author | kevin@6e1638ff-ae45-0410-89bd-df963105f760 |
Sun, 26 Oct 2008 05:32:15 +0000 | |
changeset 47 | 939a4a5b1d80 |
parent 45 | 0047a1211c3b |
child 48 | b7ade62bea27 |
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}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{sign}{supp}{maps}; |
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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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\textbf{Draft version, do not distribute.} |
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\versioninfo |
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\section*{Todo} |
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\subsection*{What else?...} |
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\begin{itemize} |
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\item higher priority |
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\begin{itemize} |
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\item K\&S: learn the state of the art in A-inf categories |
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(tensor products, Kadeishvili result, ...) |
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\item K: so-called evaluation map stuff |
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\item K: topological fields |
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\item section describing intended applications |
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\item say something about starting with semisimple n-cat (trivial?? not trivial?) |
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\item T.O.C. |
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\end{itemize} |
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\item medium priority |
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\begin{itemize} |
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\item $n=2$ examples |
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\item dimension $n+1$ (generalized Deligne conjecture?) |
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\item should be clear about PL vs Diff; probably PL is better |
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(or maybe not) |
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\item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category |
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\item something about higher derived coend things (derived 2-coend, e.g.) |
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\item shuffle product vs gluing product (?) |
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\item commutative algebra results |
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\item $A_\infty$ blob complex |
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\item connection between $A_\infty$ operad and topological $A_\infty$ cat defs |
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\end{itemize} |
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\item lower priority |
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\begin{itemize} |
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\item Derive Hochschild standard results from blob point of view? |
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\item Kh |
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\end{itemize} |
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\end{itemize} |
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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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133 |
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We then show that blob homology enjoys the following |
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\ref{property:gluing} properties. |
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\begin{property}[Functoriality] |
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\label{property:functoriality}% |
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Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association |
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\begin{equation*} |
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X \mapsto \bc_*^{\cF,\cU}(X) |
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\end{equation*} |
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is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them. |
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\scott{Do we want to or need to weaken `isomorphisms' to `homotopy equivalences' or `quasi-isomorphisms'?} |
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\end{property} |
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\begin{property}[Disjoint union] |
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\label{property:disjoint-union} |
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The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
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\begin{equation*} |
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\bc_*(X_1 \sqcup X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
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\end{equation*} |
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\end{property} |
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\begin{property}[A map for gluing] |
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\label{property:gluing-map}% |
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If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, |
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there is a chain map |
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\begin{equation*} |
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\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
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\end{equation*} |
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\end{property} |
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\begin{property}[Contractibility] |
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\label{property:contractibility}% |
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\todo{Err, requires a splitting?} |
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The blob complex for an $n$-category on an $n$-ball is quasi-isomorphic to its $0$-th homology. |
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\begin{equation} |
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\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))} |
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\end{equation} |
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|
172 |
\todo{Say that this is just the original $n$-category?} |
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173 |
\end{property} |
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174 |
|
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175 |
\begin{property}[Skein modules] |
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176 |
\label{property:skein-modules}% |
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177 |
The $0$-th blob homology of $X$ is the usual skein module associated to $X$. (See \S \ref{sec:local-relations}.) |
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178 |
\begin{equation*} |
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179 |
H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X) |
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180 |
\end{equation*} |
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181 |
\end{property} |
0 | 182 |
|
22
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183 |
\begin{property}[Hochschild homology when $X=S^1$] |
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184 |
\label{property:hochschild}% |
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185 |
The blob complex for a $1$-category $\cC$ on the circle is |
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186 |
quasi-isomorphic to the Hochschild complex. |
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187 |
\begin{equation*} |
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188 |
\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & HC_*(\cC)} |
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189 |
\end{equation*} |
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190 |
\end{property} |
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191 |
|
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192 |
\begin{property}[Evaluation map] |
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193 |
\label{property:evaluation}% |
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194 |
There is an `evaluation' chain map |
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195 |
\begin{equation*} |
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196 |
\ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). |
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197 |
\end{equation*} |
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198 |
(Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.) |
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199 |
|
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200 |
Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for |
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201 |
any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram |
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202 |
(using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
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203 |
\begin{equation*} |
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204 |
\xymatrix{ |
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205 |
\CD{X} \otimes \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) \\ |
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206 |
\CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
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207 |
\ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
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208 |
\bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
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209 |
} |
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210 |
\end{equation*} |
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211 |
\end{property} |
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212 |
|
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213 |
\begin{property}[Gluing formula] |
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214 |
\label{property:gluing}% |
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215 |
\mbox{}% <-- gets the indenting right |
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216 |
\begin{itemize} |
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217 |
\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
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218 |
naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
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219 |
|
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220 |
\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
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221 |
$A_\infty$ module for $\bc_*(Y \times I)$. |
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222 |
|
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223 |
\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
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224 |
$0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
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225 |
$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
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226 |
$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
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227 |
\begin{equation*} |
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228 |
\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} |
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229 |
\end{equation*} |
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230 |
\todo{How do you write self tensor product?} |
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231 |
\end{itemize} |
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232 |
\end{property} |
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233 |
|
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234 |
Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
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235 |
\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
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236 |
Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
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237 |
Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
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238 |
and Property \ref{property:gluing} in \S \ref{sec:gluing}. |
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239 |
|
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240 |
\section{Definitions} |
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241 |
\label{sec:definitions} |
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242 |
|
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243 |
\subsection{Systems of fields} |
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244 |
\label{sec:fields} |
0 | 245 |
|
246 |
Fix a top dimension $n$. |
|
247 |
||
8 | 248 |
A {\it system of fields} |
0 | 249 |
\nn{maybe should look for better name; but this is the name I use elsewhere} |
250 |
is a collection of functors $\cC$ from manifolds of dimension $n$ or less |
|
251 |
to sets. |
|
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252 |
These functors must satisfy various properties (see \cite{kw:tqft} for details). |
8 | 253 |
For example: |
0 | 254 |
there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$; |
255 |
there is a restriction map $\cC(X) \to \cC(\bd X)$; |
|
256 |
gluing manifolds corresponds to fibered products of fields; |
|
8 | 257 |
given a field $c \in \cC(Y)$ there is a ``product field" |
0 | 258 |
$c\times I \in \cC(Y\times I)$; ... |
259 |
\nn{should eventually include full details of definition of fields.} |
|
260 |
||
8 | 261 |
\nn{note: probably will suppress from notation the distinction |
0 | 262 |
between fields and their (orientation-reversal) duals} |
263 |
||
264 |
\nn{remark that if top dimensional fields are not already linear |
|
265 |
then we will soon linearize them(?)} |
|
266 |
||
8 | 267 |
The definition of a system of fields is intended to generalize |
0 | 268 |
the relevant properties of the following two examples of fields. |
269 |
||
270 |
The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$ |
|
8 | 271 |
is a manifold of dimension $n$ or less) to be the set of |
0 | 272 |
all maps from $X$ to $B$. |
273 |
||
274 |
The second example will take longer to explain. |
|
8 | 275 |
Given an $n$-category $C$ with the right sort of duality |
276 |
(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
|
0 | 277 |
we can construct a system of fields as follows. |
278 |
Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
|
279 |
with codimension $i$ cells labeled by $i$-morphisms of $C$. |
|
280 |
We'll spell this out for $n=1,2$ and then describe the general case. |
|
281 |
||
282 |
If $X$ has boundary, we require that the cell decompositions are in general |
|
283 |
position with respect to the boundary --- the boundary intersects each cell |
|
284 |
transversely, so cells meeting the boundary are mere half-cells. |
|
285 |
||
286 |
Put another way, the cell decompositions we consider are dual to standard cell |
|
287 |
decompositions of $X$. |
|
288 |
||
289 |
We will always assume that our $n$-categories have linear $n$-morphisms. |
|
290 |
||
291 |
For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
|
292 |
an object (0-morphism) of the 1-category $C$. |
|
293 |
A field on a 1-manifold $S$ consists of |
|
294 |
\begin{itemize} |
|
8 | 295 |
\item A cell decomposition of $S$ (equivalently, a finite collection |
0 | 296 |
of points in the interior of $S$); |
8 | 297 |
\item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
0 | 298 |
by an object (0-morphism) of $C$; |
8 | 299 |
\item a transverse orientation of each 0-cell, thought of as a choice of |
0 | 300 |
``domain" and ``range" for the two adjacent 1-cells; and |
8 | 301 |
\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
0 | 302 |
domain and range determined by the transverse orientation and the labelings of the 1-cells. |
303 |
\end{itemize} |
|
304 |
||
305 |
If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
|
8 | 306 |
of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
0 | 307 |
interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
308 |
of the algebra. |
|
309 |
||
4
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310 |
\medskip |
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311 |
|
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312 |
For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) |
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313 |
that are common in the literature. |
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314 |
We describe these carefully here. |
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315 |
|
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316 |
A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
0 | 317 |
an object of the 2-category $C$. |
318 |
A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
|
319 |
A field on a 2-manifold $Y$ consists of |
|
320 |
\begin{itemize} |
|
8 | 321 |
\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
0 | 322 |
that each component of the complement is homeomorphic to a disk); |
8 | 323 |
\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
0 | 324 |
by a 0-morphism of $C$; |
8 | 325 |
\item a transverse orientation of each 1-cell, thought of as a choice of |
0 | 326 |
``domain" and ``range" for the two adjacent 2-cells; |
8 | 327 |
\item a labeling of each 1-cell by a 1-morphism of $C$, with |
328 |
domain and range determined by the transverse orientation of the 1-cell |
|
0 | 329 |
and the labelings of the 2-cells; |
8 | 330 |
\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
0 | 331 |
of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
332 |
to $\pm 1 \in S^1$; and |
|
8 | 333 |
\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
0 | 334 |
determined by the labelings of the 1-cells and the parameterizations of the previous |
335 |
bullet. |
|
336 |
\end{itemize} |
|
337 |
\nn{need to say this better; don't try to fit everything into the bulleted list} |
|
338 |
||
339 |
For general $n$, a field on a $k$-manifold $X^k$ consists of |
|
340 |
\begin{itemize} |
|
8 | 341 |
\item A cell decomposition of $X$; |
342 |
\item an explicit general position homeomorphism from the link of each $j$-cell |
|
0 | 343 |
to the boundary of the standard $(k-j)$-dimensional bihedron; and |
8 | 344 |
\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
0 | 345 |
domain and range determined by the labelings of the link of $j$-cell. |
346 |
\end{itemize} |
|
347 |
||
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348 |
%\nn{next definition might need some work; I think linearity relations should |
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349 |
%be treated differently (segregated) from other local relations, but I'm not sure |
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|
350 |
%the next definition is the best way to do it} |
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351 |
|
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352 |
\medskip |
0 | 353 |
|
8 | 354 |
For top dimensional ($n$-dimensional) manifolds, we're actually interested |
0 | 355 |
in the linearized space of fields. |
356 |
By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is |
|
8 | 357 |
the vector space of finite |
0 | 358 |
linear combinations of fields on $X$. |
359 |
If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$. |
|
360 |
Thus the restriction (to boundary) maps are well defined because we never |
|
361 |
take linear combinations of fields with differing boundary conditions. |
|
362 |
||
363 |
In some cases we don't linearize the default way; instead we take the |
|
364 |
spaces $\cC_l(X; a)$ to be part of the data for the system of fields. |
|
365 |
In particular, for fields based on linear $n$-category pictures we linearize as follows. |
|
8 | 366 |
Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
0 | 367 |
obvious relations on 0-cell labels. |
8 | 368 |
More specifically, let $L$ be a cell decomposition of $X$ |
0 | 369 |
and let $p$ be a 0-cell of $L$. |
370 |
Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
|
371 |
$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
|
372 |
Then the subspace $K$ is generated by things of the form |
|
373 |
$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
|
374 |
to infer the meaning of $\alpha_{\lambda c + d}$. |
|
375 |
Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
|
376 |
||
8 | 377 |
\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
0 | 378 |
will do something similar below; in general, whenever a label lives in a linear |
8 | 379 |
space we do something like this; ? say something about tensor |
0 | 380 |
product of all the linear label spaces? Yes:} |
381 |
||
382 |
For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
|
383 |
Define an ``almost-field" to be a field without labels on the 0-cells. |
|
384 |
(Recall that 0-cells are labeled by $n$-morphisms.) |
|
385 |
To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
|
386 |
space determined by the labeling of the link of the 0-cell. |
|
387 |
(If the 0-cell were labeled, the label would live in this space.) |
|
388 |
We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
|
8 | 389 |
We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the |
0 | 390 |
above tensor products. |
391 |
||
392 |
||
393 |
||
394 |
\subsection{Local relations} |
|
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395 |
\label{sec:local-relations} |
0 | 396 |
|
397 |
Let $B^n$ denote the standard $n$-ball. |
|
8 | 398 |
A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ |
0 | 399 |
(for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties. |
400 |
||
8 | 401 |
\nn{Roughly, these are (1) the local relations imply (extended) isotopy; |
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|
402 |
(2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and |
8 | 403 |
(3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$). |
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404 |
See \cite{kw:tqft} for details. Need to transfer details to here.} |
0 | 405 |
|
406 |
For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$, |
|
407 |
where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
|
408 |
||
409 |
For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map |
|
410 |
$\cC_l(B^n; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
|
411 |
domain and range. |
|
412 |
||
413 |
\nn{maybe examples of local relations before general def?} |
|
414 |
||
415 |
Note that the $Y$ is an $n$-manifold which is merely homeomorphic to the standard $B^n$, |
|
416 |
then any homeomorphism $B^n \to Y$ induces the same local subspaces for $Y$. |
|
417 |
We'll denote these by $U(Y; c) \sub \cC_l(Y; c)$, $c \in \cC(\bd Y)$. |
|
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418 |
\nn{Is this true in high (smooth) dimensions? Self-diffeomorphisms of $B^n$ |
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|
419 |
rel boundary might not be isotopic to the identity. OK for PL and TOP?} |
0 | 420 |
|
421 |
Given a system of fields and local relations, we define the skein space |
|
422 |
$A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
|
423 |
the $n$-manifold $Y$ modulo local relations. |
|
424 |
The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
|
425 |
is defined to be the dual of $A(Y; c)$. |
|
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|
426 |
(See \cite{kw:tqft} or xxxx for details.) |
0 | 427 |
|
428 |
The blob complex is in some sense the derived version of $A(Y; c)$. |
|
429 |
||
430 |
||
431 |
||
432 |
\subsection{The blob complex} |
|
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433 |
\label{sec:blob-definition} |
0 | 434 |
|
435 |
Let $X$ be an $n$-manifold. |
|
436 |
Assume a fixed system of fields. |
|
437 |
In this section we will usually suppress boundary conditions on $X$ from the notation |
|
438 |
(e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$). |
|
439 |
||
8 | 440 |
We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
0 | 441 |
submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
442 |
$\overline{X \setmin Y}$. |
|
443 |
||
444 |
We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case. |
|
445 |
||
446 |
Define $\bc_0(X) = \cC_l(X)$. |
|
447 |
(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cC_l(X; c)$. |
|
448 |
We'll omit this sort of detail in the rest of this section.) |
|
449 |
In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
|
450 |
||
451 |
$\bc_1(X)$ is the space of all local relations that can be imposed on $\bc_0(X)$. |
|
452 |
More specifically, define a 1-blob diagram to consist of |
|
453 |
\begin{itemize} |
|
454 |
\item An embedded closed ball (``blob") $B \sub X$. |
|
455 |
%\nn{Does $B$ need a homeo to the standard $B^n$? I don't think so. |
|
456 |
%(See note in previous subsection.)} |
|
457 |
%\item A field (boundary condition) $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$. |
|
458 |
\item A field $r \in \cC(X \setmin B; c)$ |
|
459 |
(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
|
460 |
\item A local relation field $u \in U(B; c)$ |
|
461 |
(same $c$ as previous bullet). |
|
462 |
\end{itemize} |
|
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|
463 |
%(Note that the field $c$ is determined (implicitly) as the boundary of $u$ and/or $r$, |
0 | 464 |
%so we will omit $c$ from the notation.) |
465 |
Define $\bc_1(X)$ to be the space of all finite linear combinations of |
|
466 |
1-blob diagrams, modulo the simple relations relating labels of 0-cells and |
|
467 |
also the label ($u$ above) of the blob. |
|
468 |
\nn{maybe spell this out in more detail} |
|
469 |
(See xxxx above.) |
|
470 |
\nn{maybe restate this in terms of direct sums of tensor products.} |
|
471 |
||
472 |
There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear |
|
473 |
combination of fields on $X$ obtained by gluing $r$ to $u$. |
|
8 | 474 |
In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
0 | 475 |
just erasing the blob from the picture |
476 |
(but keeping the blob label $u$). |
|
477 |
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|
478 |
Note that the skein space $A(X)$ |
0 | 479 |
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
480 |
||
481 |
$\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$. |
|
8 | 482 |
More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
0 | 483 |
2-blob diagrams (defined below), modulo the usual linear label relations. |
484 |
\nn{and also modulo blob reordering relations?} |
|
485 |
||
486 |
\nn{maybe include longer discussion to motivate the two sorts of 2-blob diagrams} |
|
487 |
||
488 |
There are two types of 2-blob diagram: disjoint and nested. |
|
489 |
A disjoint 2-blob diagram consists of |
|
490 |
\begin{itemize} |
|
491 |
\item A pair of disjoint closed balls (blobs) $B_0, B_1 \sub X$. |
|
492 |
%\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
|
493 |
\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
|
494 |
(where $c_i \in \cC(\bd B_i)$). |
|
495 |
\item Local relation fields $u_i \in U(B_i; c_i)$. |
|
496 |
\end{itemize} |
|
497 |
Define $\bd(B_0, B_1, r, u_0, u_1) = (B_1, ru_0, u_1) - (B_0, ru_1, u_0) \in \bc_1(X)$. |
|
498 |
In other words, the boundary of a disjoint 2-blob diagram |
|
499 |
is the sum (with alternating signs) |
|
500 |
of the two ways of erasing one of the blobs. |
|
501 |
It's easy to check that $\bd^2 = 0$. |
|
502 |
||
503 |
A nested 2-blob diagram consists of |
|
504 |
\begin{itemize} |
|
505 |
\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
|
506 |
\item A field $r \in \cC(X \setmin B_0; c_0)$ |
|
507 |
(for some $c_0 \in \cC(\bd B_0)$). |
|
508 |
Let $r = r_1 \cup r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
|
509 |
(for some $c_1 \in \cC(B_1)$) and |
|
510 |
$r' \in \cC(X \setmin B_1; c_1)$. |
|
511 |
\item A local relation field $u_0 \in U(B_0; c_0)$. |
|
512 |
\end{itemize} |
|
513 |
Define $\bd(B_0, B_1, r, u_0) = (B_1, r', r_1u_0) - (B_0, r, u_0)$. |
|
514 |
Note that xxxx above guarantees that $r_1u_0 \in U(B_1)$. |
|
515 |
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
|
516 |
sum of the two ways of erasing one of the blobs. |
|
517 |
If we erase the inner blob, the outer blob inherits the label $r_1u_0$. |
|
518 |
||
519 |
Now for the general case. |
|
520 |
A $k$-blob diagram consists of |
|
521 |
\begin{itemize} |
|
522 |
\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
|
523 |
For each $i$ and $j$, we require that either $B_i \cap B_j$ is empty or |
|
524 |
$B_i \sub B_j$ or $B_j \sub B_i$. |
|
525 |
(The case $B_i = B_j$ is allowed. |
|
526 |
If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
|
527 |
If a blob has no other blobs strictly contained in it, we call it a twig blob. |
|
528 |
%\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
|
529 |
%(These are implied by the data in the next bullets, so we usually |
|
530 |
%suppress them from the notation.) |
|
531 |
%$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
|
532 |
%if the latter space is not empty. |
|
533 |
\item A field $r \in \cC(X \setmin B^t; c^t)$, |
|
534 |
where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$. |
|
535 |
\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
|
536 |
where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
|
537 |
If $B_i = B_j$ then $u_i = u_j$. |
|
538 |
\end{itemize} |
|
539 |
||
540 |
We define $\bc_k(X)$ to be the vector space of all finite linear combinations |
|
541 |
of $k$-blob diagrams, modulo the linear label relations and |
|
542 |
blob reordering relations defined in the remainder of this paragraph. |
|
543 |
Let $x$ be a blob diagram with one undetermined $n$-morphism label. |
|
544 |
The unlabeled entity is either a blob or a 0-cell outside of the twig blobs. |
|
545 |
Let $a$ and $b$ be two possible $n$-morphism labels for |
|
546 |
the unlabeled blob or 0-cell. |
|
547 |
Let $c = \lambda a + b$. |
|
548 |
Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly. |
|
549 |
Then we impose the relation |
|
550 |
\eq{ |
|
8 | 551 |
x_c = \lambda x_a + x_b . |
0 | 552 |
} |
553 |
\nn{should do this in terms of direct sums of tensor products} |
|
554 |
Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$ |
|
555 |
of their blob labelings. |
|
556 |
Then we impose the relation |
|
557 |
\eq{ |
|
8 | 558 |
x = \sign(\pi) x' . |
0 | 559 |
} |
560 |
||
561 |
(Alert readers will have noticed that for $k=2$ our definition |
|
562 |
of $\bc_k(X)$ is slightly different from the previous definition |
|
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|
563 |
of $\bc_2(X)$ --- we did not impose the reordering relations. |
0 | 564 |
The general definition takes precedence; |
565 |
the earlier definition was simplified for purposes of exposition.) |
|
566 |
||
567 |
The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
|
568 |
Let $b = (\{B_i\}, r, \{u_j\})$ be a $k$-blob diagram. |
|
569 |
Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
|
570 |
If $B_j$ is not a twig blob, this involves only decrementing |
|
571 |
the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
|
572 |
If $B_j$ is a twig blob, we have to assign new local relation labels |
|
573 |
if removing $B_j$ creates new twig blobs. |
|
574 |
If $B_l$ becomes a twig after removing $B_j$, then set $u_l = r_lu_j$, |
|
575 |
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
|
576 |
Finally, define |
|
577 |
\eq{ |
|
8 | 578 |
\bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
0 | 579 |
} |
580 |
The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
|
581 |
Thus we have a chain complex. |
|
582 |
||
583 |
\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
|
584 |
||
585 |
||
8 | 586 |
\nn{TO DO: |
587 |
expand definition to handle DGA and $A_\infty$ versions of $n$-categories; |
|
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|
588 |
relations to Chas-Sullivan string stuff} |
0 | 589 |
|
590 |
||
591 |
||
592 |
\section{Basic properties of the blob complex} |
|
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|
593 |
\label{sec:basic-properties} |
0 | 594 |
|
595 |
\begin{prop} \label{disjunion} |
|
596 |
There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
|
597 |
\end{prop} |
|
598 |
\begin{proof} |
|
599 |
Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
|
8 | 600 |
(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
0 | 601 |
blob diagram $(b_1, b_2)$ on $X \du Y$. |
602 |
Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
|
603 |
In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
|
604 |
to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
|
605 |
a pair of blob diagrams on $X$ and $Y$. |
|
606 |
These two maps are compatible with our sign conventions \nn{say more about this?} and |
|
607 |
with the linear label relations. |
|
608 |
The two maps are inverses of each other. |
|
609 |
\nn{should probably say something about sign conventions for the differential |
|
610 |
in a tensor product of chain complexes; ask Scott} |
|
611 |
\end{proof} |
|
612 |
||
613 |
For the next proposition we will temporarily restore $n$-manifold boundary |
|
614 |
conditions to the notation. |
|
615 |
||
8 | 616 |
Suppose that for all $c \in \cC(\bd B^n)$ |
617 |
we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
|
0 | 618 |
of the quotient map |
619 |
$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
|
620 |
\nn{always the case if we're working over $\c$}. |
|
621 |
Then |
|
622 |
\begin{prop} \label{bcontract} |
|
623 |
For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
|
624 |
is a chain homotopy equivalence |
|
625 |
with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
|
626 |
Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0. |
|
627 |
\end{prop} |
|
628 |
\begin{proof} |
|
629 |
By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map |
|
630 |
$h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$. |
|
631 |
For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding |
|
632 |
an $(i{+}1)$-st blob equal to all of $B^n$. |
|
633 |
In other words, add a new outermost blob which encloses all of the others. |
|
634 |
Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
|
635 |
the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
|
636 |
\nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$} |
|
637 |
\end{proof} |
|
638 |
||
8 | 639 |
(Note that for the above proof to work, we need the linear label relations |
0 | 640 |
for blob labels. |
641 |
Also we need to blob reordering relations (?).) |
|
642 |
||
643 |
(Note also that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
|
644 |
equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.) |
|
645 |
||
646 |
(For fields based on $n$-cats, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$.) |
|
647 |
||
648 |
\medskip |
|
649 |
||
650 |
As we noted above, |
|
651 |
\begin{prop} |
|
652 |
There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
|
653 |
\qed |
|
654 |
\end{prop} |
|
655 |
||
656 |
||
4
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|
657 |
% oops -- duplicate |
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|
658 |
|
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|
659 |
%\begin{prop} \label{functorialprop} |
8599e156a169
misc. edit, nothing major
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3
diff
changeset
|
660 |
%The assignment $X \mapsto \bc_*(X)$ extends to a functor from the category of |
8599e156a169
misc. edit, nothing major
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parents:
3
diff
changeset
|
661 |
%$n$-manifolds and homeomorphisms to the category of chain complexes and linear isomorphisms. |
8599e156a169
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3
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|
662 |
%\end{prop} |
8599e156a169
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3
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changeset
|
663 |
|
8599e156a169
misc. edit, nothing major
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changeset
|
664 |
%\begin{proof} |
8599e156a169
misc. edit, nothing major
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3
diff
changeset
|
665 |
%Obvious. |
8599e156a169
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3
diff
changeset
|
666 |
%\end{proof} |
8599e156a169
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3
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changeset
|
667 |
|
8599e156a169
misc. edit, nothing major
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parents:
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diff
changeset
|
668 |
%\nn{need to same something about boundaries and boundary conditions above. |
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parents:
3
diff
changeset
|
669 |
%maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.} |
8599e156a169
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parents:
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|
670 |
|
8599e156a169
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3
diff
changeset
|
671 |
|
0 | 672 |
\begin{prop} |
673 |
For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
|
8 | 674 |
of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
0 | 675 |
(chain map) isomorphisms. |
676 |
\qed |
|
677 |
\end{prop} |
|
678 |
||
4
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changeset
|
679 |
\nn{need to same something about boundaries and boundary conditions above. |
8599e156a169
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
3
diff
changeset
|
680 |
maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.} |
8599e156a169
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diff
changeset
|
681 |
|
0 | 682 |
|
683 |
In particular, |
|
684 |
\begin{prop} \label{diff0prop} |
|
685 |
There is an action of $\Diff(X)$ on $\bc_*(X)$. |
|
686 |
\qed |
|
687 |
\end{prop} |
|
688 |
||
22
ada83e7228eb
rearranging; stating all the "properties" up front
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diff
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|
689 |
The above will be greatly strengthened in Section \ref{sec:evaluation}. |
0 | 690 |
|
691 |
\medskip |
|
692 |
||
693 |
For the next proposition we will temporarily restore $n$-manifold boundary |
|
694 |
conditions to the notation. |
|
695 |
||
696 |
Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
|
697 |
Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
|
698 |
with boundary $Z\sgl$. |
|
699 |
Given compatible fields (pictures, boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
|
700 |
we have the blob complex $\bc_*(X; a, b, c)$. |
|
701 |
If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
|
702 |
$X$ to get blob diagrams on $X\sgl$: |
|
703 |
||
704 |
\begin{prop} |
|
705 |
There is a natural chain map |
|
706 |
\eq{ |
|
8 | 707 |
\gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
0 | 708 |
} |
8 | 709 |
The sum is over all fields $a$ on $Y$ compatible at their |
0 | 710 |
($n{-}2$-dimensional) boundaries with $c$. |
711 |
`Natural' means natural with respect to the actions of diffeomorphisms. |
|
712 |
\qed |
|
713 |
\end{prop} |
|
714 |
||
715 |
The above map is very far from being an isomorphism, even on homology. |
|
22
ada83e7228eb
rearranging; stating all the "properties" up front
scott@6e1638ff-ae45-0410-89bd-df963105f760
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21
diff
changeset
|
716 |
This will be fixed in Section \ref{sec:gluing} below. |
0 | 717 |
|
718 |
An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
|
719 |
and $X\sgl = X_1 \cup_Y X_2$. |
|
720 |
(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
|
721 |
For $x_i \in \bc_*(X_i)$, we introduce the notation |
|
722 |
\eq{ |
|
8 | 723 |
x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
0 | 724 |
} |
725 |
Note that we have resumed our habit of omitting boundary labels from the notation. |
|
726 |
||
727 |
||
728 |
\bigskip |
|
729 |
||
730 |
\nn{what else?} |
|
731 |
||
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
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diff
changeset
|
732 |
\section{Hochschild homology when $n=1$} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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diff
changeset
|
733 |
\label{sec:hochschild} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
13
diff
changeset
|
734 |
\input{text/hochschild} |
7
4ef2f77a4652
small to medium sized changes
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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diff
changeset
|
735 |
|
22
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rearranging; stating all the "properties" up front
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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diff
changeset
|
736 |
\section{Action of $\CD{X}$} |
ada83e7228eb
rearranging; stating all the "properties" up front
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parents:
21
diff
changeset
|
737 |
\label{sec:evaluation} |
0 | 738 |
|
739 |
Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of |
|
740 |
the space of diffeomorphisms |
|
741 |
of the $n$-manifold $X$ (fixed on $\bd X$). |
|
742 |
For convenience, we will permit the singular cells generating $CD_*(X)$ to be more general |
|
743 |
than simplices --- they can be based on any linear polyhedron. |
|
744 |
\nn{be more restrictive here? does more need to be said?} |
|
745 |
||
746 |
\begin{prop} \label{CDprop} |
|
747 |
For each $n$-manifold $X$ there is a chain map |
|
748 |
\eq{ |
|
8 | 749 |
e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) . |
0 | 750 |
} |
751 |
On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$ |
|
752 |
(Proposition (\ref{diff0prop})). |
|
753 |
For any splitting $X = X_1 \cup X_2$, the following diagram commutes |
|
754 |
\eq{ \xymatrix{ |
|
8 | 755 |
CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\ |
756 |
CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
|
757 |
\ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & |
|
758 |
\bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl} |
|
0 | 759 |
} } |
760 |
Any other map satisfying the above two properties is homotopic to $e_X$. |
|
761 |
\end{prop} |
|
762 |
||
18
aac9fd8d6bc6
finished with evaluation map stuff for now.
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
17
diff
changeset
|
763 |
\nn{need to rewrite for self-gluing instead of gluing two pieces together} |
aac9fd8d6bc6
finished with evaluation map stuff for now.
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
17
diff
changeset
|
764 |
|
16
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
15
diff
changeset
|
765 |
\nn{Should say something stronger about uniqueness. |
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
15
diff
changeset
|
766 |
Something like: there is |
19 | 767 |
a contractible subcomplex of the complex of chain maps |
16
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
15
diff
changeset
|
768 |
$CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), |
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
15
diff
changeset
|
769 |
and all choices in the construction lie in the 0-cells of this |
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
15
diff
changeset
|
770 |
contractible subcomplex. |
19 | 771 |
Or maybe better to say any two choices are homotopic, and |
16
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
15
diff
changeset
|
772 |
any two homotopies and second order homotopic, and so on.} |
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
15
diff
changeset
|
773 |
|
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
15
diff
changeset
|
774 |
\nn{Also need to say something about associativity. |
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
15
diff
changeset
|
775 |
Put it in the above prop or make it a separate prop? |
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
15
diff
changeset
|
776 |
I lean toward the latter.} |
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
15
diff
changeset
|
777 |
\medskip |
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
15
diff
changeset
|
778 |
|
0 | 779 |
The proof will occupy the remainder of this section. |
18
aac9fd8d6bc6
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|
780 |
\nn{unless we put associativity prop at end} |
0 | 781 |
|
782 |
\medskip |
|
783 |
||
784 |
Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. |
|
785 |
We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all |
|
42
9744833c9b90
some improvements to c-star-diff section; probably a few more changes will follow soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
41
diff
changeset
|
786 |
$x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if there is a family of diffeomorphisms $f' : P \times S \to S$ and a `background' |
40
b7bc1a931b73
cleaning up and writing a little more on topological A_\infty categories
scott@6e1638ff-ae45-0410-89bd-df963105f760
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37
diff
changeset
|
787 |
diffeomorphism $f_0 : X \to X$ so that |
b7bc1a931b73
cleaning up and writing a little more on topological A_\infty categories
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
37
diff
changeset
|
788 |
\begin{align} |
42
9744833c9b90
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
41
diff
changeset
|
789 |
f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ |
40
b7bc1a931b73
cleaning up and writing a little more on topological A_\infty categories
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
37
diff
changeset
|
790 |
\intertext{and} |
42
9744833c9b90
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
41
diff
changeset
|
791 |
f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}. |
40
b7bc1a931b73
cleaning up and writing a little more on topological A_\infty categories
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
37
diff
changeset
|
792 |
\end{align} |
0 | 793 |
Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. |
794 |
||
795 |
Let $\cU = \{U_\alpha\}$ be an open cover of $X$. |
|
796 |
A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is |
|
797 |
{\it adapted to $\cU$} if there is a factorization |
|
798 |
\eq{ |
|
8 | 799 |
P = P_1 \times \cdots \times P_m |
0 | 800 |
} |
801 |
(for some $m \le k$) |
|
802 |
and families of diffeomorphisms |
|
803 |
\eq{ |
|
8 | 804 |
f_i : P_i \times X \to X |
0 | 805 |
} |
8 | 806 |
such that |
0 | 807 |
\begin{itemize} |
42
9744833c9b90
some improvements to c-star-diff section; probably a few more changes will follow soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
41
diff
changeset
|
808 |
\item each $f_i$ is supported on some connected $V_i \sub X$; |
40
b7bc1a931b73
cleaning up and writing a little more on topological A_\infty categories
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
37
diff
changeset
|
809 |
\item the sets $V_i$ are mutually disjoint; |
8 | 810 |
\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, |
0 | 811 |
where $k_i = \dim(P_i)$; and |
42
9744833c9b90
some improvements to c-star-diff section; probably a few more changes will follow soon
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
41
diff
changeset
|
812 |
\item $f(p, \cdot) = g \circ f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot)$ |
9744833c9b90
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41
diff
changeset
|
813 |
for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. |
0 | 814 |
\end{itemize} |
7
4ef2f77a4652
small to medium sized changes
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
5
diff
changeset
|
815 |
A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum |
0 | 816 |
of singular cells, each of which is adapted to $\cU$. |
817 |
||
818 |
\begin{lemma} \label{extension_lemma} |
|
819 |
Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
|
820 |
Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
18
aac9fd8d6bc6
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
17
diff
changeset
|
821 |
Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
0 | 822 |
\end{lemma} |
823 |
||
22
ada83e7228eb
rearranging; stating all the "properties" up front
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
21
diff
changeset
|
824 |
The proof will be given in Section \ref{sec:localising}. |
0 | 825 |
|
826 |
\medskip |
|
827 |
||
16
9ae2fd41b903
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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15
diff
changeset
|
828 |
The strategy for the proof of Proposition \ref{CDprop} is as follows. |
9ae2fd41b903
begin reworking/completion of evaluation map stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
829 |
We will identify a subcomplex |
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|
830 |
\[ |
19 | 831 |
G_* \sub CD_*(X) \otimes \bc_*(X) |
16
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|
832 |
\] |
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|
833 |
on which the evaluation map is uniquely determined (up to homotopy) by the conditions |
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|
834 |
in \ref{CDprop}. |
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|
835 |
We then show that the inclusion of $G_*$ into the full complex |
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|
836 |
is an equivalence in the appropriate sense. |
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|
837 |
\nn{need to be more specific here} |
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changeset
|
838 |
|
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|
839 |
Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$. |
19 | 840 |
Roughly speaking, $p\otimes b$ is in $G_*$ if each component $V$ of the support of $p$ |
16
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|
841 |
intersects at most one blob $B$ of $b$. |
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changeset
|
842 |
Since $V \cup B$ might not itself be a ball, we need a more careful and complicated definition. |
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|
843 |
Choose a metric for $X$. |
19 | 844 |
We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that |
845 |
$\supp(p) \cup N_\epsilon(b)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon |
|
16
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|
846 |
neighborhood of the support of $b$. |
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|
847 |
\nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.} |
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|
848 |
|
42
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|
849 |
\nn{need to worry about case where the intrinsic support of $p$ is not a union of balls. |
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|
850 |
probably we can just stipulate that it is (i.e. only consider families of diffeos with this property). |
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|
851 |
maybe we should build into the definition of ``adapted" that support takes up all of $U_i$.} |
16
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|
852 |
|
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|
853 |
\nn{need to eventually show independence of choice of metric. maybe there's a better way than |
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|
854 |
choosing a metric. perhaps just choose a nbd of each ball, but I think I see problems |
17
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|
855 |
with that as well. |
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|
856 |
the bottom line is that we need a scheme for choosing unions of balls |
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|
857 |
which satisfies the $C$, $C'$, $C''$ claim made a few paragraphs below.} |
16
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|
858 |
|
17
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|
859 |
Next we define the evaluation map $e_X$ on $G_*$. |
16
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|
860 |
We'll proceed inductively on $G_i$. |
17
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|
861 |
The induction starts on $G_0$, where the evaluation map is determined |
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|
862 |
by the action of $\Diff(X)$ on $\bc_*(X)$ |
16
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|
863 |
because $G_0 \sub CD_0\otimes \bc_0$. |
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|
864 |
Assume we have defined the evaluation map up to $G_{k-1}$ and |
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|
865 |
let $p\otimes b$ be a generator of $G_k$. |
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|
866 |
Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$. |
42
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|
867 |
There is a factorization $p = g \circ p'$, where $g\in \Diff(X)$ and $p'$ is a family of diffeomorphisms which is the identity outside of $C$. |
17
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|
868 |
Let $b = b'\bullet b''$, where $b' \in \bc_*(C)$ and $b'' \in \bc_0(X\setmin C)$. |
42
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|
869 |
We may assume inductively |
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|
870 |
(cf the end of this paragraph) |
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|
871 |
that $e_X(\bd(p\otimes b))$ has a similar factorization $x\bullet g(b'')$, where |
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|
872 |
$x \in \bc_*(g(C))$ and $\bd x = 0$. |
17
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|
873 |
Since $\bc_*(g(C))$ is contractible, there exists $y \in \bc_*(g(C))$ such that $\bd y = x$. |
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|
874 |
Define $e_X(p\otimes b) = y\bullet g(b'')$. |
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changeset
|
875 |
|
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16
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changeset
|
876 |
We now show that $e_X$ on $G_*$ is, up to homotopy, independent of the various choices made. |
19 | 877 |
If we make a different series of choice of the chain $y$ in the previous paragraph, |
17
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changeset
|
878 |
we can inductively construct a homotopy between the two sets of choices, |
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16
diff
changeset
|
879 |
again relying on the contractibility of $\bc_*(g(G))$. |
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continuing work of evaluation map proof
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16
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changeset
|
880 |
A similar argument shows that this homotopy is unique up to second order homotopy, and so on. |
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diff
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|
881 |
|
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16
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|
882 |
Given a different set of choices $\{C'\}$ of the unions of balls $\{C\}$, |
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16
diff
changeset
|
883 |
we can find a third set of choices $\{C''\}$ such that $C, C' \sub C''$. |
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diff
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|
884 |
The argument now proceeds as in the previous paragraph. |
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16
diff
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|
885 |
\nn{should maybe say more here; also need to back up claim about third set of choices} |
42
9744833c9b90
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|
886 |
\nn{this definitely needs reworking} |
17
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16
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changeset
|
887 |
|
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16
diff
changeset
|
888 |
Next we show that given $x \in CD_*(X) \otimes \bc_*(X)$ with $\bd x \in G_*$, there exists |
c73e8beb4a20
continuing work of evaluation map proof
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16
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|
889 |
a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and |
c73e8beb4a20
continuing work of evaluation map proof
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16
diff
changeset
|
890 |
this homotopy are unique up to iterated homotopy. |
16
9ae2fd41b903
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15
diff
changeset
|
891 |
|
19 | 892 |
Given $k>0$ and a blob diagram $b$, we say that a cover $\cU$ of $X$ is $k$-compatible with |
893 |
$b$ if, for any $\{U_1, \ldots, U_k\} \sub \cU$, the union |
|
18
aac9fd8d6bc6
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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17
diff
changeset
|
894 |
$U_1\cup\cdots\cup U_k$ is a union of balls which satisfies the condition used to define $G_*$ above. |
42
9744833c9b90
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41
diff
changeset
|
895 |
It follows from Lemma \ref{extension_lemma} |
9744833c9b90
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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41
diff
changeset
|
896 |
that if $\cU$ is $k$-compatible with $b$ and |
9744833c9b90
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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41
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changeset
|
897 |
$p$ is a $k$-parameter family of diffeomorphisms which is adapted to $\cU$, then |
9744833c9b90
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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41
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changeset
|
898 |
$p\otimes b \in G_*$. |
18
aac9fd8d6bc6
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17
diff
changeset
|
899 |
\nn{maybe emphasize this more; it's one of the main ideas in the proof} |
16
9ae2fd41b903
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15
diff
changeset
|
900 |
|
18
aac9fd8d6bc6
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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17
diff
changeset
|
901 |
Let $k$ be the degree of $x$ and choose a cover $\cU$ of $X$ such that $\cU$ is |
aac9fd8d6bc6
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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17
diff
changeset
|
902 |
$k$-compatible with each of the (finitely many) blob diagrams occurring in $x$. |
19 | 903 |
We will use Lemma \ref{extension_lemma} with respect to the cover $\cU$ to |
18
aac9fd8d6bc6
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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17
diff
changeset
|
904 |
construct the homotopy to $G_*$. |
aac9fd8d6bc6
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17
diff
changeset
|
905 |
First we will construct a homotopy $h \in G_*$ from $\bd x$ to a cycle $z$ such that |
aac9fd8d6bc6
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diff
changeset
|
906 |
each family of diffeomorphisms $p$ occurring in $z$ is adapted to $\cU$. |
aac9fd8d6bc6
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17
diff
changeset
|
907 |
Then we will construct a homotopy (rel boundary) $r$ from $x + h$ to $y$ such that |
aac9fd8d6bc6
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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17
diff
changeset
|
908 |
each family of diffeomorphisms $p$ occurring in $y$ is adapted to $\cU$. |
aac9fd8d6bc6
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17
diff
changeset
|
909 |
This implies that $y \in G_*$. |
19 | 910 |
The homotopy $r$ can also be thought of as a homotopy from $x$ to $y-h \in G_*$, and this is the homotopy we seek. |
16
9ae2fd41b903
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15
diff
changeset
|
911 |
|
19 | 912 |
We will define $h$ inductively on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$. |
18
aac9fd8d6bc6
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17
diff
changeset
|
913 |
Define $h$ to be zero on bidegree $(0, k-1)$. |
aac9fd8d6bc6
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17
diff
changeset
|
914 |
Let $p\otimes b$ be a generator occurring in $\bd x$ with bidegree $(1, k-2)$. |
44
1b9b2aab1f35
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diff
changeset
|
915 |
Using Lemma \ref{extension_lemma}, construct a homotopy (rel $\bd$) $q$ from $p$ to $p'$ which is adapted to $\cU$. |
18
aac9fd8d6bc6
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diff
changeset
|
916 |
Define $h$ at $p\otimes b$ to be $q\otimes b$. |
aac9fd8d6bc6
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changeset
|
917 |
Let $p'\otimes b'$ be a generator occurring in $\bd x$ with bidegree $(2, k-3)$. |
44
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|
918 |
Let $s$ denote the sum of the $q$'s from the previous step for generators |
1b9b2aab1f35
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diff
changeset
|
919 |
adjacent to $(\bd p')\otimes b'$. |
1b9b2aab1f35
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changeset
|
920 |
\nn{need to say more here} |
1b9b2aab1f35
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diff
changeset
|
921 |
Apply Lemma \ref{extension_lemma} to $p'+s$ |
18
aac9fd8d6bc6
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changeset
|
922 |
yielding a family of diffeos $q'$. |
aac9fd8d6bc6
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changeset
|
923 |
Define $h$ at $p'\otimes b'$ to be $q'\otimes b'$. |
aac9fd8d6bc6
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|
924 |
Continuing in this way, we define all of $h$. |
0 | 925 |
|
18
aac9fd8d6bc6
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diff
changeset
|
926 |
The homotopy $r$ is constructed similarly. |
aac9fd8d6bc6
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diff
changeset
|
927 |
|
19 | 928 |
\nn{need to say something about uniqueness of $r$, $h$ etc. |
18
aac9fd8d6bc6
finished with evaluation map stuff for now.
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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17
diff
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|
929 |
postpone this until second draft.} |
0 | 930 |
|
18
aac9fd8d6bc6
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17
diff
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|
931 |
At this point, we have finished defining the evaluation map. |
aac9fd8d6bc6
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17
diff
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|
932 |
The uniqueness statement in the proposition is clear from the method of proof. |
aac9fd8d6bc6
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17
diff
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|
933 |
All that remains is to show that the evaluation map gets along well with cutting and gluing, |
aac9fd8d6bc6
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17
diff
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|
934 |
as claimed in the proposition. |
aac9fd8d6bc6
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17
diff
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|
935 |
This is in fact not difficult, since the myriad choices involved in defining the |
aac9fd8d6bc6
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17
diff
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|
936 |
evaluation map can be made in parallel for the top and bottom |
aac9fd8d6bc6
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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17
diff
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|
937 |
arrows in the commutative diagram. |
aac9fd8d6bc6
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17
diff
changeset
|
938 |
|
aac9fd8d6bc6
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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17
diff
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|
939 |
This completes the proof of Proposition \ref{CDprop}. |
0 | 940 |
|
7
4ef2f77a4652
small to medium sized changes
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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5
diff
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|
941 |
\medskip |
0 | 942 |
|
18
aac9fd8d6bc6
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17
diff
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|
943 |
\nn{say something about associativity here} |
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17
diff
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|
944 |
|
22
ada83e7228eb
rearranging; stating all the "properties" up front
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21
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|
945 |
\section{Gluing} |
ada83e7228eb
rearranging; stating all the "properties" up front
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21
diff
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|
946 |
\label{sec:gluing}% |
ada83e7228eb
rearranging; stating all the "properties" up front
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21
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|
947 |
|
40
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37
diff
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|
948 |
We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction |
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37
diff
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|
949 |
\begin{itemize} |
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37
diff
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|
950 |
%\mbox{}% <-- gets the indenting right |
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37
diff
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|
951 |
\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is |
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37
diff
changeset
|
952 |
naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. |
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37
diff
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|
953 |
|
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37
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|
954 |
\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an |
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37
diff
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|
955 |
$A_\infty$ module for $\bc_*(Y \times I)$. |
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37
diff
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|
956 |
|
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37
diff
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|
957 |
\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
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scott@6e1638ff-ae45-0410-89bd-df963105f760
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37
diff
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|
958 |
$0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
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37
diff
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|
959 |
$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
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37
diff
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|
960 |
$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
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37
diff
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|
961 |
\begin{equation*} |
b7bc1a931b73
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37
diff
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|
962 |
\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} |
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37
diff
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|
963 |
\end{equation*} |
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parents:
37
diff
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|
964 |
\todo{How do you write self tensor product?} |
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37
diff
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|
965 |
\end{itemize} |
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37
diff
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|
966 |
|
b7bc1a931b73
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37
diff
changeset
|
967 |
Although this gluing formula is stated in terms of $A_\infty$ categories and their (bi-)modules, it will be more natural for us to give alternative |
b7bc1a931b73
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scott@6e1638ff-ae45-0410-89bd-df963105f760
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37
diff
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|
968 |
definitions of `topological' $A_\infty$-categories and their bimodules, explain how to translate between the `algebraic' and `topological' definitions, |
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37
diff
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|
969 |
and then prove the gluing formula in the topological langauge. Section \ref{sec:topological-A-infty} below explains these definitions, and establishes |
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37
diff
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|
970 |
the desired equivalence. This is quite involved, and in particular requires us to generalise the definition of blob homology to allow $A_\infty$ algebras |
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37
diff
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|
971 |
as inputs, and to re-establish many of the properties of blob homology in this generality. Many readers may prefer to read the |
b7bc1a931b73
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37
diff
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|
972 |
Definitions \ref{defn:topological-algebra} and \ref{defn:topological-module} of `topological' $A_\infty$-categories, and Definition \ref{???} of the |
b7bc1a931b73
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37
diff
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|
973 |
self-tensor product of a `topological' $A_\infty$-bimodule, then skip to \S \ref{sec:boundary-action} and \S \ref{sec:gluing-formula} for the proofs |
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37
diff
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|
974 |
of the gluing formula in the topological context. |
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|
975 |
|
22
ada83e7228eb
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21
diff
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|
976 |
\subsection{`Topological' $A_\infty$ $n$-categories} |
ada83e7228eb
rearranging; stating all the "properties" up front
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diff
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|
977 |
\label{sec:topological-A-infty}% |
ada83e7228eb
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21
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|
978 |
|
23
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|
979 |
This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$-$n$-category}. |
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|
980 |
The main result of this section is |
22
ada83e7228eb
rearranging; stating all the "properties" up front
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21
diff
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|
981 |
|
ada83e7228eb
rearranging; stating all the "properties" up front
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21
diff
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|
982 |
\begin{thm} |
40
b7bc1a931b73
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|
983 |
Topological $A_\infty$-$1$-categories are equivalent to the usual notion of |
23
7b0a43bdd3c4
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diff
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|
984 |
$A_\infty$-$1$-categories. |
22
ada83e7228eb
rearranging; stating all the "properties" up front
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21
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|
985 |
\end{thm} |
18
aac9fd8d6bc6
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17
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|
986 |
|
37
2f677e283c26
adding some things to the bibliography
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36
diff
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|
987 |
Before proving this theorem, we embark upon a long string of definitions. |
40
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diff
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|
988 |
For expository purposes, we begin with the $n=1$ special cases,\scott{Why are we treating the $n>1$ cases at all?} and define |
23
7b0a43bdd3c4
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diff
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|
989 |
first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn |
7b0a43bdd3c4
writing definitions of topological a_\infty categories, modules, etc.
scott@6e1638ff-ae45-0410-89bd-df963105f760
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diff
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|
990 |
to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. |
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writing definitions of topological a_\infty categories, modules, etc.
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|
991 |
\nn{Something about duals?} |
7b0a43bdd3c4
writing definitions of topological a_\infty categories, modules, etc.
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diff
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|
992 |
\todo{Explain that we're not making contact with any previous notions for the general $n$ case?} |
32
538f38ddf395
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|
993 |
\kevin{probably we should say something about the relation |
37
2f677e283c26
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|
994 |
to [framed] $E_\infty$ algebras |
2f677e283c26
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|
995 |
} |
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diff
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|
996 |
|
2f677e283c26
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|
997 |
\todo{} |
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diff
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|
998 |
Various citations we might want to make: |
2f677e283c26
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diff
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|
999 |
\begin{itemize} |
2f677e283c26
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diff
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|
1000 |
\item \cite{MR2061854} McClure and Smith's review article |
2f677e283c26
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|
1001 |
\item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad) |
2f677e283c26
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|
1002 |
\item \cite{MR0236922,MR0420609} Boardman and Vogt |
2f677e283c26
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|
1003 |
\item \cite{MR1256989} definition of framed little-discs operad |
2f677e283c26
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|
1004 |
\end{itemize} |
0 | 1005 |
|
23
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|
1006 |
\begin{defn} |
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|
1007 |
\label{defn:topological-algebra}% |
32
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|
1008 |
A ``topological $A_\infty$-algebra'' $A$ consists of the following data. |
23
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|
1009 |
\begin{enumerate} |
37
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|
1010 |
\item For each $1$-manifold $J$ diffeomorphic to the standard interval |
32
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|
1011 |
$I=\left[0,1\right]$, a complex of vector spaces $A(J)$. |
25 | 1012 |
% either roll functoriality into the evaluation map |
37
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|
1013 |
\item For each pair of intervals $J,J'$ an `evaluation' chain map |
32
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|
1014 |
$\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$. |
538f38ddf395
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|
1015 |
\item For each decomposition of intervals $J = J'\cup J''$, |
40
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|
1016 |
a gluing map $\gl_{J',J''} : A(J') \tensor A(J'') \to A(J)$. |
25 | 1017 |
% or do it as two separate pieces of data |
1018 |
%\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, |
|
1019 |
%\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, |
|
1020 |
%\item and for each pair of intervals $J,J'$ a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$, |
|
23
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|
1021 |
\end{enumerate} |
32
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|
1022 |
This data is required to satisfy the following conditions. |
23
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|
1023 |
\begin{itemize} |
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|
1024 |
\item The evaluation chain map is associative, in that the diagram |
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|
1025 |
\begin{equation*} |
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|
1026 |
\xymatrix{ |
40
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|
1027 |
& \quad \mathclap{\CD{J' \to J''} \tensor \CD{J \to J'} \tensor A(J)} \quad \ar[dr]^{\id \tensor \ev_{J \to J'}} \ar[dl]_{\compose \tensor \id} & \\ |
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|
1028 |
\CD{J' \to J''} \tensor A(J') \ar[dr]^{\ev_{J' \to J''}} & & \CD{J \to J''} \tensor A(J) \ar[dl]_{\ev_{J \to J''}} \\ |
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|
1029 |
& A(J'') & |
23
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|
1030 |
} |
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|
1031 |
\end{equation*} |
40
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|
1032 |
commutes up to homotopy. |
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|
1033 |
Here the map $$\compose : \CD{J' \to J''} \tensor \CD{J \to J'} \to \CD{J \to J''}$$ is a composition: take products of singular chains first, then compose diffeomorphisms. |
25 | 1034 |
%% or the version for separate pieces of data: |
1035 |
%\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. |
|
1036 |
%\item The evaluation chain map is associative, in that the diagram |
|
1037 |
%\begin{equation*} |
|
1038 |
%\xymatrix{ |
|
40
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|
1039 |
%\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\id \tensor \ev_J} \ar[d]_{\compose \tensor \id} & |
25 | 1040 |
%\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\ |
1041 |
%\CD{J} \tensor A(J) \ar[r]_{\ev_J} & |
|
1042 |
%A(J) |
|
1043 |
%} |
|
1044 |
%\end{equation*} |
|
1045 |
%commutes. (Here the map $\compose : \CD{J} \tensor \CD{J} \to \CD{J}$ is a composition: take products of singular chains first, then use the group multiplication in $\Diff(J)$.) |
|
23
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|
1046 |
\item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram |
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|
1047 |
\begin{equation*} |
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|
1048 |
\xymatrix{ |
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|
1049 |
A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \id} \ar[d]_{\id \tensor \gl_{J',J''}} && |
23
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|
1050 |
A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\ |
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|
1051 |
A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} && |
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|
1052 |
A(J \cup J' \cup J'') |
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|
1053 |
} |
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|
1054 |
\end{equation*} |
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|
1055 |
commutes. |
34 | 1056 |
\item The gluing and evaluation maps are compatible. |
1057 |
\nn{give diagram, or just say ``in the obvious way", or refer to diagram in blob eval map section?} |
|
23
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|
1058 |
\end{itemize} |
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|
1059 |
\end{defn} |
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|
1060 |
|
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|
1061 |
\begin{rem} |
26 | 1062 |
We can restrict the evaluation map to $0$-chains, and see that $J \mapsto A(J)$ and $(\phi:J \to J') \mapsto \ev_{J \to J'}(\phi, \bullet)$ together |
25 | 1063 |
constitute a functor from the category of intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
1064 |
Further, once this functor has been specified, we only need to know how the evaluation map acts when $J = J'$. |
|
23
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|
1065 |
\end{rem} |
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|
1066 |
|
25 | 1067 |
%% if we do things separately, we should say this: |
1068 |
%\begin{rem} |
|
1069 |
%Of course, the first and third pieces of data (the complexes, and the isomorphisms) together just constitute a functor from the category of |
|
1070 |
%intervals and diffeomorphisms between them to the category of complexes of vector spaces. |
|
1071 |
%Further, one can combine the second and third pieces of data, asking instead for a map |
|
1072 |
%\begin{equation*} |
|
1073 |
%\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J'). |
|
1074 |
%\end{equation*} |
|
1075 |
%(Any $k$-parameter family of diffeomorphisms in $C_k(\Diff(J \to J'))$ factors into a single diffeomorphism $J \to J'$ and a $k$-parameter family of |
|
1076 |
%diffeomorphisms in $\CD{J'}$.) |
|
1077 |
%\end{rem} |
|
1078 |
||
23
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|
1079 |
To generalise the definition to that of a category, we simply introduce a set of objects which we call $A(pt)$. Now we associate complexes to each |
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|
1080 |
interval with boundary conditions $(J, c_-, c_+)$, with $c_-, c_+ \in A(pt)$, and only ask for gluing maps when the boundary conditions match up: |
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|
1081 |
\begin{equation*} |
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|
1082 |
\gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). |
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|
1083 |
\end{equation*} |
25 | 1084 |
The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions. |
40
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|
1085 |
\todo{we presumably need to say something about $\id_c \in A(J, c, c)$.} |
23
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|
1086 |
|
25 | 1087 |
At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$. |
1088 |
\begin{defn} |
|
26 | 1089 |
Define the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ by |
25 | 1090 |
\begin{enumerate} |
1091 |
\item $A(J) = C_*(\Maps(J \to M))$, singular chains on the space of smooth maps from $J$ to $M$, |
|
26 | 1092 |
\item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition |
1093 |
\begin{align*} |
|
1094 |
\CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), |
|
1095 |
\end{align*} |
|
40
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|
1096 |
where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism, |
25 | 1097 |
\item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. |
1098 |
\end{enumerate} |
|
1099 |
The associativity conditions are trivially satisfied. |
|
1100 |
\end{defn} |
|
1101 |
||
1102 |
The second example is simply the blob complex of $Y \times J$, for any $n-1$ manifold $Y$. We define $A(J) = \bc_*(Y \times J)$. |
|
1103 |
Observe $\Diff(J \to J')$ embeds into $\Diff(Y \times J \to Y \times J')$. The evaluation and gluing maps then come directly from Properties |
|
40
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|
1104 |
\ref{property:evaluation} and \ref{property:gluing-map} respectively. We'll often write $bc_*(Y)$ for this algebra. |
25 | 1105 |
|
23
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|
1106 |
The definition of a module follows closely the definition of an algebra or category. |
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|
1107 |
\begin{defn} |
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|
1108 |
\label{defn:topological-module}% |
37
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|
1109 |
A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ |
33
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|
1110 |
consists of the following data. |
23
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|
1111 |
\begin{enumerate} |
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|
1112 |
\item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with the upper boundary point `marked', to complexes of vector spaces. |
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|
1113 |
\item For each pair of such marked intervals, |
33
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|
1114 |
an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$. |
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|
1115 |
\item For each decomposition $K = J\cup K'$ of the marked interval |
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|
1116 |
$K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map |
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|
1117 |
$\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$. |
23
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|
1118 |
\end{enumerate} |
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|
1119 |
The above data is required to satisfy |
33
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|
1120 |
conditions analogous to those in Definition \ref{defn:topological-algebra}. |
23
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|
1121 |
\end{defn} |
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|
1122 |
|
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|
1123 |
For any manifold $X$ with $\bdy X = Y$ (or indeed just with $Y$ a codimension $0$-submanifold of $\bdy X$) we can think of $\bc_*(X)$ as |
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|
1124 |
a topological $A_\infty$ module over $\bc_*(Y)$, the topological $A_\infty$ category described above. |
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|
1125 |
For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. |
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|
1126 |
(Here we glue $Y \times pt$ to $Y \subset \bdy X$, where $pt$ is the marked point of $K$.) Again, the evaluation and gluing maps come directly from Properties |
26 | 1127 |
\ref{property:evaluation} and \ref{property:gluing-map} respectively. |
1128 |
||
34 | 1129 |
The definition of a bimodule is like the definition of a module, |
1130 |
except that we have two disjoint marked intervals $K$ and $L$, one with a marked point |
|
1131 |
on the upper boundary and the other with a marked point on the lower boundary. |
|
1132 |
There are evaluation maps corresponding to gluing unmarked intervals |
|
1133 |
to the unmarked ends of $K$ and $L$. |
|
1134 |
||
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|
1135 |
Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a |
34 | 1136 |
codimension-0 submanifold of $\bdy X$. |
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|
1137 |
Then the the assignment $K,L \mapsto \bc_*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the |
34 | 1138 |
structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$. |
1139 |
||
1140 |
Next we define the coend |
|
1141 |
(or gluing or tensor product or self tensor product, depending on the context) |
|
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|
1142 |
$\gl(M)$ of a topological $A_\infty$ bimodule $M$. This will be an `initial' or `universal' object satisfying various properties. |
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|
1143 |
\begin{defn} |
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|
1144 |
We define a category $\cG(M)$. Objects consist of the following data. |
35 | 1145 |
\begin{itemize} |
1146 |
\item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N). |
|
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|
1147 |
\item For each pair of intervals $N,N'$ an evaluation chain map |
35 | 1148 |
$\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$. |
1149 |
\item For each decomposition of intervals $N = K\cup L$, |
|
1150 |
a gluing map $\gl_{K,L} : M(K,L) \to C(N)$. |
|
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|
1151 |
\end{itemize} |
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|
1152 |
This data must satisfy the following conditions. |
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|
1153 |
\begin{itemize} |
35 | 1154 |
\item The evaluation maps are associative. |
1155 |
\nn{up to homotopy?} |
|
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|
1156 |
\item Gluing is strictly associative. |
35 | 1157 |
That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to |
1158 |
$K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$ |
|
1159 |
agree. |
|
1160 |
\item the gluing and evaluation maps are compatible. |
|
1161 |
\end{itemize} |
|
34 | 1162 |
|
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|
1163 |
A morphism $f$ between such objects $C$ and $C'$ is a chain map $f_N : C(N) \to C'(N)$ for each interval $N$ with both endpoints marked, |
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|
1164 |
satisfying the following conditions. |
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|
1165 |
\begin{itemize} |
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|
1166 |
\item For each pair of intervals $N,N'$, the diagram |
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|
1167 |
\begin{equation*} |
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|
1168 |
\xymatrix{ |
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|
1169 |
\CD{N \to N'} \tensor C(N) \ar[d]_{\ev} \ar[r]^{\id \tensor f_N} & \CD{N \to N'} \tensor C'(N) \ar[d]^{\ev} \\ |
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|
1170 |
C(N) \ar[r]_{f_N} & C'(N) |
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|
1171 |
} |
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|
1172 |
\end{equation*} |
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|
1173 |
commutes. |
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|
1174 |
\item For each decomposition of intervals $N = K \cup L$, the gluing map for $C'$, $\gl'_{K,L} : M(K,L) \to C'(N)$ is the composition |
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|
1175 |
$$M(K,L) \xto{\gl_{K,L}} C(N) \xto{f_N} C'(N).$$ |
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|
1176 |
\end{itemize} |
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|
1177 |
\end{defn} |
34 | 1178 |
|
40
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|
1179 |
We now define $\gl(M)$ to be an initial object in the category $\cG{M}$. This just says that for any other object $C'$ in $\cG{M}$, |
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|
1180 |
there are chain maps $f_N: \gl(M)(N) \to C'(N)$, compatible with the action of families of diffeomorphisms, so that the gluing maps $M(K,L) \to C'(N)$ |
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|
1181 |
factor through the gluing maps for $\gl(M)$. |
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|
1182 |
|
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|
1183 |
We return to our two favourite examples. First, the coend of the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ as a bimodule over itself |
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|
1184 |
is essentially $C_*(\Maps(S^1 \to M))$. \todo{} |
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|
1185 |
|
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|
1186 |
For the second example, given $X$ and $Y\du -Y \sub \bdy X$, the assignment |
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|
1187 |
$$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y))$$ clearly gives an object in $\cG{M}$. |
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|
1188 |
Showing that it is an initial object is the content of the gluing theorem proved below. |
34 | 1189 |
|
35 | 1190 |
The definitions for a topological $A_\infty$-$n$-category are very similar to the above |
1191 |
$n=1$ case. |
|
1192 |
One replaces intervals with manifolds diffeomorphic to the ball $B^n$. |
|
1193 |
Marked points are replaced by copies of $B^{n-1}$ in $\bdy B^n$. |
|
1194 |
||
1195 |
\nn{give examples: $A(J^n) = \bc_*(Z\times J)$ and $A(J^n) = C_*(\Maps(J \to M))$.} |
|
23
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|
1196 |
|
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|
1197 |
\todo{the motivating example $C_*(\maps(X, M))$} |
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|
1198 |
|
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|
1199 |
|
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|
1200 |
|
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|
1201 |
\newcommand{\skel}[1]{\operatorname{skeleton}(#1)} |
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|
1202 |
|
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|
1203 |
Given a topological $A_\infty$-category $\cC$, we can construct an `algebraic' $A_\infty$ category $\skel{\cC}$. First, pick your |
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|
1204 |
favorite diffeomorphism $\phi: I \cup I \to I$. |
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|
1205 |
\begin{defn} |
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|
1206 |
We'll write $\skel{\cC} = (A, m_k)$. Define $A = \cC(I)$, and $m_2 : A \tensor A \to A$ by |
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|
1207 |
\begin{equation*} |
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|
1208 |
m_2 \cC(I) \tensor \cC(I) \xrightarrow{\gl_{I,I}} \cC(I \cup I) \xrightarrow{\cC(\phi)} \cC(I). |
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|
1209 |
\end{equation*} |
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|
1210 |
Next, we define all the `higher associators' $m_k$ by |
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|
1211 |
\todo{} |
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|
1212 |
\end{defn} |
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|
1213 |
|
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|
1214 |
Give an `algebraic' $A_\infty$ category $(A, m_k)$, we can construct a topological $A_\infty$-category, which we call $\bc_*^A$. You should |
24 | 1215 |
think of this as a generalisation of the blob complex, although the construction we give will \emph{not} specialise to exactly the usual definition |
23
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|
1216 |
in the case the $A$ is actually an associative category. |
26 | 1217 |
|
1218 |
We'll first define $\cT_{k,n}$ to be the set of planar forests consisting of $n-k$ trees, with a total of $n$ leaves. Thus |
|
1219 |
\todo{$\cT_{0,n}$ has 1 element, with $n$ vertical lines, $\cT_{1,n}$ has $n-1$ elements, each with a single trivalent vertex, $\cT_{2,n}$ etc...} |
|
1220 |
\begin{align*} |
|
1221 |
\end{align*} |
|
1222 |
||
23
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|
1223 |
\begin{defn} |
26 | 1224 |
The topological $A_\infty$ category $\bc_*^A$ is doubly graded, by `blob degree' and `internal degree'. We'll write $\bc_k^A$ for the blob degree $k$ piece. |
1225 |
The homological degree of an element $a \in \bc_*^A(J)$ |
|
1226 |
is the sum of the blob degree and the internal degree. |
|
1227 |
||
1228 |
We first define $\bc_0^A(J)$ as a vector space by |
|
1229 |
\begin{equation*} |
|
30 | 1230 |
\bc_0^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A). |
26 | 1231 |
\end{equation*} |
1232 |
(That is, for each division of $J$ into finitely many subintervals, |
|
1233 |
we have the tensor product of chains of diffeomorphisms from each subinterval to the standard interval, |
|
1234 |
and a copy of $A$ for each subinterval.) |
|
1235 |
The internal degree of an element $(f_1 \tensor a_1, \ldots, f_n \tensor a_n)$ is the sum of the dimensions of the singular chains |
|
1236 |
plus the sum of the homological degrees of the elements of $A$. |
|
1237 |
The differential is defined just by the graded Leibniz rule and the differentials on $\CD{J_i \to I}$ and on $A$. |
|
1238 |
||
1239 |
Next, |
|
1240 |
\begin{equation*} |
|
30 | 1241 |
\bc_1^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \DirectSum_{T \in \cT_{1,n}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A). |
26 | 1242 |
\end{equation*} |
23
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|
1243 |
\end{defn} |
0 | 1244 |
|
30 | 1245 |
\begin{figure}[!ht] |
1246 |
\begin{equation*} |
|
1247 |
\mathfig{0.7}{associahedron/A4-vertices} |
|
1248 |
\end{equation*} |
|
1249 |
\caption{The vertices of the $k$-dimensional associahedron are indexed by binary trees on $k+2$ leaves.} |
|
1250 |
\label{fig:A4-vertices} |
|
1251 |
\end{figure} |
|
26 | 1252 |
|
30 | 1253 |
\begin{figure}[!ht] |
1254 |
\begin{equation*} |
|
1255 |
\mathfig{0.7}{associahedron/A4-faces} |
|
1256 |
\end{equation*} |
|
1257 |
\caption{The faces of the $k$-dimensional associahedron are indexed by trees with $2$ vertices on $k+2$ leaves.} |
|
1258 |
\label{fig:A4-vertices} |
|
1259 |
\end{figure} |
|
1260 |
||
1261 |
\newcommand{\tm}{\widetilde{m}} |
|
26 | 1262 |
|
1263 |
Let $\tm_1(a) = a$. |
|
1264 |
||
30 | 1265 |
We now define $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$, first giving an opaque formula, then explaining the combinatorics behind it. |
1266 |
\begin{align} |
|
1267 |
\notag \bdy(\tm_k(a_1 & \tensor \cdots \tensor a_k)) = \\ |
|
36 | 1268 |
\label{eq:bdy-tm-k-1} & \phantom{+} \sum_{\ell'=0}^{k-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_k(a_1 \tensor \cdots \tensor \bdy a_{\ell'+1} \tensor \cdots \tensor a_k) + \\ |
30 | 1269 |
\label{eq:bdy-tm-k-2} & + \sum_{\ell=1}^{k-1} \tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k) + \\ |
36 | 1270 |
\label{eq:bdy-tm-k-3} & + \sum_{\ell=1}^{k-1} \sum_{\ell'=0}^{l-1} (-1)^{\abs{\tm_k}+\sum_{j=1}^{\ell'} \abs{a_j}} \tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell + 1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell + 1}) \tensor \cdots \tensor a_k) |
30 | 1271 |
\end{align} |
1272 |
The first set of terms in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ just have $\bdy$ acting on each argument $a_i$. |
|
1273 |
The terms appearing in \eqref{eq:bdy-tm-k-2} and \eqref{eq:bdy-tm-k-3} are indexed by trees with $2$ vertices on $k+1$ leaves. |
|
1274 |
Note here that we have one more leaf than there arguments of $\tm_k$. |
|
1275 |
(See Figure \ref{fig:A4-vertices}, in which the rightmost branches are helpfully drawn in red.) |
|
1276 |
We will treat the vertices which involve a rightmost (red) branch differently from the vertices which only involve the first $k$ leaves. |
|
1277 |
The terms in \eqref{eq:bdy-tm-k-2} arise in the cases in which both |
|
1278 |
vertices are rightmost, and the corresponding term in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ is a tensor product of the form |
|
1279 |
$$\tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k)$$ |
|
1280 |
where $\ell + 1$ and $k - \ell + 1$ are the number of branches entering the vertices. |
|
1281 |
If only one vertex is rightmost, we get the term $$\tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell+1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell}) \tensor \cdots \tensor a_k)$$ |
|
1282 |
in \eqref{eq:bdy-tm-k-3}, |
|
1283 |
where again $\ell + 1$ is the number of branches entering the rightmost vertex, $k-\ell+1$ is the number of branches entering the other vertex, and $\ell'$ is the number of edges meeting the rightmost vertex which start to the left of the other vertex. |
|
1284 |
For example, we have |
|
26 | 1285 |
\begin{align*} |
36 | 1286 |
\bdy(\tm_2(a \tensor b)) & = \left(\tm_2(\bdy a \tensor b) + (-1)^{\abs{a}} \tm_2(a \tensor \bdy b)\right) + \\ |
1287 |
& \qquad - a \tensor b + m_2(a \tensor b) \\ |
|
1288 |
\bdy(\tm_3(a \tensor b \tensor c)) & = \left(- \tm_3(\bdy a \tensor b \tensor c) + (-1)^{\abs{a} + 1} \tm_3(a \tensor \bdy b \tensor c) + (-1)^{\abs{a} + \abs{b} + 1} \tm_3(a \tensor b \tensor \bdy c)\right) + \\ |
|
1289 |
& \qquad + \left(- \tm_2(a \tensor b) \tensor c + a \tensor \tm_2(b \tensor c)\right) + \\ |
|
1290 |
& \qquad + \left(- \tm_2(m_2(a \tensor b) \tensor c) + \tm_2(a, m_2(b \tensor c)) + m_3(a \tensor b \tensor c)\right) |
|
30 | 1291 |
\end{align*} |
1292 |
\begin{align*} |
|
1293 |
\bdy(& \tm_4(a \tensor b \tensor c \tensor d)) = \left(\tm_4(\bdy a \tensor b \tensor c \tensor d) + \cdots + \tm_4(a \tensor b \tensor c \tensor \bdy d)\right) + \\ |
|
1294 |
& + \left(\tm_3(a \tensor b \tensor c) \tensor d + \tm_2(a \tensor b) \tensor \tm_2(c \tensor d) + a \tensor \tm_3(b \tensor c \tensor d)\right) + \\ |
|
1295 |
& + \left(\tm_3(m_2(a \tensor b) \tensor c \tensor d) + \tm_3(a \tensor m_2(b \tensor c) \tensor d) + \tm_3(a \tensor b \tensor m_2(c \tensor d))\right. + \\ |
|
1296 |
& + \left.\tm_2(m_3(a \tensor b \tensor c) \tensor d) + \tm_2(a \tensor m_3(b \tensor c \tensor d)) + m_4(a \tensor b \tensor c \tensor d)\right) \\ |
|
26 | 1297 |
\end{align*} |
30 | 1298 |
See Figure \ref{fig:A4-terms}, comparing it against Figure \ref{fig:A4-faces}, to see this illustrated in the case $k=4$. There the $3$ faces closest |
1299 |
to the top of the diagram have two rightmost vertices, while the other $6$ faces have only one. |
|
1300 |
||
1301 |
\begin{figure}[!ht] |
|
1302 |
\begin{equation*} |
|
1303 |
\mathfig{1.0}{associahedron/A4-terms} |
|
1304 |
\end{equation*} |
|
1305 |
\caption{The terms of $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ correspond to the faces of the $k-1$ dimensional associahedron.} |
|
1306 |
\label{fig:A4-terms} |
|
1307 |
\end{figure} |
|
26 | 1308 |
|
31
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|
1309 |
\begin{lem} |
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|
1310 |
This definition actually results in a chain complex, that is $\bdy^2 = 0$. |
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|
1311 |
\end{lem} |
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|
1312 |
\begin{proof} |
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|
1313 |
\newcommand{\T}{\text{---}} |
e155c518ce31
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|
1314 |
\newcommand{\ssum}[1]{{\sum}^{(#1)}} |
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|
1315 |
For the duration of this proof, inside a summation over variables $l_1, \ldots, l_m$, an expression with $m$ dashes will be interpreted |
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|
1316 |
by replacing each dash with contiguous factors from $a_1 \tensor \cdots \tensor a_k$, so the first dash takes the first $l_1$ factors, the second |
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|
1317 |
takes the next $l_2$ factors, and so on. Further, we'll write $\ssum{m}$ for $\sum_{\sum_{i=1}^m l_i = k}$. |
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|
1318 |
In this notation, the formula for the differential becomes |
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|
1319 |
\begin{align} |
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|
1320 |
\notag |
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|
1321 |
\bdy \tm(\T) & = \ssum{2} \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} + \ssum{3} \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\ |
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|
1322 |
\intertext{and we calculate} |
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|
1323 |
\notag |
36 | 1324 |
\bdy^2 \tm(\T) & = \ssum{2} \bdy \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2} \\ |
1325 |
\notag & \qquad + \ssum{2} \tm(\T) \tensor \bdy \tm(\T) \times \sigma_{0;l_1,l_2} \\ |
|
31
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|
1326 |
\notag & \qquad + \ssum{3} \bdy \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3} \\ |
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|
1327 |
\label{eq:d21} & = \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2,l_3} \sigma_{0;l_1,l_2} \\ |
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|
1328 |
\label{eq:d22} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \sigma_{0;l_1+l_2+l_3,l_4} \tau_{0;l_1,l_2,l_3} \\ |
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|
1329 |
\label{eq:d23} & \qquad + \ssum{3} \tm(\T) \tensor \tm(\T) \tensor \tm(\T) \times \sigma_{0;l_1,l_2+l_3} \sigma_{l_1;l_2,l_3} \\ |
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|
1330 |
\label{eq:d24} & \qquad + \ssum{4} \tm(\T) \tensor \tm(\T \tensor m(\T) \tensor \T) \times \sigma_{0;l_1,l_2+l_3+l_4} \tau_{l_1;l_2,l_3,l_4} \\ |
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|
1331 |
\label{eq:d25} & \qquad + \ssum{4} \tm(\T \tensor m(\T) \tensor \T) \tensor \tm(\T) \times \tau_{0;l_1,l_2,l_3+l_4} ??? \\ |
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|
1332 |
\label{eq:d26} & \qquad + \ssum{4} \tm(\T) \tensor \tm(\T \tensor m(\T) \tensor \T) \times \tau_{0;l_1+l_2,l_3,l_4} \sigma_{0;l_1,l_2} \\ |
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|
1333 |
\label{eq:d27} & \qquad + \ssum{5} \tm(\T \tensor m(\T) \tensor \T \tensor m(\T) \tensor \T) \times \tau_{0;l_1+l_2+l_3,l_4,l_5} \tau_{0;l_1,l_2,l_3} \\ |
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|
1334 |
\label{eq:d28} & \qquad + \ssum{5} \tm(\T \tensor m(\T \tensor m(\T) \tensor \T) \tensor \T) \times \tau_{0;l_1,l_2+l_3+l_4,l_5} ??? \\ |
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|
1335 |
\label{eq:d29} & \qquad + \ssum{5} \tm(\T \tensor m(\T) \tensor \T \tensor m(\T) \tensor \T) \times \tau_{0;l_1,l_2,l_3+l_4+l_5} ??? |
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|
1336 |
\end{align} |
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|
1337 |
Now, we see the the expressions on the right hand side of line \eqref{eq:d21} and those on \eqref{eq:d23} cancel. Similarly, line \eqref{eq:d22} cancels |
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|
1338 |
with \eqref{eq:d25}, \eqref{eq:d24} with \eqref{eq:d26}, and \eqref{eq:d27} with \eqref{eq:d29}. Finally, we need to see that \eqref{eq:d28} gives $0$, |
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|
1339 |
by the usual relations between the $m_k$ in an $A_\infty$ algebra. |
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|
1340 |
\end{proof} |
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|
1341 |
|
22
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1342 |
\nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
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|
1343 |
$n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty |
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|
1344 |
easy, I think, so maybe it should be done earlier??} |
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1345 |
|
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|
1346 |
\bigskip |
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1347 |
|
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|
1348 |
Outline: |
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|
1349 |
\begin{itemize} |
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1350 |
\item recall defs of $A_\infty$ category (1-category only), modules, (self-) tensor product. |
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|
1351 |
use graphical/tree point of view, rather than following Keller exactly |
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1352 |
\item define blob complex in $A_\infty$ case; fat mapping cones? tree decoration? |
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1353 |
\item topological $A_\infty$ cat def (maybe this should go first); also modules gluing |
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1354 |
\item motivating example: $C_*(\maps(X, M))$ |
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1355 |
\item maybe incorporate dual point of view (for $n=1$), where points get |
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|
1356 |
object labels and intervals get 1-morphism labels |
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|
1357 |
\end{itemize} |
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1358 |
|
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|
1359 |
|
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1360 |
\subsection{$A_\infty$ action on the boundary} |
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|
1361 |
\label{sec:boundary-action}% |
22
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|
1362 |
Let $Y$ be an $n{-}1$-manifold. |
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|
1363 |
The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary |
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|
1364 |
conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure |
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|
1365 |
of an $A_\infty$ category. |
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|
1366 |
|
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|
1367 |
Composition of morphisms (multiplication) depends of a choice of homeomorphism |
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|
1368 |
$I\cup I \cong I$. Given this choice, gluing gives a map |
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|
1369 |
\eq{ |
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|
1370 |
\bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) |
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1371 |
\cong \bc_*(Y\times I; a, c) |
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|
1372 |
} |
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|
1373 |
Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various |
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|
1374 |
higher associators of the $A_\infty$ structure, more or less canonically. |
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|
1375 |
|
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|
1376 |
\nn{is this obvious? does more need to be said?} |
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|
1377 |
|
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|
1378 |
Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. |
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|
1379 |
|
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|
1380 |
Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism |
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|
1381 |
$(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ |
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|
1382 |
(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the |
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|
1383 |
$A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. |
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|
1384 |
Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood |
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|
1385 |
of $Y$ in $X$. |
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|
1386 |
|
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|
1387 |
In the next section we use the above $A_\infty$ actions to state and prove |
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|
1388 |
a gluing theorem for the blob complexes of $n$-manifolds. |
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|
1389 |
|
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|
1390 |
|
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|
1391 |
\subsection{The gluing formula} |
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|
1392 |
\label{sec:gluing-formula}% |
22
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|
1393 |
Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy |
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|
1394 |
of $Y \du -Y$ contained in its boundary. |
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|
1395 |
Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. |
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|
1396 |
We wish to describe the blob complex of $X\sgl$ in terms of the blob complex |
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|
1397 |
of $X$. |
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|
1398 |
More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, |
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|
1399 |
where $c\sgl \in \cC(\bd X\sgl)$, |
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|
1400 |
in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation |
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|
1401 |
of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. |
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|
1402 |
|
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|
1403 |
\begin{thm} |
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|
1404 |
$\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product |
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|
1405 |
of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. |
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|
1406 |
\end{thm} |
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|
1407 |
|
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|
1408 |
The proof will occupy the remainder of this section. |
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|
1409 |
|
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|
1410 |
\nn{...} |
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|
1411 |
|
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|
1412 |
\bigskip |
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|
1413 |
|
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|
1414 |
\nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} |
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|
1415 |
|
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|
1416 |
|
47
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|
1417 |
\section{Commutative algebras as $n$-categories} |
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|
1418 |
|
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|
1419 |
\nn{this should probably not be a section by itself. i'm just trying to write down the outline |
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|
1420 |
while it's still fresh in my mind.} |
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|
1421 |
|
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|
1422 |
If $C$ is a commutative algebra it |
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|
1423 |
can (and will) also be thought of as an $n$-category with trivial $j$-morphisms for |
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|
1424 |
$j<n$ and $n$-morphisms are $C$. |
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|
1425 |
The goal of this \nn{subsection?} is to compute |
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|
1426 |
$\bc_*(M^n, C)$ for various commutative algebras $C$. |
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|
1427 |
|
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|
1428 |
Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
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|
1429 |
|
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|
1430 |
Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
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|
1431 |
unlabeled points in $M$. |
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|
1432 |
Note that $\Sigma^i(M)$ is a point. |
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|
1433 |
Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
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|
1434 |
|
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|
1435 |
Let $C_*(X)$ denote the singular chain complex of the space $X$. |
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|
1436 |
|
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|
1437 |
\begin{prop} |
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|
1438 |
$\bc_*(M^n, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M))$. |
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|
1439 |
\end{prop} |
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|
1440 |
|
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|
1441 |
\begin{proof} |
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|
1442 |
To define the chain maps between the two complexes we will use the following lemma: |
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|
1443 |
|
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|
1444 |
\begin{lemma} |
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|
1445 |
Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with |
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|
1446 |
a basis (e.g.\ blob diagrams or singular simplices). |
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|
1447 |
For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ |
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|
1448 |
such that $R(c') \sub R(c)$ whenever $c'$ is a basis element which is part of $\bd c$. |
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|
1449 |
Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that |
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|
1450 |
$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
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|
1451 |
\end{lemma} |
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|
1452 |
|
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|
1453 |
\begin{proof} |
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|
1454 |
\nn{easy, but should probably write the details eventually} |
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|
1455 |
\end{proof} |
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|
1456 |
|
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|
1457 |
\nn{...} |
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|
1458 |
|
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|
1459 |
\end{proof} |
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|
1460 |
|
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|
1461 |
|
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|
1462 |
|
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|
1463 |
|
22
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|
1464 |
|
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|
1465 |
|
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|
1466 |
\appendix |
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|
1467 |
|
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|
1468 |
\section{Families of Diffeomorphisms} \label{sec:localising} |
0 | 1469 |
|
1470 |
||
1471 |
Lo, the proof of Lemma (\ref{extension_lemma}): |
|
1472 |
||
1473 |
\nn{should this be an appendix instead?} |
|
1474 |
||
1475 |
\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in |
|
1476 |
later draft} |
|
1477 |
||
1478 |
\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry |
|
1479 |
about boundary later} |
|
1480 |
||
8 | 1481 |
Recall that we are given |
0 | 1482 |
an open cover $\cU = \{U_\alpha\}$ and an |
1483 |
$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. |
|
1484 |
We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
|
1485 |
||
1486 |
Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. |
|
1487 |
||
1488 |
As a first approximation to the argument we will eventually make, let's replace $x$ |
|
8 | 1489 |
with a single singular cell |
0 | 1490 |
\eq{ |
8 | 1491 |
f: P \times X \to X . |
0 | 1492 |
} |
1493 |
Also, we'll ignore for now issues around $\bd P$. |
|
1494 |
||
1495 |
Our homotopy will have the form |
|
1496 |
\eqar{ |
|
8 | 1497 |
F: I \times P \times X &\to& X \\ |
1498 |
(t, p, x) &\mapsto& f(u(t, p, x), x) |
|
0 | 1499 |
} |
1500 |
for some function |
|
1501 |
\eq{ |
|
8 | 1502 |
u : I \times P \times X \to P . |
0 | 1503 |
} |
1504 |
First we describe $u$, then we argue that it does what we want it to do. |
|
1505 |
||
1506 |
For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$. |
|
1507 |
The various $K_\alpha$ should be in general position with respect to each other. |
|
1508 |
We will see below that the $K_\alpha$'s need to be sufficiently fine in order |
|
1509 |
to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not |
|
1510 |
merely a homotopy through maps $X\to X$. |
|
1511 |
||
1512 |
Let $L$ be the union of all the $K_\alpha$'s. |
|
1513 |
$L$ is itself a cell decomposition of $P$. |
|
1514 |
\nn{next two sentences not needed?} |
|
1515 |
To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$, |
|
1516 |
where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$. |
|
1517 |
Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$. |
|
1518 |
||
1519 |
Let $J$ denote the handle decomposition of $P$ corresponding to $L$. |
|
1520 |
Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and, |
|
1521 |
more importantly, a $k{-}i$-dimensional normal coordinate. |
|
1522 |
||
7
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|
1523 |
For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$. |
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|
1524 |
Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding |
0 | 1525 |
$k$-cell of $L$. |
1526 |
To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s |
|
1527 |
which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$. |
|
1528 |
||
1529 |
For $p \in D$ we define |
|
1530 |
\eq{ |
|
8 | 1531 |
u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . |
0 | 1532 |
} |
1533 |
(Recall that $P$ is a single linear cell, so the weighted average of points of $P$ |
|
1534 |
makes sense.) |
|
1535 |
||
1536 |
So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. |
|
8 | 1537 |
For handles of $J$ of index less than $k$, we will define $u$ to |
0 | 1538 |
interpolate between the values on $k$-handles defined above. |
1539 |
||
8 | 1540 |
If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate |
0 | 1541 |
of $E$. |
1542 |
In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ |
|
1543 |
with a $k$-handle. |
|
1544 |
Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell |
|
1545 |
corresponding to $E$. |
|
1546 |
Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$ |
|
1547 |
adjacent to the $k{-}1$-cell corresponding to $E$. |
|
1548 |
For $p \in E$, define |
|
1549 |
\eq{ |
|
8 | 1550 |
u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} |
1551 |
+ r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . |
|
0 | 1552 |
} |
1553 |
||
1554 |
In general, for $E$ a $k{-}j$-handle, there is a normal coordinate |
|
1555 |
$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron. |
|
1556 |
The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$. |
|
1557 |
If we triangulate $R$ (without introducing new vertices), we can linearly extend |
|
1
8174b33dda66
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|
1558 |
a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$. |
0 | 1559 |
Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets |
1560 |
the $k{-}j$-cell corresponding to $E$. |
|
1561 |
For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. |
|
1562 |
Now define, for $p \in E$, |
|
1563 |
\eq{ |
|
8 | 1564 |
u(t, p, x) = (1-t)p + t \left( |
1565 |
\sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} |
|
1566 |
+ \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) |
|
1567 |
\right) . |
|
0 | 1568 |
} |
1569 |
Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension |
|
1570 |
mentioned above. |
|
1571 |
||
1572 |
This completes the definition of $u: I \times P \times X \to P$. |
|
1573 |
||
1574 |
\medskip |
|
1575 |
||
1576 |
Next we verify that $u$ has the desired properties. |
|
1577 |
||
1578 |
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 $x$. |
|
1579 |
Therefore $F$ is a homotopy from $f$ to something. |
|
1580 |
||
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|
1581 |
Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions, |
0 | 1582 |
then $F$ is a homotopy through diffeomorphisms. |
1583 |
We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. |
|
1584 |
We have |
|
1585 |
\eq{ |
|
8 | 1586 |
% \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) . |
1587 |
\pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . |
|
0 | 1588 |
} |
1589 |
Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and |
|
1590 |
\nn{bounded away from zero, or something like that}. |
|
1591 |
(Recall that $X$ and $P$ are compact.) |
|
1592 |
Also, $\pd{f}{p}$ is bounded. |
|
1593 |
So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done. |
|
1594 |
It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$ |
|
7
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|
1595 |
(which is bounded) |
0 | 1596 |
and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s. |
1597 |
These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine. |
|
1598 |
This completes the proof that $F$ is a homotopy through diffeomorphisms. |
|
1599 |
||
1600 |
\medskip |
|
1601 |
||
1602 |
Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$ |
|
1603 |
is a singular cell adapted to $\cU$. |
|
1604 |
This will complete the proof of the lemma. |
|
1605 |
\nn{except for boundary issues and the `$P$ is a cell' assumption} |
|
1606 |
||
8 | 1607 |
Let $j$ be the codimension of $D$. |
0 | 1608 |
(Or rather, the codimension of its corresponding cell. From now on we will not make a distinction |
1609 |
between handle and corresponding cell.) |
|
1610 |
Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, |
|
1611 |
where the $j_i$'s are the codimensions of the $K_\alpha$ |
|
1612 |
cells of codimension greater than 0 which intersect to form $D$. |
|
1613 |
We will show that |
|
1614 |
if the relevant $U_\alpha$'s are disjoint, then |
|
1615 |
$F(1, \cdot, \cdot) : D\times X \to X$ |
|
1616 |
is a product of singular cells of dimensions $j_1, \ldots, j_m$. |
|
1617 |
If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular |
|
1618 |
cells whose dimensions correspond to a partition of the $j_i$'s. |
|
1619 |
We will consider some simple special cases first, then do the general case. |
|
1620 |
||
1621 |
First consider the case $j=0$ (and $m=0$). |
|
1622 |
A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$, |
|
1623 |
is independent of $p \in P$. |
|
1624 |
So the corresponding map $D \to \Diff(X)$ is constant. |
|
1625 |
||
1626 |
Next consider the case $j = 1$ (and $m=1$, $j_1=1$). |
|
1627 |
Now Equation yyyy applies. |
|
1628 |
We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$. |
|
1629 |
It follows that the singular cell $D \to \Diff(X)$ can be written as a product |
|
1630 |
of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$. |
|
1631 |
The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. |
|
1632 |
||
1633 |
Next case: $j=2$, $m=1$, $j_1 = 2$. |
|
8 | 1634 |
This is similar to the previous case, except that the normal bundle is 2-dimensional instead of |
0 | 1635 |
1-dimensional. |
1636 |
We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell |
|
1637 |
and a 2-cell with support $U_\beta$. |
|
1638 |
||
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|
1639 |
Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$. |
0 | 1640 |
In this case the codimension 2 cell $D$ is the intersection of two |
1641 |
codimension 1 cells, from $K_\beta$ and $K_\gamma$. |
|
1642 |
We can write $D = D' \times I \times I$, where the normal coordinates are constant |
|
1643 |
on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$. |
|
1644 |
If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and |
|
1645 |
two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively. |
|
1646 |
If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and |
|
1647 |
a 2-cell supported on $U_\beta \cup U_\gamma$. |
|
1648 |
\nn{need to check that this is true} |
|
1649 |
||
1650 |
\nn{finally, general case...} |
|
1651 |
||
1652 |
\nn{this completes proof} |
|
1653 |
||
13 | 1654 |
\input{text/explicit.tex} |
0 | 1655 |
|
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|
1656 |
|
22
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|
1657 |
% ---------------------------------------------------------------- |
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7b0a43bdd3c4
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|
1658 |
%\newcommand{\urlprefix}{} |
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|
1659 |
\bibliographystyle{plain} |
22
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|
1660 |
%Included for winedt: |
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|
1661 |
%input "bibliography/bibliography.bib" |
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|
1662 |
\bibliography{bibliography/bibliography} |
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|
1663 |
% ---------------------------------------------------------------- |
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|
1664 |
|
22
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|
1665 |
This paper is available online at \arxiv{?????}, and at |
47
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diff
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|
1666 |
\url{http://tqft.net/blobs}, |
939a4a5b1d80
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|
1667 |
and at \url{http://canyon23.net/math/}. |
7
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|
1668 |
|
22
ada83e7228eb
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|
1669 |
% A GTART necessity: |
ada83e7228eb
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|
1670 |
% \Addresses |
ada83e7228eb
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|
1671 |
% ---------------------------------------------------------------- |
ada83e7228eb
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|
1672 |
\end{document} |
ada83e7228eb
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|
1673 |
% ---------------------------------------------------------------- |
0 | 1674 |
|
1675 |
||
1676 |
||
1677 |
||
1678 |
%Recall that for $n$-category picture fields there is an evaluation map |
|
1679 |
%$m: \bc_0(B^n; c, c') \to \mor(c, c')$. |
|
1680 |
%If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain |
|
1681 |
%map $m: \bc_*(B^n; c, c') \to \mor(c, c')$. |