author | kevin@6e1638ff-ae45-0410-89bd-df963105f760 |
Tue, 24 Jun 2008 19:46:06 +0000 | |
changeset 17 | c73e8beb4a20 |
parent 16 | 9ae2fd41b903 |
child 18 | aac9fd8d6bc6 |
permissions | -rw-r--r-- |
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\documentclass[11pt,leqno]{amsart} |
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\newcommand{\pathtotrunk}{./} |
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\input{text/article_preamble.tex} |
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\input{text/top_matter.tex} |
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% test edit #3 |
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%%%%% excerpts from my include file of standard macros |
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\def\bc{{\mathcal B}} |
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\def\z{\mathbb{Z}} |
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\def\r{\mathbb{R}} |
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\def\c{\mathbb{C}} |
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\def\t{\mathbb{T}} |
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\def\du{\sqcup} |
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\def\bd{\partial} |
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\def\sub{\subset} |
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\def\sup{\supset} |
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%\def\setmin{\smallsetminus} |
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\def\setmin{\setminus} |
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\def\ep{\epsilon} |
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\def\sgl{_\mathrm{gl}} |
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\def\op{^\mathrm{op}} |
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\def\deq{\stackrel{\mathrm{def}}{=}} |
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\def\pd#1#2{\frac{\partial #1}{\partial #2}} |
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\def\nn#1{{{\it \small [#1]}}} |
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% equations |
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\newcommand{\eq}[1]{\begin{displaymath}#1\end{displaymath}} |
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\newcommand{\eqar}[1]{\begin{eqnarray*}#1\end{eqnarray*}} |
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\newcommand{\eqspl}[1]{\begin{displaymath}\begin{split}#1\end{split}\end{displaymath}} |
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% tricky way to iterate macros over a list |
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\def\semicolon{;} |
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\def\applytolist#1{ |
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\expandafter\def\csname multi#1\endcsname##1{ |
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\def\multiack{##1}\ifx\multiack\semicolon |
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\def\next{\relax} |
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\else |
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\csname #1\endcsname{##1} |
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\def\next{\csname multi#1\endcsname} |
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\fi |
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\next} |
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\csname multi#1\endcsname} |
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% \def\cA{{\cal A}} for A..Z |
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\def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}} |
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\applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM; |
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% \DeclareMathOperator{\pr}{pr} etc. |
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\def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} |
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\applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Diff}{sign}{supp}; |
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%%%%%% end excerpt |
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\title{Blob Homology} |
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\begin{document} |
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\makeatletter |
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\@addtoreset{equation}{section} |
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\gdef\theequation{\thesection.\arabic{equation}} |
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\makeatother |
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\maketitle |
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\section{Introduction} |
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(motivation, summary/outline, etc.) |
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(motivation: |
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(1) restore exactness in pictures-mod-relations; |
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(1') add relations-amongst-relations etc. to pictures-mod-relations; |
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(2) want answer independent of handle decomp (i.e. don't |
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just go from coend to derived coend (e.g. Hochschild homology)); |
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(3) ... |
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) |
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\section{Definitions} |
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\subsection{Fields} |
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Fix a top dimension $n$. |
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A {\it system of fields} |
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\nn{maybe should look for better name; but this is the name I use elsewhere} |
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is a collection of functors $\cC$ from manifolds of dimension $n$ or less |
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to sets. |
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These functors must satisfy various properties (see KW TQFT notes for details). |
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For example: |
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there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$; |
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there is a restriction map $\cC(X) \to \cC(\bd X)$; |
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gluing manifolds corresponds to fibered products of fields; |
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given a field $c \in \cC(Y)$ there is a ``product field" |
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$c\times I \in \cC(Y\times I)$; ... |
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\nn{should eventually include full details of definition of fields.} |
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\nn{note: probably will suppress from notation the distinction |
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between fields and their (orientation-reversal) duals} |
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\nn{remark that if top dimensional fields are not already linear |
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then we will soon linearize them(?)} |
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The definition of a system of fields is intended to generalize |
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the relevant properties of the following two examples of fields. |
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The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$ |
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is a manifold of dimension $n$ or less) to be the set of |
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all maps from $X$ to $B$. |
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The second example will take longer to explain. |
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Given an $n$-category $C$ with the right sort of duality |
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(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
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we can construct a system of fields as follows. |
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Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
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with codimension $i$ cells labeled by $i$-morphisms of $C$. |
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We'll spell this out for $n=1,2$ and then describe the general case. |
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If $X$ has boundary, we require that the cell decompositions are in general |
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position with respect to the boundary --- the boundary intersects each cell |
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transversely, so cells meeting the boundary are mere half-cells. |
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Put another way, the cell decompositions we consider are dual to standard cell |
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decompositions of $X$. |
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We will always assume that our $n$-categories have linear $n$-morphisms. |
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For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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an object (0-morphism) of the 1-category $C$. |
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A field on a 1-manifold $S$ consists of |
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\begin{itemize} |
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\item A cell decomposition of $S$ (equivalently, a finite collection |
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of points in the interior of $S$); |
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\item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
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by an object (0-morphism) of $C$; |
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\item a transverse orientation of each 0-cell, thought of as a choice of |
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``domain" and ``range" for the two adjacent 1-cells; and |
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\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
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domain and range determined by the transverse orientation and the labelings of the 1-cells. |
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\end{itemize} |
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If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
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of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
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interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
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of the algebra. |
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\medskip |
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For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) |
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that are common in the literature. |
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We describe these carefully here. |
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A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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an object of the 2-category $C$. |
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A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
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A field on a 2-manifold $Y$ consists of |
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\begin{itemize} |
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\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
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that each component of the complement is homeomorphic to a disk); |
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\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
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by a 0-morphism of $C$; |
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\item a transverse orientation of each 1-cell, thought of as a choice of |
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``domain" and ``range" for the two adjacent 2-cells; |
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\item a labeling of each 1-cell by a 1-morphism of $C$, with |
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domain and range determined by the transverse orientation of the 1-cell |
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and the labelings of the 2-cells; |
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\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
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of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
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to $\pm 1 \in S^1$; and |
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\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
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determined by the labelings of the 1-cells and the parameterizations of the previous |
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bullet. |
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\end{itemize} |
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\nn{need to say this better; don't try to fit everything into the bulleted list} |
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For general $n$, a field on a $k$-manifold $X^k$ consists of |
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\begin{itemize} |
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\item A cell decomposition of $X$; |
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\item an explicit general position homeomorphism from the link of each $j$-cell |
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to the boundary of the standard $(k-j)$-dimensional bihedron; and |
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\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
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domain and range determined by the labelings of the link of $j$-cell. |
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\end{itemize} |
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%\nn{next definition might need some work; I think linearity relations should |
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%be treated differently (segregated) from other local relations, but I'm not sure |
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%the next definition is the best way to do it} |
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\medskip |
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For top dimensional ($n$-dimensional) manifolds, we're actually interested |
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in the linearized space of fields. |
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By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is |
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the vector space of finite |
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linear combinations of fields on $X$. |
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If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$. |
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Thus the restriction (to boundary) maps are well defined because we never |
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take linear combinations of fields with differing boundary conditions. |
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In some cases we don't linearize the default way; instead we take the |
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spaces $\cC_l(X; a)$ to be part of the data for the system of fields. |
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In particular, for fields based on linear $n$-category pictures we linearize as follows. |
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Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
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obvious relations on 0-cell labels. |
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More specifically, let $L$ be a cell decomposition of $X$ |
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and let $p$ be a 0-cell of $L$. |
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Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
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$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
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Then the subspace $K$ is generated by things of the form |
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$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
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to infer the meaning of $\alpha_{\lambda c + d}$. |
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Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
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\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
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will do something similar below; in general, whenever a label lives in a linear |
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space we do something like this; ? say something about tensor |
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product of all the linear label spaces? Yes:} |
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For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
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Define an ``almost-field" to be a field without labels on the 0-cells. |
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(Recall that 0-cells are labeled by $n$-morphisms.) |
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To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
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space determined by the labeling of the link of the 0-cell. |
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(If the 0-cell were labeled, the label would live in this space.) |
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We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
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We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the |
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above tensor products. |
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\subsection{Local relations} |
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Let $B^n$ denote the standard $n$-ball. |
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A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ |
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(for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties. |
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\nn{Roughly, these are (1) the local relations imply (extended) isotopy; |
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(2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and |
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(3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$). |
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See KW TQFT notes for details. Need to transfer details to here.} |
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For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$, |
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where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
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For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map |
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$\cC_l(B^n; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
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domain and range. |
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\nn{maybe examples of local relations before general def?} |
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Note that the $Y$ is an $n$-manifold which is merely homeomorphic to the standard $B^n$, |
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then any homeomorphism $B^n \to Y$ induces the same local subspaces for $Y$. |
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We'll denote these by $U(Y; c) \sub \cC_l(Y; c)$, $c \in \cC(\bd Y)$. |
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\nn{Is this true in high (smooth) dimensions? Self-diffeomorphisms of $B^n$ |
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rel boundary might not be isotopic to the identity. OK for PL and TOP?} |
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Given a system of fields and local relations, we define the skein space |
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$A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
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the $n$-manifold $Y$ modulo local relations. |
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The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
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is defined to be the dual of $A(Y; c)$. |
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(See KW TQFT notes or xxxx for details.) |
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The blob complex is in some sense the derived version of $A(Y; c)$. |
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\subsection{The blob complex} |
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Let $X$ be an $n$-manifold. |
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Assume a fixed system of fields. |
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In this section we will usually suppress boundary conditions on $X$ from the notation |
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(e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$). |
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We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
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submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
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$\overline{X \setmin Y}$. |
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We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case. |
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Define $\bc_0(X) = \cC_l(X)$. |
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(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cC_l(X; c)$. |
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We'll omit this sort of detail in the rest of this section.) |
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In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
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$\bc_1(X)$ is the space of all local relations that can be imposed on $\bc_0(X)$. |
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More specifically, define a 1-blob diagram to consist of |
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\begin{itemize} |
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\item An embedded closed ball (``blob") $B \sub X$. |
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%\nn{Does $B$ need a homeo to the standard $B^n$? I don't think so. |
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%(See note in previous subsection.)} |
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%\item A field (boundary condition) $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$. |
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\item A field $r \in \cC(X \setmin B; c)$ |
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(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
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\item A local relation field $u \in U(B; c)$ |
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(same $c$ as previous bullet). |
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\end{itemize} |
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%(Note that the field $c$ is determined (implicitly) as the boundary of $u$ and/or $r$, |
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%so we will omit $c$ from the notation.) |
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Define $\bc_1(X)$ to be the space of all finite linear combinations of |
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1-blob diagrams, modulo the simple relations relating labels of 0-cells and |
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also the label ($u$ above) of the blob. |
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\nn{maybe spell this out in more detail} |
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(See xxxx above.) |
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\nn{maybe restate this in terms of direct sums of tensor products.} |
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There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear |
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combination of fields on $X$ obtained by gluing $r$ to $u$. |
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In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
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just erasing the blob from the picture |
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(but keeping the blob label $u$). |
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Note that the skein space $A(X)$ |
0 | 328 |
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
329 |
||
330 |
$\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$. |
|
8 | 331 |
More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
0 | 332 |
2-blob diagrams (defined below), modulo the usual linear label relations. |
333 |
\nn{and also modulo blob reordering relations?} |
|
334 |
||
335 |
\nn{maybe include longer discussion to motivate the two sorts of 2-blob diagrams} |
|
336 |
||
337 |
There are two types of 2-blob diagram: disjoint and nested. |
|
338 |
A disjoint 2-blob diagram consists of |
|
339 |
\begin{itemize} |
|
340 |
\item A pair of disjoint closed balls (blobs) $B_0, B_1 \sub X$. |
|
341 |
%\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
|
342 |
\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
|
343 |
(where $c_i \in \cC(\bd B_i)$). |
|
344 |
\item Local relation fields $u_i \in U(B_i; c_i)$. |
|
345 |
\end{itemize} |
|
346 |
Define $\bd(B_0, B_1, r, u_0, u_1) = (B_1, ru_0, u_1) - (B_0, ru_1, u_0) \in \bc_1(X)$. |
|
347 |
In other words, the boundary of a disjoint 2-blob diagram |
|
348 |
is the sum (with alternating signs) |
|
349 |
of the two ways of erasing one of the blobs. |
|
350 |
It's easy to check that $\bd^2 = 0$. |
|
351 |
||
352 |
A nested 2-blob diagram consists of |
|
353 |
\begin{itemize} |
|
354 |
\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
|
355 |
\item A field $r \in \cC(X \setmin B_0; c_0)$ |
|
356 |
(for some $c_0 \in \cC(\bd B_0)$). |
|
357 |
Let $r = r_1 \cup r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
|
358 |
(for some $c_1 \in \cC(B_1)$) and |
|
359 |
$r' \in \cC(X \setmin B_1; c_1)$. |
|
360 |
\item A local relation field $u_0 \in U(B_0; c_0)$. |
|
361 |
\end{itemize} |
|
362 |
Define $\bd(B_0, B_1, r, u_0) = (B_1, r', r_1u_0) - (B_0, r, u_0)$. |
|
363 |
Note that xxxx above guarantees that $r_1u_0 \in U(B_1)$. |
|
364 |
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
|
365 |
sum of the two ways of erasing one of the blobs. |
|
366 |
If we erase the inner blob, the outer blob inherits the label $r_1u_0$. |
|
367 |
||
368 |
Now for the general case. |
|
369 |
A $k$-blob diagram consists of |
|
370 |
\begin{itemize} |
|
371 |
\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
|
372 |
For each $i$ and $j$, we require that either $B_i \cap B_j$ is empty or |
|
373 |
$B_i \sub B_j$ or $B_j \sub B_i$. |
|
374 |
(The case $B_i = B_j$ is allowed. |
|
375 |
If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
|
376 |
If a blob has no other blobs strictly contained in it, we call it a twig blob. |
|
377 |
%\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
|
378 |
%(These are implied by the data in the next bullets, so we usually |
|
379 |
%suppress them from the notation.) |
|
380 |
%$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
|
381 |
%if the latter space is not empty. |
|
382 |
\item A field $r \in \cC(X \setmin B^t; c^t)$, |
|
383 |
where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$. |
|
384 |
\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
|
385 |
where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
|
386 |
If $B_i = B_j$ then $u_i = u_j$. |
|
387 |
\end{itemize} |
|
388 |
||
389 |
We define $\bc_k(X)$ to be the vector space of all finite linear combinations |
|
390 |
of $k$-blob diagrams, modulo the linear label relations and |
|
391 |
blob reordering relations defined in the remainder of this paragraph. |
|
392 |
Let $x$ be a blob diagram with one undetermined $n$-morphism label. |
|
393 |
The unlabeled entity is either a blob or a 0-cell outside of the twig blobs. |
|
394 |
Let $a$ and $b$ be two possible $n$-morphism labels for |
|
395 |
the unlabeled blob or 0-cell. |
|
396 |
Let $c = \lambda a + b$. |
|
397 |
Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly. |
|
398 |
Then we impose the relation |
|
399 |
\eq{ |
|
8 | 400 |
x_c = \lambda x_a + x_b . |
0 | 401 |
} |
402 |
\nn{should do this in terms of direct sums of tensor products} |
|
403 |
Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$ |
|
404 |
of their blob labelings. |
|
405 |
Then we impose the relation |
|
406 |
\eq{ |
|
8 | 407 |
x = \sign(\pi) x' . |
0 | 408 |
} |
409 |
||
410 |
(Alert readers will have noticed that for $k=2$ our definition |
|
411 |
of $\bc_k(X)$ is slightly different from the previous definition |
|
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|
412 |
of $\bc_2(X)$ --- we did not impose the reordering relations. |
0 | 413 |
The general definition takes precedence; |
414 |
the earlier definition was simplified for purposes of exposition.) |
|
415 |
||
416 |
The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
|
417 |
Let $b = (\{B_i\}, r, \{u_j\})$ be a $k$-blob diagram. |
|
418 |
Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
|
419 |
If $B_j$ is not a twig blob, this involves only decrementing |
|
420 |
the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
|
421 |
If $B_j$ is a twig blob, we have to assign new local relation labels |
|
422 |
if removing $B_j$ creates new twig blobs. |
|
423 |
If $B_l$ becomes a twig after removing $B_j$, then set $u_l = r_lu_j$, |
|
424 |
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
|
425 |
Finally, define |
|
426 |
\eq{ |
|
8 | 427 |
\bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
0 | 428 |
} |
429 |
The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
|
430 |
Thus we have a chain complex. |
|
431 |
||
432 |
\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
|
433 |
||
434 |
||
8 | 435 |
\nn{TO DO: |
436 |
expand definition to handle DGA and $A_\infty$ versions of $n$-categories; |
|
4
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|
437 |
relations to Chas-Sullivan string stuff} |
0 | 438 |
|
439 |
||
440 |
||
441 |
\section{Basic properties of the blob complex} |
|
442 |
||
443 |
\begin{prop} \label{disjunion} |
|
444 |
There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
|
445 |
\end{prop} |
|
446 |
\begin{proof} |
|
447 |
Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
|
8 | 448 |
(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
0 | 449 |
blob diagram $(b_1, b_2)$ on $X \du Y$. |
450 |
Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
|
451 |
In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
|
452 |
to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
|
453 |
a pair of blob diagrams on $X$ and $Y$. |
|
454 |
These two maps are compatible with our sign conventions \nn{say more about this?} and |
|
455 |
with the linear label relations. |
|
456 |
The two maps are inverses of each other. |
|
457 |
\nn{should probably say something about sign conventions for the differential |
|
458 |
in a tensor product of chain complexes; ask Scott} |
|
459 |
\end{proof} |
|
460 |
||
461 |
For the next proposition we will temporarily restore $n$-manifold boundary |
|
462 |
conditions to the notation. |
|
463 |
||
8 | 464 |
Suppose that for all $c \in \cC(\bd B^n)$ |
465 |
we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
|
0 | 466 |
of the quotient map |
467 |
$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
|
468 |
\nn{always the case if we're working over $\c$}. |
|
469 |
Then |
|
470 |
\begin{prop} \label{bcontract} |
|
471 |
For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
|
472 |
is a chain homotopy equivalence |
|
473 |
with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
|
474 |
Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0. |
|
475 |
\end{prop} |
|
476 |
\begin{proof} |
|
477 |
By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map |
|
478 |
$h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$. |
|
479 |
For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding |
|
480 |
an $(i{+}1)$-st blob equal to all of $B^n$. |
|
481 |
In other words, add a new outermost blob which encloses all of the others. |
|
482 |
Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
|
483 |
the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
|
484 |
\nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$} |
|
485 |
\end{proof} |
|
486 |
||
8 | 487 |
(Note that for the above proof to work, we need the linear label relations |
0 | 488 |
for blob labels. |
489 |
Also we need to blob reordering relations (?).) |
|
490 |
||
491 |
(Note also that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
|
492 |
equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.) |
|
493 |
||
494 |
(For fields based on $n$-cats, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$.) |
|
495 |
||
496 |
\medskip |
|
497 |
||
498 |
As we noted above, |
|
499 |
\begin{prop} |
|
500 |
There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
|
501 |
\qed |
|
502 |
\end{prop} |
|
503 |
||
504 |
||
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|
505 |
% oops -- duplicate |
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|
506 |
|
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|
507 |
%\begin{prop} \label{functorialprop} |
8599e156a169
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
3
diff
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|
508 |
%The assignment $X \mapsto \bc_*(X)$ extends to a functor from the category of |
8599e156a169
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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diff
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|
509 |
%$n$-manifolds and homeomorphisms to the category of chain complexes and linear isomorphisms. |
8599e156a169
misc. edit, nothing major
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parents:
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diff
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|
510 |
%\end{prop} |
8599e156a169
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3
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changeset
|
511 |
|
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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3
diff
changeset
|
512 |
%\begin{proof} |
8599e156a169
misc. edit, nothing major
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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3
diff
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|
513 |
%Obvious. |
8599e156a169
misc. edit, nothing major
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
514 |
%\end{proof} |
8599e156a169
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3
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|
515 |
|
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misc. edit, nothing major
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parents:
3
diff
changeset
|
516 |
%\nn{need to same something about boundaries and boundary conditions above. |
8599e156a169
misc. edit, nothing major
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
3
diff
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|
517 |
%maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.} |
8599e156a169
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|
518 |
|
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|
519 |
|
0 | 520 |
\begin{prop} |
521 |
For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
|
8 | 522 |
of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
0 | 523 |
(chain map) isomorphisms. |
524 |
\qed |
|
525 |
\end{prop} |
|
526 |
||
4
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changeset
|
527 |
\nn{need to same something about boundaries and boundary conditions above. |
8599e156a169
misc. edit, nothing major
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|
528 |
maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.} |
8599e156a169
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|
529 |
|
0 | 530 |
|
531 |
In particular, |
|
532 |
\begin{prop} \label{diff0prop} |
|
533 |
There is an action of $\Diff(X)$ on $\bc_*(X)$. |
|
534 |
\qed |
|
535 |
\end{prop} |
|
536 |
||
537 |
The above will be greatly strengthened in Section \ref{diffsect}. |
|
538 |
||
539 |
\medskip |
|
540 |
||
541 |
For the next proposition we will temporarily restore $n$-manifold boundary |
|
542 |
conditions to the notation. |
|
543 |
||
544 |
Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
|
545 |
Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
|
546 |
with boundary $Z\sgl$. |
|
547 |
Given compatible fields (pictures, boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
|
548 |
we have the blob complex $\bc_*(X; a, b, c)$. |
|
549 |
If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
|
550 |
$X$ to get blob diagrams on $X\sgl$: |
|
551 |
||
552 |
\begin{prop} |
|
553 |
There is a natural chain map |
|
554 |
\eq{ |
|
8 | 555 |
\gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
0 | 556 |
} |
8 | 557 |
The sum is over all fields $a$ on $Y$ compatible at their |
0 | 558 |
($n{-}2$-dimensional) boundaries with $c$. |
559 |
`Natural' means natural with respect to the actions of diffeomorphisms. |
|
560 |
\qed |
|
561 |
\end{prop} |
|
562 |
||
563 |
The above map is very far from being an isomorphism, even on homology. |
|
564 |
This will be fixed in Section \ref{gluesect} below. |
|
565 |
||
566 |
An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
|
567 |
and $X\sgl = X_1 \cup_Y X_2$. |
|
568 |
(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
|
569 |
For $x_i \in \bc_*(X_i)$, we introduce the notation |
|
570 |
\eq{ |
|
8 | 571 |
x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
0 | 572 |
} |
573 |
Note that we have resumed our habit of omitting boundary labels from the notation. |
|
574 |
||
575 |
||
576 |
\bigskip |
|
577 |
||
578 |
\nn{what else?} |
|
579 |
||
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|
580 |
\section{Hochschild homology when $n=1$} |
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scott@6e1638ff-ae45-0410-89bd-df963105f760
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|
581 |
\label{sec:hochschild} |
7340ab80db25
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|
582 |
\input{text/hochschild} |
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|
583 |
|
0 | 584 |
\section{Action of $C_*(\Diff(X))$} \label{diffsect} |
585 |
||
586 |
Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of |
|
587 |
the space of diffeomorphisms |
|
588 |
of the $n$-manifold $X$ (fixed on $\bd X$). |
|
589 |
For convenience, we will permit the singular cells generating $CD_*(X)$ to be more general |
|
590 |
than simplices --- they can be based on any linear polyhedron. |
|
591 |
\nn{be more restrictive here? does more need to be said?} |
|
592 |
||
593 |
\begin{prop} \label{CDprop} |
|
594 |
For each $n$-manifold $X$ there is a chain map |
|
595 |
\eq{ |
|
8 | 596 |
e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) . |
0 | 597 |
} |
598 |
On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$ |
|
599 |
(Proposition (\ref{diff0prop})). |
|
600 |
For any splitting $X = X_1 \cup X_2$, the following diagram commutes |
|
601 |
\eq{ \xymatrix{ |
|
8 | 602 |
CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\ |
603 |
CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
|
604 |
\ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & |
|
605 |
\bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl} |
|
0 | 606 |
} } |
607 |
Any other map satisfying the above two properties is homotopic to $e_X$. |
|
608 |
\end{prop} |
|
609 |
||
16
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|
610 |
\nn{Should say something stronger about uniqueness. |
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diff
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|
611 |
Something like: there is |
9ae2fd41b903
begin reworking/completion of evaluation map stuff
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diff
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|
612 |
a contractible subcomplex of the complex of chain maps |
9ae2fd41b903
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|
613 |
$CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), |
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614 |
and all choices in the construction lie in the 0-cells of this |
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615 |
contractible subcomplex. |
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|
616 |
Or maybe better to say any two choices are homotopic, and |
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617 |
any two homotopies and second order homotopic, and so on.} |
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|
618 |
|
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|
619 |
\nn{Also need to say something about associativity. |
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|
620 |
Put it in the above prop or make it a separate prop? |
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|
621 |
I lean toward the latter.} |
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|
622 |
\medskip |
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623 |
|
0 | 624 |
The proof will occupy the remainder of this section. |
625 |
||
626 |
\medskip |
|
627 |
||
628 |
Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. |
|
629 |
We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
|
630 |
$x \notin S$ and $p, q \in P$. |
|
631 |
Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
|
632 |
||
633 |
Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
|
634 |
A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is |
|
635 |
{\it adapted to $\cU$} if there is a factorization |
|
636 |
\eq{ |
|
8 | 637 |
P = P_1 \times \cdots \times P_m |
0 | 638 |
} |
639 |
(for some $m \le k$) |
|
640 |
and families of diffeomorphisms |
|
641 |
\eq{ |
|
8 | 642 |
f_i : P_i \times X \to X |
0 | 643 |
} |
8 | 644 |
such that |
0 | 645 |
\begin{itemize} |
646 |
\item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$; |
|
647 |
\item the $V_i$'s are mutually disjoint; |
|
8 | 648 |
\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
0 | 649 |
where $k_i = \dim(P_i)$; and |
7
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|
650 |
\item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot) \circ g$ |
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|
651 |
for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
0 | 652 |
\end{itemize} |
7
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|
653 |
A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum |
0 | 654 |
of singular cells, each of which is adapted to $\cU$. |
655 |
||
656 |
\begin{lemma} \label{extension_lemma} |
|
657 |
Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
|
658 |
Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
659 |
\end{lemma} |
|
660 |
||
661 |
The proof will be given in Section \ref{fam_diff_sect}. |
|
662 |
||
663 |
\medskip |
|
664 |
||
16
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|
665 |
The strategy for the proof of Proposition \ref{CDprop} is as follows. |
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|
666 |
We will identify a subcomplex |
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|
667 |
\[ |
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|
668 |
G_* \sub CD_*(X) \otimes \bc_*(X) |
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|
669 |
\] |
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|
670 |
on which the evaluation map is uniquely determined (up to homotopy) by the conditions |
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|
671 |
in \ref{CDprop}. |
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|
672 |
We then show that the inclusion of $G_*$ into the full complex |
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|
673 |
is an equivalence in the appropriate sense. |
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|
674 |
\nn{need to be more specific here} |
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|
675 |
|
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|
676 |
Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
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|
677 |
Roughly speaking, $p\otimes b$ is in $G_*$ if each component $V$ of the support of $p$ |
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|
678 |
intersects at most one blob $B$ of $b$. |
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|
679 |
Since $V \cup B$ might not itself be a ball, we need a more careful and complicated definition. |
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|
680 |
Choose a metric for $X$. |
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|
681 |
We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that |
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|
682 |
$N_\epsilon(b) \cup \supp(p)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon |
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|
683 |
neighborhood of the support of $b$. |
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|
684 |
\nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.} |
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|
685 |
|
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|
686 |
\nn{need to worry about case where the intrinsic support of $p$ is not a union of balls} |
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|
687 |
|
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|
688 |
\nn{need to eventually show independence of choice of metric. maybe there's a better way than |
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|
689 |
choosing a metric. perhaps just choose a nbd of each ball, but I think I see problems |
17
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|
690 |
with that as well. |
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|
691 |
the bottom line is that we need a scheme for choosing unions of balls |
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|
692 |
which satisfies the $C$, $C'$, $C''$ claim made a few paragraphs below.} |
16
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|
693 |
|
17
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|
694 |
Next we define the evaluation map $e_X$ on $G_*$. |
16
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|
695 |
We'll proceed inductively on $G_i$. |
17
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|
696 |
The induction starts on $G_0$, where the evaluation map is determined |
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|
697 |
by the action of $\Diff(X)$ on $\bc_*(X)$ |
16
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|
698 |
because $G_0 \sub CD_0\otimes \bc_0$. |
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|
699 |
Assume we have defined the evaluation map up to $G_{k-1}$ and |
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|
700 |
let $p\otimes b$ be a generator of $G_k$. |
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|
701 |
Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$. |
17
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|
702 |
There is a factorization $p = p' \circ g$, where $g\in \Diff(X)$ and $p'$ is a family of diffeomorphisms which is the identity outside of $C$. |
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|
703 |
Let $b = b'\bullet b''$, where $b' \in \bc_*(C)$ and $b'' \in \bc_0(X\setmin C)$. |
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|
704 |
We may assume inductively that $e_X(\bd(p\otimes b))$ has the form $x\bullet g(b'')$, where |
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|
705 |
$x \in \bc_*(g(C))$. |
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|
706 |
Since $\bc_*(g(C))$ is contractible, there exists $y \in \bc_*(g(C))$ such that $\bd y = x$. |
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|
707 |
\nn{need to say more if degree of $x$ is 0} |
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|
708 |
Define $e_X(p\otimes b) = y\bullet g(b'')$. |
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|
709 |
|
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|
710 |
We now show that $e_X$ on $G_*$ is, up to homotopy, independent of the various choices made. |
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|
711 |
If we make a different series of choice of the chain $y$ in the previous paragraph, |
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|
712 |
we can inductively construct a homotopy between the two sets of choices, |
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|
713 |
again relying on the contractibility of $\bc_*(g(G))$. |
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|
714 |
A similar argument shows that this homotopy is unique up to second order homotopy, and so on. |
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|
715 |
|
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|
716 |
Given a different set of choices $\{C'\}$ of the unions of balls $\{C\}$, |
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|
717 |
we can find a third set of choices $\{C''\}$ such that $C, C' \sub C''$. |
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|
718 |
The argument now proceeds as in the previous paragraph. |
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|
719 |
\nn{should maybe say more here; also need to back up claim about third set of choices} |
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|
720 |
|
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|
721 |
Next we show that given $x \in CD_*(X) \otimes \bc_*(X)$ with $\bd x \in G_*$, there exists |
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|
722 |
a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and |
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|
723 |
this homotopy are unique up to iterated homotopy. |
16
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|
724 |
|
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|
725 |
|
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|
726 |
|
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|
727 |
|
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|
728 |
|
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|
729 |
\medskip |
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|
730 |
\hrule |
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|
731 |
\medskip |
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|
732 |
\hrule |
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|
733 |
\medskip |
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|
734 |
\nn{older stuff:} |
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|
735 |
|
8 | 736 |
Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$ |
0 | 737 |
(e.g.~the support of a blob diagram). |
738 |
We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if |
|
739 |
$f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$ |
|
740 |
either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$. |
|
8 | 741 |
A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells, |
0 | 742 |
each of which is compatible. |
743 |
(Note that we could strengthen the definition of compatibility to incorporate |
|
744 |
a factorization condition, similar to the definition of ``adapted to" above. |
|
745 |
The weaker definition given here will suffice for our needs below.) |
|
746 |
||
747 |
\begin{cor} \label{extension_lemma_2} |
|
748 |
Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is compatible with $\{B_j\}$. |
|
749 |
Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$. |
|
750 |
\end{cor} |
|
751 |
\begin{proof} |
|
8 | 752 |
This will follow from Lemma \ref{extension_lemma} for |
0 | 753 |
appropriate choice of cover $\cU = \{U_\alpha\}$. |
754 |
Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let |
|
755 |
$V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$. |
|
756 |
Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$ |
|
757 |
either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$. |
|
758 |
||
8 | 759 |
Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$, |
0 | 760 |
with the (compatible) support of $f_i$ in place of $X$. |
761 |
This insures that the resulting homotopy $h_i$ is compatible. |
|
762 |
Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$. |
|
763 |
\nn{actually, need to start with the 0-skeleton of $\bd x$, then 1-skeleton, etc.; fix this} |
|
764 |
\end{proof} |
|
765 |
||
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|
766 |
\medskip |
0 | 767 |
|
7
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|
768 |
((argument continues roughly as follows: up to homotopy, there is only one way to define $e_X$ |
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|
769 |
on compatible $x\otimes y \in CD_*(X)\otimes \bc_*(X)$. |
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|
770 |
This is because $x$ is the gluing of $x'$ and $x''$, where $x'$ has degree zero and is defined on |
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|
771 |
the complement of the $D_i$'s, and $x''$ is defined on the $D_i$'s. |
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|
772 |
We have no choice on $x'$, since we already know the map on 0-parameter families of diffeomorphisms. |
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|
773 |
We have no choice, up to homotopy, on $x''$, since the target chain complex is contractible.)) |
0 | 774 |
|
775 |
||
776 |
\section{Families of Diffeomorphisms} \label{fam_diff_sect} |
|
777 |
||
778 |
||
779 |
Lo, the proof of Lemma (\ref{extension_lemma}): |
|
780 |
||
781 |
\nn{should this be an appendix instead?} |
|
782 |
||
783 |
\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
|
784 |
later draft} |
|
785 |
||
786 |
\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
|
787 |
about boundary later} |
|
788 |
||
8 | 789 |
Recall that we are given |
0 | 790 |
an open cover $\cU = \{U_\alpha\}$ and an |
791 |
$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
|
792 |
We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
793 |
||
794 |
Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. |
|
795 |
||
796 |
As a first approximation to the argument we will eventually make, let's replace $x$ |
|
8 | 797 |
with a single singular cell |
0 | 798 |
\eq{ |
8 | 799 |
f: P \times X \to X . |
0 | 800 |
} |
801 |
Also, we'll ignore for now issues around $\bd P$. |
|
802 |
||
803 |
Our homotopy will have the form |
|
804 |
\eqar{ |
|
8 | 805 |
F: I \times P \times X &\to& X \\ |
806 |
(t, p, x) &\mapsto& f(u(t, p, x), x) |
|
0 | 807 |
} |
808 |
for some function |
|
809 |
\eq{ |
|
8 | 810 |
u : I \times P \times X \to P . |
0 | 811 |
} |
812 |
First we describe $u$, then we argue that it does what we want it to do. |
|
813 |
||
814 |
For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. |
|
815 |
The various $K_\alpha$ should be in general position with respect to each other. |
|
816 |
We will see below that the $K_\alpha$'s need to be sufficiently fine in order |
|
817 |
to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not |
|
818 |
merely a homotopy through maps $X\to X$. |
|
819 |
||
820 |
Let $L$ be the union of all the $K_\alpha$'s. |
|
821 |
$L$ is itself a cell decomposition of $P$. |
|
822 |
\nn{next two sentences not needed?} |
|
823 |
To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$, |
|
824 |
where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$. |
|
825 |
Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$. |
|
826 |
||
827 |
Let $J$ denote the handle decomposition of $P$ corresponding to $L$. |
|
828 |
Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and, |
|
829 |
more importantly, a $k{-}i$-dimensional normal coordinate. |
|
830 |
||
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|
831 |
For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$. |
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|
832 |
Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding |
0 | 833 |
$k$-cell of $L$. |
834 |
To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
|
835 |
which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
|
836 |
||
837 |
For $p \in D$ we define |
|
838 |
\eq{ |
|
8 | 839 |
u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
0 | 840 |
} |
841 |
(Recall that $P$ is a single linear cell, so the weighted average of points of $P$ |
|
842 |
makes sense.) |
|
843 |
||
844 |
So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. |
|
8 | 845 |
For handles of $J$ of index less than $k$, we will define $u$ to |
0 | 846 |
interpolate between the values on $k$-handles defined above. |
847 |
||
8 | 848 |
If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
0 | 849 |
of $E$. |
850 |
In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
|
851 |
with a $k$-handle. |
|
852 |
Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell |
|
853 |
corresponding to $E$. |
|
854 |
Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$ |
|
855 |
adjacent to the $k{-}1$-cell corresponding to $E$. |
|
856 |
For $p \in E$, define |
|
857 |
\eq{ |
|
8 | 858 |
u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} |
859 |
+ r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . |
|
0 | 860 |
} |
861 |
||
862 |
In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
|
863 |
$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
|
864 |
The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
|
865 |
If we triangulate $R$ (without introducing new vertices), we can linearly extend |
|
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|
866 |
a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
0 | 867 |
Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
868 |
the $k{-}j$-cell corresponding to $E$. |
|
869 |
For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
|
870 |
Now define, for $p \in E$, |
|
871 |
\eq{ |
|
8 | 872 |
u(t, p, x) = (1-t)p + t \left( |
873 |
\sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
|
874 |
+ \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
|
875 |
\right) . |
|
0 | 876 |
} |
877 |
Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
|
878 |
mentioned above. |
|
879 |
||
880 |
This completes the definition of $u: I \times P \times X \to P$. |
|
881 |
||
882 |
\medskip |
|
883 |
||
884 |
Next we verify that $u$ has the desired properties. |
|
885 |
||
886 |
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 $x$. |
|
887 |
Therefore $F$ is a homotopy from $f$ to something. |
|
888 |
||
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|
889 |
Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
0 | 890 |
then $F$ is a homotopy through diffeomorphisms. |
891 |
We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
|
892 |
We have |
|
893 |
\eq{ |
|
8 | 894 |
% \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) . |
895 |
\pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
|
0 | 896 |
} |
897 |
Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
|
898 |
\nn{bounded away from zero, or something like that}. |
|
899 |
(Recall that $X$ and $P$ are compact.) |
|
900 |
Also, $\pd{f}{p}$ is bounded. |
|
901 |
So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
|
902 |
It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
|
7
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|
903 |
(which is bounded) |
0 | 904 |
and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s. |
905 |
These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
|
906 |
This completes the proof that $F$ is a homotopy through diffeomorphisms. |
|
907 |
||
908 |
\medskip |
|
909 |
||
910 |
Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ |
|
911 |
is a singular cell adapted to $\cU$. |
|
912 |
This will complete the proof of the lemma. |
|
913 |
\nn{except for boundary issues and the `$P$ is a cell' assumption} |
|
914 |
||
8 | 915 |
Let $j$ be the codimension of $D$. |
0 | 916 |
(Or rather, the codimension of its corresponding cell. From now on we will not make a distinction |
917 |
between handle and corresponding cell.) |
|
918 |
Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, |
|
919 |
where the $j_i$'s are the codimensions of the $K_\alpha$ |
|
920 |
cells of codimension greater than 0 which intersect to form $D$. |
|
921 |
We will show that |
|
922 |
if the relevant $U_\alpha$'s are disjoint, then |
|
923 |
$F(1, \cdot, \cdot) : D\times X \to X$ |
|
924 |
is a product of singular cells of dimensions $j_1, \ldots, j_m$. |
|
925 |
If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular |
|
926 |
cells whose dimensions correspond to a partition of the $j_i$'s. |
|
927 |
We will consider some simple special cases first, then do the general case. |
|
928 |
||
929 |
First consider the case $j=0$ (and $m=0$). |
|
930 |
A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$, |
|
931 |
is independent of $p \in P$. |
|
932 |
So the corresponding map $D \to \Diff(X)$ is constant. |
|
933 |
||
934 |
Next consider the case $j = 1$ (and $m=1$, $j_1=1$). |
|
935 |
Now Equation yyyy applies. |
|
936 |
We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$. |
|
937 |
It follows that the singular cell $D \to \Diff(X)$ can be written as a product |
|
938 |
of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. |
|
939 |
The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. |
|
940 |
||
941 |
Next case: $j=2$, $m=1$, $j_1 = 2$. |
|
8 | 942 |
This is similar to the previous case, except that the normal bundle is 2-dimensional instead of |
0 | 943 |
1-dimensional. |
944 |
We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell |
|
945 |
and a 2-cell with support $U_\beta$. |
|
946 |
||
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|
947 |
Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$. |
0 | 948 |
In this case the codimension 2 cell $D$ is the intersection of two |
949 |
codimension 1 cells, from $K_\beta$ and $K_\gamma$. |
|
950 |
We can write $D = D' \times I \times I$, where the normal coordinates are constant |
|
951 |
on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$. |
|
952 |
If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and |
|
953 |
two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively. |
|
954 |
If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and |
|
955 |
a 2-cell supported on $U_\beta \cup U_\gamma$. |
|
956 |
\nn{need to check that this is true} |
|
957 |
||
958 |
\nn{finally, general case...} |
|
959 |
||
960 |
\nn{this completes proof} |
|
961 |
||
13 | 962 |
\input{text/explicit.tex} |
0 | 963 |
|
964 |
||
965 |
\section{$A_\infty$ action on the boundary} |
|
966 |
||
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|
967 |
Let $Y$ be an $n{-}1$-manifold. |
8 | 968 |
The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary |
7
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|
969 |
conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure |
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|
970 |
of an $A_\infty$ category. |
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|
971 |
|
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|
972 |
Composition of morphisms (multiplication) depends of a choice of homeomorphism |
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|
973 |
$I\cup I \cong I$. Given this choice, gluing gives a map |
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|
974 |
\eq{ |
8 | 975 |
\bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) |
976 |
\cong \bc_*(Y\times I; a, c) |
|
7
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|
977 |
} |
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|
978 |
Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various |
4ef2f77a4652
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|
979 |
higher associators of the $A_\infty$ structure, more or less canonically. |
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|
980 |
|
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|
981 |
\nn{is this obvious? does more need to be said?} |
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|
982 |
|
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|
983 |
Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. |
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|
984 |
|
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|
985 |
Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism |
4ef2f77a4652
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|
986 |
$(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ |
8 | 987 |
(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the |
7
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|
988 |
$A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. |
4ef2f77a4652
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|
989 |
Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood |
4ef2f77a4652
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|
990 |
of $Y$ in $X$. |
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|
991 |
|
4ef2f77a4652
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|
992 |
In the next section we use the above $A_\infty$ actions to state and prove |
4ef2f77a4652
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|
993 |
a gluing theorem for the blob complexes of $n$-manifolds. |
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|
994 |
|
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|
995 |
|
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996 |
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997 |
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998 |
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999 |
|
0 | 1000 |
|
1001 |
\section{Gluing} \label{gluesect} |
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1002 |
||
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1003 |
Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy |
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1004 |
of $Y \du -Y$ contained in its boundary. |
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1005 |
Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. |
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1006 |
We wish to describe the blob complex of $X\sgl$ in terms of the blob complex |
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1007 |
of $X$. |
8 | 1008 |
More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, |
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1009 |
where $c\sgl \in \cC(\bd X\sgl)$, |
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1010 |
in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation |
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1011 |
of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. |
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1012 |
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1013 |
\begin{thm} |
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1014 |
$\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product |
8 | 1015 |
of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. |
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1016 |
\end{thm} |
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1017 |
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1018 |
The proof will occupy the remainder of this section. |
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1019 |
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1020 |
\nn{...} |
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1021 |
|
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1022 |
\bigskip |
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1023 |
|
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1024 |
\nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} |
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1025 |
|
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1026 |
|
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|
1027 |
|
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1028 |
|
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1029 |
|
0 | 1030 |
\section{Extension to ...} |
1031 |
||
8 | 1032 |
\nn{Need to let the input $n$-category $C$ be a graded thing |
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1033 |
(e.g.~DGA or $A_\infty$ $n$-category).} |
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1034 |
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1035 |
\nn{maybe this should be done earlier in the exposition? |
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1036 |
if we can plausibly claim that the various proofs work almost |
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1037 |
the same with the extended def, then maybe it's better to extend late (here)} |
0 | 1038 |
|
1039 |
||
1040 |
\section{What else?...} |
|
1041 |
||
1042 |
\begin{itemize} |
|
1043 |
\item Derive Hochschild standard results from blob point of view? |
|
1044 |
\item $n=2$ examples |
|
1045 |
\item Kh |
|
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1046 |
\item dimension $n+1$ (generalized Deligne conjecture?) |
0 | 1047 |
\item should be clear about PL vs Diff; probably PL is better |
1048 |
(or maybe not) |
|
1049 |
\item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category |
|
1050 |
\item something about higher derived coend things (derived 2-coend, e.g.) |
|
1051 |
\end{itemize} |
|
1052 |
||
1053 |
||
1054 |
||
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1055 |
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1056 |
|
0 | 1057 |
\end{document} |
1058 |
||
1059 |
||
1060 |
||
1061 |
%Recall that for $n$-category picture fields there is an evaluation map |
|
1062 |
%$m: \bc_0(B^n; c, c') \to \mor(c, c')$. |
|
1063 |
%If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain |
|
1064 |
%map $m: \bc_*(B^n; c, c') \to \mor(c, c')$. |