author | scott@6e1638ff-ae45-0410-89bd-df963105f760 |
Thu, 24 Apr 2008 02:56:48 +0000 | |
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parent 8 | 15e6335ff1d4 |
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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}; |
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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 $A(X)$ |
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is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
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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; |
|
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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} |
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|
508 |
%The assignment $X \mapsto \bc_*(X)$ extends to a functor from the category of |
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|
509 |
%$n$-manifolds and homeomorphisms to the category of chain complexes and linear isomorphisms. |
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|
510 |
%\end{prop} |
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|
511 |
|
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|
512 |
%\begin{proof} |
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|
513 |
%Obvious. |
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|
514 |
%\end{proof} |
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|
515 |
|
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|
516 |
%\nn{need to same something about boundaries and boundary conditions above. |
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|
517 |
%maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.} |
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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 |
||
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|
527 |
\nn{need to same something about boundaries and boundary conditions above. |
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|
528 |
maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.} |
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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 |
||
580 |
||
581 |
||
582 |
||
583 |
\section{$n=1$ and Hochschild homology} |
|
584 |
||
585 |
In this section we analyze the blob complex in dimension $n=1$ |
|
8 | 586 |
and find that for $S^1$ the homology of the blob complex is the |
0 | 587 |
Hochschild homology of the category (algebroid) that we started with. |
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|
588 |
\nn{or maybe say here that the complexes are quasi-isomorphic? in general, |
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|
589 |
should perhaps put more emphasis on the complexes and less on the homology.} |
0 | 590 |
|
591 |
Notation: $HB_i(X) = H_i(\bc_*(X))$. |
|
592 |
||
8 | 593 |
Let us first note that there is no loss of generality in assuming that our system of |
0 | 594 |
fields comes from a category. |
595 |
(Or maybe (???) there {\it is} a loss of generality. |
|
8 | 596 |
Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be |
0 | 597 |
thought of as the morphisms of a 1-category $C$. |
598 |
More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$ |
|
599 |
are $A(I; a, b)$, and composition is given by gluing. |
|
600 |
If we instead take our fields to be $C$-pictures, the $\cC(pt)$ does not change |
|
601 |
and neither does $A(I; a, b) = HB_0(I; a, b)$. |
|
602 |
But what about $HB_i(I; a, b)$ for $i > 0$? |
|
603 |
Might these higher blob homology groups be different? |
|
604 |
Seems unlikely, but I don't feel like trying to prove it at the moment. |
|
605 |
In any case, we'll concentrate on the case of fields based on 1-category |
|
606 |
pictures for the rest of this section.) |
|
607 |
||
608 |
(Another question: $\bc_*(I)$ is an $A_\infty$-category. |
|
609 |
How general of an $A_\infty$-category is it? |
|
610 |
Given an arbitrary $A_\infty$-category can one find fields and local relations so |
|
611 |
that $\bc_*(I)$ is in some sense equivalent to the original $A_\infty$-category? |
|
612 |
Probably not, unless we generalize to the case where $n$-morphisms are complexes.) |
|
613 |
||
614 |
Continuing... |
|
615 |
||
616 |
Let $C$ be a *-1-category. |
|
617 |
Then specializing the definitions from above to the case $n=1$ we have: |
|
618 |
\begin{itemize} |
|
619 |
\item $\cC(pt) = \ob(C)$ . |
|
620 |
\item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
|
8 | 621 |
Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
0 | 622 |
points in the interior |
623 |
of $R$, each labeled by a morphism of $C$. |
|
624 |
The intervals between the points are labeled by objects of $C$, consistent with |
|
625 |
the boundary condition $c$ and the domains and ranges of the point labels. |
|
626 |
\item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by |
|
627 |
composing the morphism labels of the points. |
|
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|
628 |
Note that we also need the * of *-1-category here in order to make all the morphisms point |
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|
629 |
the same way. |
0 | 630 |
\item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single |
631 |
point (at some standard location) labeled by $x$. |
|
8 | 632 |
Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
0 | 633 |
form $y - \chi(e(y))$. |
634 |
Thus we can, if we choose, restrict the blob twig labels to things of this form. |
|
635 |
\end{itemize} |
|
636 |
||
8 | 637 |
We want to show that $HB_*(S^1)$ is naturally isomorphic to the |
0 | 638 |
Hochschild homology of $C$. |
639 |
\nn{Or better that the complexes are homotopic |
|
640 |
or quasi-isomorphic.} |
|
641 |
In order to prove this we will need to extend the blob complex to allow points to also |
|
642 |
be labeled by elements of $C$-$C$-bimodules. |
|
643 |
%Given an interval (1-ball) so labeled, there is an evaluation map to some tensor product |
|
644 |
%(over $C$) of $C$-$C$-bimodules. |
|
645 |
%Define the local relations $U(I; a, b)$ to be the direct sum of the kernels of these maps. |
|
646 |
%Now we can define the blob complex for $S^1$. |
|
647 |
%This complex is the sum of complexes with a fixed cyclic tuple of bimodules present. |
|
648 |
%If $M$ is a $C$-$C$-bimodule, let $G_*(M)$ denote the summand of $\bc_*(S^1)$ corresponding |
|
649 |
%to the cyclic 1-tuple $(M)$. |
|
650 |
%In other words, $G_*(M)$ is a blob-like complex where exactly one point is labeled |
|
651 |
%by an element of $M$ and the remaining points are labeled by morphisms of $C$. |
|
652 |
%It's clear that $G_*(C)$ is isomorphic to the original bimodule-less |
|
653 |
%blob complex for $S^1$. |
|
654 |
%\nn{Is it really so clear? Should say more.} |
|
655 |
||
656 |
%\nn{alternative to the above paragraph:} |
|
657 |
Fix points $p_1, \ldots, p_k \in S^1$ and $C$-$C$-bimodules $M_1, \ldots M_k$. |
|
658 |
We define a blob-like complex $F_*(S^1, (p_i), (M_i))$. |
|
659 |
The fields have elements of $M_i$ labeling $p_i$ and elements of $C$ labeling |
|
660 |
other points. |
|
661 |
The blob twig labels lie in kernels of evaluation maps. |
|
662 |
(The range of these evaluation maps is a tensor product (over $C$) of $M_i$'s.) |
|
663 |
Let $F_*(M) = F_*(S^1, (*), (M))$, where $* \in S^1$ is some standard base point. |
|
664 |
In other words, fields for $F_*(M)$ have an element of $M$ at the fixed point $*$ |
|
665 |
and elements of $C$ at variable other points. |
|
666 |
||
667 |
We claim that the homology of $F_*(M)$ is isomorphic to the Hochschild |
|
668 |
homology of $M$. |
|
669 |
\nn{Or maybe we should claim that $M \to F_*(M)$ is the/a derived coend. |
|
670 |
Or maybe that $F_*(M)$ is quasi-isomorphic (or perhaps homotopic) to the Hochschild |
|
671 |
complex of $M$.} |
|
672 |
This follows from the following lemmas: |
|
673 |
\begin{itemize} |
|
674 |
\item $F_*(M_1 \oplus M_2) \cong F_*(M_1) \oplus F_*(M_2)$. |
|
675 |
\item An exact sequence $0 \to M_1 \to M_2 \to M_3 \to 0$ |
|
676 |
gives rise to an exact sequence $0 \to F_*(M_1) \to F_*(M_2) \to F_*(M_3) \to 0$. |
|
677 |
(See below for proof.) |
|
678 |
\item $F_*(C\otimes C)$ (the free $C$-$C$-bimodule with one generator) is |
|
7
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679 |
quasi-isomorphic to the 0-step complex $C$. |
0 | 680 |
(See below for proof.) |
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|
681 |
\item $F_*(C)$ (here $C$ is wearing its $C$-$C$-bimodule hat) is quasi-isomorphic to $\bc_*(S^1)$. |
0 | 682 |
(See below for proof.) |
683 |
\end{itemize} |
|
684 |
||
685 |
First we show that $F_*(C\otimes C)$ is |
|
7
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686 |
quasi-isomorphic to the 0-step complex $C$. |
0 | 687 |
|
8 | 688 |
Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of |
0 | 689 |
the point $*$ is $1 \otimes 1 \in C\otimes C$. |
690 |
We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism. |
|
691 |
||
692 |
Fix a small $\ep > 0$. |
|
693 |
Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. |
|
8 | 694 |
Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex |
7
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|
695 |
generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from |
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|
696 |
or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$. |
0 | 697 |
For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ |
698 |
labeled by $1\otimes 1$ and the only other labeled points at distance $\pm\ep/2$ from $*$. |
|
699 |
(See Figure xxxx.) |
|
7
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700 |
Note that $y - s_\ep(y) \in U(B_\ep)$. |
0 | 701 |
\nn{maybe it's simpler to assume that there are no labeled points, other than $*$, in $B_\ep$.} |
702 |
||
703 |
Define a degree 1 chain map $j_\ep : F^\ep_* \to F^\ep_*$ as follows. |
|
704 |
Let $x \in F^\ep_*$ be a blob diagram. |
|
705 |
If $*$ is not contained in any twig blob, $j_\ep(x)$ is obtained by adding $B_\ep$ to |
|
706 |
$x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. |
|
707 |
If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. |
|
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708 |
Let $y_i$ be the restriction of $z_i$ to $B_\ep$. |
8 | 709 |
Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, |
0 | 710 |
and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. |
711 |
Define $j_\ep(x) = \sum x_i$. |
|
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|
712 |
\nn{need to check signs coming from blob complex differential} |
0 | 713 |
|
714 |
Note that if $x \in F'_* \cap F^\ep_*$ then $j_\ep(x) \in F'_*$ also. |
|
715 |
||
716 |
The key property of $j_\ep$ is |
|
717 |
\eq{ |
|
8 | 718 |
\bd j_\ep + j_\ep \bd = \id - \sigma_\ep , |
0 | 719 |
} |
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|
720 |
where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field |
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|
721 |
mentioned in $x \in F^\ep_*$ with $s_\ep(y)$. |
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|
722 |
Note that $\sigma_\ep(x) \in F'_*$. |
0 | 723 |
|
724 |
If $j_\ep$ were defined on all of $F_*(C\otimes C)$, it would show that $\sigma_\ep$ |
|
725 |
is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$. |
|
726 |
One strategy would be to try to stitch together various $j_\ep$ for progressively smaller |
|
727 |
$\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$. |
|
8 | 728 |
Instead, we'll be less ambitious and just show that |
0 | 729 |
$F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
730 |
||
8 | 731 |
If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have |
0 | 732 |
$x \in F_*^\ep$. |
733 |
(This is true for any chain in $F_*(C\otimes C)$, since chains are sums of |
|
734 |
finitely many blob diagrams.) |
|
735 |
Then $x$ is homologous to $s_\ep(x)$, which is in $F'_*$, so the inclusion map |
|
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|
736 |
$F'_* \sub F_*(C\otimes C)$ is surjective on homology. |
0 | 737 |
If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$ |
738 |
and |
|
739 |
\eq{ |
|
8 | 740 |
\bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . |
0 | 741 |
} |
742 |
Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. |
|
743 |
This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. |
|
744 |
||
745 |
\medskip |
|
746 |
||
747 |
Let $F''_* \sub F'_*$ be the subcomplex of $F'_*$ where $*$ is not contained in any blob. |
|
748 |
We will show that the inclusion $i: F''_* \to F'_*$ is a homotopy equivalence. |
|
749 |
||
750 |
First, a lemma: Let $G''_*$ and $G'_*$ be defined the same as $F''_*$ and $F'_*$, except with |
|
751 |
$S^1$ replaced some (any) neighborhood of $* \in S^1$. |
|
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752 |
Then $G''_*$ and $G'_*$ are both contractible |
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|
753 |
and the inclusion $G''_* \sub G'_*$ is a homotopy equivalence. |
0 | 754 |
For $G'_*$ the proof is the same as in (\ref{bcontract}), except that the splitting |
755 |
$G'_0 \to H_0(G'_*)$ concentrates the point labels at two points to the right and left of $*$. |
|
756 |
For $G''_*$ we note that any cycle is supported \nn{need to establish terminology for this; maybe |
|
757 |
in ``basic properties" section above} away from $*$. |
|
758 |
Thus any cycle lies in the image of the normal blob complex of a disjoint union |
|
759 |
of two intervals, which is contractible by (\ref{bcontract}) and (\ref{disjunion}). |
|
760 |
Actually, we need the further (easy) result that the inclusion |
|
761 |
$G''_* \to G'_*$ induces an isomorphism on $H_0$. |
|
762 |
||
763 |
Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that |
|
764 |
for all $x \in F'_*$ we have |
|
765 |
\eq{ |
|
8 | 766 |
x - \bd h(x) - h(\bd x) \in F''_* . |
0 | 767 |
} |
768 |
Since $F'_0 = F''_0$, we can take $h_0 = 0$. |
|
769 |
Let $x \in F'_1$, with single blob $B \sub S^1$. |
|
770 |
If $* \notin B$, then $x \in F''_1$ and we define $h_1(x) = 0$. |
|
771 |
If $* \in B$, then we work in the image of $G'_*$ and $G''_*$ (with respect to $B$). |
|
772 |
Choose $x'' \in G''_1$ such that $\bd x'' = \bd x$. |
|
773 |
Since $G'_*$ is contractible, there exists $y \in G'_2$ such that $\bd y = x - x''$. |
|
774 |
Define $h_1(x) = y$. |
|
775 |
The general case is similar, except that we have to take lower order homotopies into account. |
|
776 |
Let $x \in F'_k$. |
|
777 |
If $*$ is not contained in any of the blobs of $x$, then define $h_k(x) = 0$. |
|
778 |
Otherwise, let $B$ be the outermost blob of $x$ containing $*$. |
|
779 |
By xxxx above, $x = x' \bullet p$, where $x'$ is supported on $B$ and $p$ is supported away from $B$. |
|
780 |
So $x' \in G'_l$ for some $l \le k$. |
|
781 |
Choose $x'' \in G''_l$ such that $\bd x'' = \bd (x' - h_{l-1}\bd x')$. |
|
782 |
Choose $y \in G'_{l+1}$ such that $\bd y = x' - x'' - h_{l-1}\bd x'$. |
|
783 |
Define $h_k(x) = y \bullet p$. |
|
784 |
This completes the proof that $i: F''_* \to F'_*$ is a homotopy equivalence. |
|
785 |
\nn{need to say above more clearly and settle on notation/terminology} |
|
786 |
||
787 |
Finally, we show that $F''_*$ is contractible. |
|
788 |
\nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now} |
|
789 |
Let $x$ be a cycle in $F''_*$. |
|
8 | 790 |
The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a |
0 | 791 |
ball $B \subset S^1$ containing the union of the supports and not containing $*$. |
792 |
Adding $B$ as a blob to $x$ gives a contraction. |
|
793 |
\nn{need to say something else in degree zero} |
|
794 |
||
795 |
This completes the proof that $F_*(C\otimes C)$ is |
|
796 |
homotopic to the 0-step complex $C$. |
|
797 |
||
798 |
\medskip |
|
799 |
||
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|
800 |
Next we show that $F_*(C)$ is quasi-isomorphic to $\bc_*(S^1)$. |
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|
801 |
$F_*(C)$ differs from $\bc_*(S^1)$ only in that the base point * |
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|
802 |
is always a labeled point in $F_*(C)$, while in $\bc_*(S^1)$ it may or may not be. |
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|
803 |
In other words, there is an inclusion map $i: F_*(C) \to \bc_*(S^1)$. |
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|
804 |
|
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|
805 |
We define a quasi-inverse \nn{right term?} $s: \bc_*(S^1) \to F_*(C)$ to the inclusion as follows. |
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|
806 |
If $y$ is a field defined on a neighborhood of *, define $s(y) = y$ if |
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|
807 |
* is a labeled point in $y$. |
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|
808 |
Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. |
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|
809 |
Let $x \in \bc_*(S^1)$. |
8 | 810 |
Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in |
7
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|
811 |
$x$ with $y$. |
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|
812 |
It is easy to check that $s$ is a chain map and $s \circ i = \id$. |
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|
813 |
|
8 | 814 |
Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points |
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|
815 |
in a neighborhood $B_\ep$ of *, except perhaps *. |
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|
816 |
Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$. |
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|
817 |
\nn{rest of argument goes similarly to above} |
0 | 818 |
|
819 |
\bigskip |
|
820 |
||
821 |
\nn{still need to prove exactness claim} |
|
822 |
||
823 |
\nn{What else needs to be said to establish quasi-isomorphism to Hochschild complex? |
|
824 |
Do we need a map from hoch to blob? |
|
825 |
Does the above exactness and contractibility guarantee such a map without writing it |
|
826 |
down explicitly? |
|
827 |
Probably it's worth writing down an explicit map even if we don't need to.} |
|
828 |
||
829 |
||
8 | 830 |
We can also describe explicitly a map from the standard Hochschild |
831 |
complex to the blob complex on the circle. \nn{What properties does this |
|
832 |
map have?} |
|
0 | 833 |
|
8 | 834 |
\begin{figure}% |
835 |
$$\mathfig{0.6}{barycentric/barycentric}$$ |
|
836 |
\caption{The Hochschild chain $a \tensor b \tensor c$ is sent to |
|
837 |
the sum of six blob $2$-chains, corresponding to a barycentric subdivision of a $2$-simplex.} |
|
838 |
\label{fig:Hochschild-example}% |
|
839 |
\end{figure} |
|
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840 |
|
8 | 841 |
As an example, Figure \ref{fig:Hochschild-example} shows the image of the Hochschild chain $a \tensor b \tensor c$. Only the $0$-cells are shown explicitly. |
842 |
The edges marked $x, y$ and $z$ carry the $1$-chains |
|
843 |
\begin{align*} |
|
844 |
x & = \mathfig{0.1}{barycentric/ux} & u_x = \mathfig{0.1}{barycentric/ux_ca} - \mathfig{0.1}{barycentric/ux_c-a} \\ |
|
845 |
y & = \mathfig{0.1}{barycentric/uy} & u_y = \mathfig{0.1}{barycentric/uy_cab} - \mathfig{0.1}{barycentric/uy_ca-b} \\ |
|
846 |
z & = \mathfig{0.1}{barycentric/uz} & u_z = \mathfig{0.1}{barycentric/uz_c-a-b} - \mathfig{0.1}{barycentric/uz_cab} |
|
847 |
\end{align*} |
|
848 |
and the $2$-chain labelled $A$ is |
|
849 |
\begin{equation*} |
|
850 |
A = \mathfig{0.1}{barycentric/Ax}+\mathfig{0.1}{barycentric/Ay}. |
|
851 |
\end{equation*} |
|
852 |
Note that we then have |
|
853 |
\begin{equation*} |
|
854 |
\bdy A = x+y+z. |
|
855 |
\end{equation*} |
|
856 |
||
857 |
In general, the Hochschild chain $\Tensor_{i=1}^n a_i$ is sent to the sum of $n!$ blob $(n-1)$-chains, indexed by permutations, |
|
858 |
$$\phi\left(\Tensor_{i=1}^n a_i) = \sum_{\pi} \phi^\pi(a_1, \ldots, a_n)$$ |
|
859 |
with ... |
|
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|
860 |
|
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|
861 |
|
0 | 862 |
\section{Action of $C_*(\Diff(X))$} \label{diffsect} |
863 |
||
864 |
Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of |
|
865 |
the space of diffeomorphisms |
|
866 |
of the $n$-manifold $X$ (fixed on $\bd X$). |
|
867 |
For convenience, we will permit the singular cells generating $CD_*(X)$ to be more general |
|
868 |
than simplices --- they can be based on any linear polyhedron. |
|
869 |
\nn{be more restrictive here? does more need to be said?} |
|
870 |
||
871 |
\begin{prop} \label{CDprop} |
|
872 |
For each $n$-manifold $X$ there is a chain map |
|
873 |
\eq{ |
|
8 | 874 |
e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) . |
0 | 875 |
} |
876 |
On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$ |
|
877 |
(Proposition (\ref{diff0prop})). |
|
878 |
For any splitting $X = X_1 \cup X_2$, the following diagram commutes |
|
879 |
\eq{ \xymatrix{ |
|
8 | 880 |
CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\ |
881 |
CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
|
882 |
\ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & |
|
883 |
\bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl} |
|
0 | 884 |
} } |
885 |
Any other map satisfying the above two properties is homotopic to $e_X$. |
|
886 |
\end{prop} |
|
887 |
||
888 |
The proof will occupy the remainder of this section. |
|
889 |
||
890 |
\medskip |
|
891 |
||
892 |
Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. |
|
893 |
We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
|
894 |
$x \notin S$ and $p, q \in P$. |
|
895 |
Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
|
896 |
||
897 |
Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
|
898 |
A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is |
|
899 |
{\it adapted to $\cU$} if there is a factorization |
|
900 |
\eq{ |
|
8 | 901 |
P = P_1 \times \cdots \times P_m |
0 | 902 |
} |
903 |
(for some $m \le k$) |
|
904 |
and families of diffeomorphisms |
|
905 |
\eq{ |
|
8 | 906 |
f_i : P_i \times X \to X |
0 | 907 |
} |
8 | 908 |
such that |
0 | 909 |
\begin{itemize} |
910 |
\item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$; |
|
911 |
\item the $V_i$'s are mutually disjoint; |
|
8 | 912 |
\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
0 | 913 |
where $k_i = \dim(P_i)$; and |
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|
914 |
\item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot) \circ g$ |
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|
915 |
for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
0 | 916 |
\end{itemize} |
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|
917 |
A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum |
0 | 918 |
of singular cells, each of which is adapted to $\cU$. |
919 |
||
920 |
\begin{lemma} \label{extension_lemma} |
|
921 |
Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
|
922 |
Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
923 |
\end{lemma} |
|
924 |
||
925 |
The proof will be given in Section \ref{fam_diff_sect}. |
|
926 |
||
927 |
\medskip |
|
928 |
||
8 | 929 |
Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$ |
0 | 930 |
(e.g.~the support of a blob diagram). |
931 |
We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if |
|
932 |
$f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$ |
|
933 |
either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$. |
|
8 | 934 |
A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells, |
0 | 935 |
each of which is compatible. |
936 |
(Note that we could strengthen the definition of compatibility to incorporate |
|
937 |
a factorization condition, similar to the definition of ``adapted to" above. |
|
938 |
The weaker definition given here will suffice for our needs below.) |
|
939 |
||
940 |
\begin{cor} \label{extension_lemma_2} |
|
941 |
Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is compatible with $\{B_j\}$. |
|
942 |
Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$. |
|
943 |
\end{cor} |
|
944 |
\begin{proof} |
|
8 | 945 |
This will follow from Lemma \ref{extension_lemma} for |
0 | 946 |
appropriate choice of cover $\cU = \{U_\alpha\}$. |
947 |
Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let |
|
948 |
$V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$. |
|
949 |
Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$ |
|
950 |
either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$. |
|
951 |
||
8 | 952 |
Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$, |
0 | 953 |
with the (compatible) support of $f_i$ in place of $X$. |
954 |
This insures that the resulting homotopy $h_i$ is compatible. |
|
955 |
Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$. |
|
956 |
\nn{actually, need to start with the 0-skeleton of $\bd x$, then 1-skeleton, etc.; fix this} |
|
957 |
\end{proof} |
|
958 |
||
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|
959 |
\medskip |
0 | 960 |
|
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|
961 |
((argument continues roughly as follows: up to homotopy, there is only one way to define $e_X$ |
4ef2f77a4652
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|
962 |
on compatible $x\otimes y \in CD_*(X)\otimes \bc_*(X)$. |
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|
963 |
This is because $x$ is the gluing of $x'$ and $x''$, where $x'$ has degree zero and is defined on |
4ef2f77a4652
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|
964 |
the complement of the $D_i$'s, and $x''$ is defined on the $D_i$'s. |
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|
965 |
We have no choice on $x'$, since we already know the map on 0-parameter families of diffeomorphisms. |
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|
966 |
We have no choice, up to homotopy, on $x''$, since the target chain complex is contractible.)) |
0 | 967 |
|
968 |
||
969 |
\section{Families of Diffeomorphisms} \label{fam_diff_sect} |
|
970 |
||
971 |
||
972 |
Lo, the proof of Lemma (\ref{extension_lemma}): |
|
973 |
||
974 |
\nn{should this be an appendix instead?} |
|
975 |
||
976 |
\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
|
977 |
later draft} |
|
978 |
||
979 |
\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
|
980 |
about boundary later} |
|
981 |
||
8 | 982 |
Recall that we are given |
0 | 983 |
an open cover $\cU = \{U_\alpha\}$ and an |
984 |
$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
|
985 |
We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
986 |
||
987 |
Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. |
|
988 |
||
989 |
As a first approximation to the argument we will eventually make, let's replace $x$ |
|
8 | 990 |
with a single singular cell |
0 | 991 |
\eq{ |
8 | 992 |
f: P \times X \to X . |
0 | 993 |
} |
994 |
Also, we'll ignore for now issues around $\bd P$. |
|
995 |
||
996 |
Our homotopy will have the form |
|
997 |
\eqar{ |
|
8 | 998 |
F: I \times P \times X &\to& X \\ |
999 |
(t, p, x) &\mapsto& f(u(t, p, x), x) |
|
0 | 1000 |
} |
1001 |
for some function |
|
1002 |
\eq{ |
|
8 | 1003 |
u : I \times P \times X \to P . |
0 | 1004 |
} |
1005 |
First we describe $u$, then we argue that it does what we want it to do. |
|
1006 |
||
1007 |
For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. |
|
1008 |
The various $K_\alpha$ should be in general position with respect to each other. |
|
1009 |
We will see below that the $K_\alpha$'s need to be sufficiently fine in order |
|
1010 |
to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not |
|
1011 |
merely a homotopy through maps $X\to X$. |
|
1012 |
||
1013 |
Let $L$ be the union of all the $K_\alpha$'s. |
|
1014 |
$L$ is itself a cell decomposition of $P$. |
|
1015 |
\nn{next two sentences not needed?} |
|
1016 |
To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$, |
|
1017 |
where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$. |
|
1018 |
Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$. |
|
1019 |
||
1020 |
Let $J$ denote the handle decomposition of $P$ corresponding to $L$. |
|
1021 |
Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and, |
|
1022 |
more importantly, a $k{-}i$-dimensional normal coordinate. |
|
1023 |
||
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|
1024 |
For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$. |
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|
1025 |
Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding |
0 | 1026 |
$k$-cell of $L$. |
1027 |
To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
|
1028 |
which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
|
1029 |
||
1030 |
For $p \in D$ we define |
|
1031 |
\eq{ |
|
8 | 1032 |
u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
0 | 1033 |
} |
1034 |
(Recall that $P$ is a single linear cell, so the weighted average of points of $P$ |
|
1035 |
makes sense.) |
|
1036 |
||
1037 |
So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. |
|
8 | 1038 |
For handles of $J$ of index less than $k$, we will define $u$ to |
0 | 1039 |
interpolate between the values on $k$-handles defined above. |
1040 |
||
8 | 1041 |
If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
0 | 1042 |
of $E$. |
1043 |
In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
|
1044 |
with a $k$-handle. |
|
1045 |
Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell |
|
1046 |
corresponding to $E$. |
|
1047 |
Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$ |
|
1048 |
adjacent to the $k{-}1$-cell corresponding to $E$. |
|
1049 |
For $p \in E$, define |
|
1050 |
\eq{ |
|
8 | 1051 |
u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} |
1052 |
+ r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . |
|
0 | 1053 |
} |
1054 |
||
1055 |
In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
|
1056 |
$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
|
1057 |
The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
|
1058 |
If we triangulate $R$ (without introducing new vertices), we can linearly extend |
|
1
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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0
diff
changeset
|
1059 |
a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
0 | 1060 |
Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
1061 |
the $k{-}j$-cell corresponding to $E$. |
|
1062 |
For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
|
1063 |
Now define, for $p \in E$, |
|
1064 |
\eq{ |
|
8 | 1065 |
u(t, p, x) = (1-t)p + t \left( |
1066 |
\sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
|
1067 |
+ \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
|
1068 |
\right) . |
|
0 | 1069 |
} |
1070 |
Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
|
1071 |
mentioned above. |
|
1072 |
||
1073 |
This completes the definition of $u: I \times P \times X \to P$. |
|
1074 |
||
1075 |
\medskip |
|
1076 |
||
1077 |
Next we verify that $u$ has the desired properties. |
|
1078 |
||
1079 |
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 $x$. |
|
1080 |
Therefore $F$ is a homotopy from $f$ to something. |
|
1081 |
||
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|
1082 |
Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
0 | 1083 |
then $F$ is a homotopy through diffeomorphisms. |
1084 |
We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
|
1085 |
We have |
|
1086 |
\eq{ |
|
8 | 1087 |
% \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) . |
1088 |
\pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
|
0 | 1089 |
} |
1090 |
Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
|
1091 |
\nn{bounded away from zero, or something like that}. |
|
1092 |
(Recall that $X$ and $P$ are compact.) |
|
1093 |
Also, $\pd{f}{p}$ is bounded. |
|
1094 |
So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
|
1095 |
It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
|
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|
1096 |
(which is bounded) |
0 | 1097 |
and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s. |
1098 |
These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
|
1099 |
This completes the proof that $F$ is a homotopy through diffeomorphisms. |
|
1100 |
||
1101 |
\medskip |
|
1102 |
||
1103 |
Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ |
|
1104 |
is a singular cell adapted to $\cU$. |
|
1105 |
This will complete the proof of the lemma. |
|
1106 |
\nn{except for boundary issues and the `$P$ is a cell' assumption} |
|
1107 |
||
8 | 1108 |
Let $j$ be the codimension of $D$. |
0 | 1109 |
(Or rather, the codimension of its corresponding cell. From now on we will not make a distinction |
1110 |
between handle and corresponding cell.) |
|
1111 |
Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, |
|
1112 |
where the $j_i$'s are the codimensions of the $K_\alpha$ |
|
1113 |
cells of codimension greater than 0 which intersect to form $D$. |
|
1114 |
We will show that |
|
1115 |
if the relevant $U_\alpha$'s are disjoint, then |
|
1116 |
$F(1, \cdot, \cdot) : D\times X \to X$ |
|
1117 |
is a product of singular cells of dimensions $j_1, \ldots, j_m$. |
|
1118 |
If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular |
|
1119 |
cells whose dimensions correspond to a partition of the $j_i$'s. |
|
1120 |
We will consider some simple special cases first, then do the general case. |
|
1121 |
||
1122 |
First consider the case $j=0$ (and $m=0$). |
|
1123 |
A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$, |
|
1124 |
is independent of $p \in P$. |
|
1125 |
So the corresponding map $D \to \Diff(X)$ is constant. |
|
1126 |
||
1127 |
Next consider the case $j = 1$ (and $m=1$, $j_1=1$). |
|
1128 |
Now Equation yyyy applies. |
|
1129 |
We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$. |
|
1130 |
It follows that the singular cell $D \to \Diff(X)$ can be written as a product |
|
1131 |
of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. |
|
1132 |
The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. |
|
1133 |
||
1134 |
Next case: $j=2$, $m=1$, $j_1 = 2$. |
|
8 | 1135 |
This is similar to the previous case, except that the normal bundle is 2-dimensional instead of |
0 | 1136 |
1-dimensional. |
1137 |
We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell |
|
1138 |
and a 2-cell with support $U_\beta$. |
|
1139 |
||
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|
1140 |
Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$. |
0 | 1141 |
In this case the codimension 2 cell $D$ is the intersection of two |
1142 |
codimension 1 cells, from $K_\beta$ and $K_\gamma$. |
|
1143 |
We can write $D = D' \times I \times I$, where the normal coordinates are constant |
|
1144 |
on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$. |
|
1145 |
If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and |
|
1146 |
two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively. |
|
1147 |
If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and |
|
1148 |
a 2-cell supported on $U_\beta \cup U_\gamma$. |
|
1149 |
\nn{need to check that this is true} |
|
1150 |
||
1151 |
\nn{finally, general case...} |
|
1152 |
||
1153 |
\nn{this completes proof} |
|
1154 |
||
1155 |
||
1156 |
||
1157 |
||
1158 |
\section{$A_\infty$ action on the boundary} |
|
1159 |
||
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changeset
|
1160 |
Let $Y$ be an $n{-}1$-manifold. |
8 | 1161 |
The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary |
7
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|
1162 |
conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure |
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|
1163 |
of an $A_\infty$ category. |
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|
1164 |
|
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|
1165 |
Composition of morphisms (multiplication) depends of a choice of homeomorphism |
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|
1166 |
$I\cup I \cong I$. Given this choice, gluing gives a map |
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|
1167 |
\eq{ |
8 | 1168 |
\bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) |
1169 |
\cong \bc_*(Y\times I; a, c) |
|
7
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|
1170 |
} |
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|
1171 |
Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various |
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|
1172 |
higher associators of the $A_\infty$ structure, more or less canonically. |
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|
1173 |
|
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|
1174 |
\nn{is this obvious? does more need to be said?} |
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|
1175 |
|
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|
1176 |
Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. |
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|
1177 |
|
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|
1178 |
Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism |
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|
1179 |
$(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ |
8 | 1180 |
(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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|
1181 |
$A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. |
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|
1182 |
Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood |
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|
1183 |
of $Y$ in $X$. |
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|
1184 |
|
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|
1185 |
In the next section we use the above $A_\infty$ actions to state and prove |
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|
1186 |
a gluing theorem for the blob complexes of $n$-manifolds. |
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|
1187 |
|
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|
1188 |
|
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|
1189 |
|
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|
1190 |
|
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|
1191 |
|
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|
1192 |
|
0 | 1193 |
|
1194 |
\section{Gluing} \label{gluesect} |
|
1195 |
||
7
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|
1196 |
Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy |
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|
1197 |
of $Y \du -Y$ contained in its boundary. |
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|
1198 |
Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. |
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|
1199 |
We wish to describe the blob complex of $X\sgl$ in terms of the blob complex |
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|
1200 |
of $X$. |
8 | 1201 |
More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, |
7
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|
1202 |
where $c\sgl \in \cC(\bd X\sgl)$, |
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|
1203 |
in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation |
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|
1204 |
of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. |
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|
1205 |
|
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|
1206 |
\begin{thm} |
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|
1207 |
$\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product |
8 | 1208 |
of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. |
7
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|
1209 |
\end{thm} |
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|
1210 |
|
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|
1211 |
The proof will occupy the remainder of this section. |
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|
1212 |
|
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|
1213 |
\nn{...} |
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|
1214 |
|
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|
1215 |
\bigskip |
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|
1216 |
|
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|
1217 |
\nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} |
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|
1218 |
|
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|
1219 |
|
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|
1220 |
|
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|
1221 |
|
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|
1222 |
|
0 | 1223 |
\section{Extension to ...} |
1224 |
||
8 | 1225 |
\nn{Need to let the input $n$-category $C$ be a graded thing |
7
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|
1226 |
(e.g.~DGA or $A_\infty$ $n$-category).} |
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|
1227 |
|
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|
1228 |
\nn{maybe this should be done earlier in the exposition? |
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|
1229 |
if we can plausibly claim that the various proofs work almost |
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|
1230 |
the same with the extended def, then maybe it's better to extend late (here)} |
0 | 1231 |
|
1232 |
||
1233 |
\section{What else?...} |
|
1234 |
||
1235 |
\begin{itemize} |
|
1236 |
\item Derive Hochschild standard results from blob point of view? |
|
1237 |
\item $n=2$ examples |
|
1238 |
\item Kh |
|
7
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|
1239 |
\item dimension $n+1$ (generalized Deligne conjecture?) |
0 | 1240 |
\item should be clear about PL vs Diff; probably PL is better |
1241 |
(or maybe not) |
|
1242 |
\item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category |
|
1243 |
\item something about higher derived coend things (derived 2-coend, e.g.) |
|
1244 |
\end{itemize} |
|
1245 |
||
1246 |
||
1247 |
||
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|
1248 |
|
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|
1249 |
|
0 | 1250 |
\end{document} |
1251 |
||
1252 |
||
1253 |
||
1254 |
%Recall that for $n$-category picture fields there is an evaluation map |
|
1255 |
%$m: \bc_0(B^n; c, c') \to \mor(c, c')$. |
|
1256 |
%If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain |
|
1257 |
%map $m: \bc_*(B^n; c, c') \to \mor(c, c')$. |