$F(1, \cdot, \cdot) : D\times X \to X$
+is a product of singular cells of dimensions $j_1, \ldots, j_m$.
+If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular
+cells whose dimensions correspond to a partition of the $j_i$'s.
+We will consider some simple special cases first, then do the general case.
+
+First consider the case $j=0$ (and $m=0$).
+A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$,
+is independent of $p \in P$.
+So the corresponding map $D \to \Diff(X)$ is constant.
+
+Next consider the case $j = 1$ (and $m=1$, $j_1=1$).
+Now Equation yyyy applies.
+We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$.
+It follows that the singular cell $D \to \Diff(X)$ can be written as a product
+of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$.
+The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set.
+
+Next case: $j=2$, $m=1$, $j_1 = 2$.
+This is similar to the previous case, except that the normal bundle is 2-dimensional instead of
+1-dimensional.
+We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell
+and a 2-cell with support $U_\beta$.
+
+Next case: $j=2$, $m=2$, $j_1 = j_2 = 2$.
+In this case the codimension 2 cell $D$ is the intersection of two
+codimension 1 cells, from $K_\beta$ and $K_\gamma$.
+We can write $D = D' \times I \times I$, where the normal coordinates are constant
+on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$.
+If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and
+two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively.
+If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and
+a 2-cell supported on $U_\beta \cup U_\gamma$.
+\nn{need to check that this is true}
+
+\nn{finally, general case...}
+
+\nn{this completes proof}
+
+
+
+
+\section{$A_\infty$ action on the boundary}
+
+
+\section{Gluing} \label{gluesect}
+
+\section{Extension to ...}
+
+(Need to let the input $n$-category $C$ be a graded thing
+(e.g.~DGA or $A_\infty$ $n$-category).)
+
+
+\section{What else?...}
+
+\begin{itemize}
+\item Derive Hochschild standard results from blob point of view?
+\item $n=2$ examples
+\item Kh
+\item dimension $n+1$
+\item should be clear about PL vs Diff; probably PL is better
+(or maybe not)
+\item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category
+\item something about higher derived coend things (derived 2-coend, e.g.)
+\end{itemize}
+
+
+
+\end{document}
+
+
+
+%Recall that for $n$-category picture fields there is an evaluation map
+%$m: \bc_0(B^n; c, c') \to \mor(c, c')$.
+%If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain
+%map $m: \bc_*(B^n; c, c') \to \mor(c, c')$.
+
+
+