# HG changeset patch # User scott@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1202518863 0 # Node ID f4fc8028aacb10954301a0af0dec859ef2cc0b5d ... diff -r 000000000000 -r f4fc8028aacb blob1.pdf Binary file blob1.pdf has changed diff -r 000000000000 -r f4fc8028aacb blob1.tex --- /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 $m: \bc_*(B^n; c, c') \to \mor(c, c')$. + + +