diff -r b7812497643a -r ada83e7228eb blob1.tex --- a/blob1.tex Mon Jun 30 20:20:52 2008 +0000 +++ b/blob1.tex Tue Jul 01 01:53:15 2008 +0000 @@ -54,7 +54,7 @@ % \DeclareMathOperator{\pr}{pr} etc. \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} -\applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Diff}{sign}{supp}{maps}; +\applytolist{declaremathop}{pr}{im}{id}{gl}{ev}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Diff}{sign}{supp}{maps}; @@ -78,7 +78,25 @@ \maketitle -\textbf{Draft version, do not distribute. \versioninfo} +\textbf{Draft version, do not distribute.} + +\versioninfo + +\section*{Todo} + +\subsection*{What else?...} + +\begin{itemize} +\item Derive Hochschild standard results from blob point of view? +\item $n=2$ examples +\item Kh +\item dimension $n+1$ (generalized Deligne conjecture?) +\item should be clear about PL vs Diff; probably PL is better +(or maybe not) +\item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category +\item something about higher derived coend things (derived 2-coend, e.g.) +\end{itemize} + \section{Introduction} @@ -92,9 +110,118 @@ (3) ... ) -\section{Definitions} + + +We then show that blob homology enjoys the following +\ref{property:gluing} properties. + +\begin{property}[Functoriality] +\label{property:functoriality}% +Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association +\begin{equation*} +X \mapsto \bc_*^{\cF,\cU}(X) +\end{equation*} +is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them. +\scott{Do we want to or need to weaken `isomorphisms' to `homotopy equivalences' or `quasi-isomorphisms'?} +\end{property} + +\begin{property}[Disjoint union] +\label{property:disjoint-union} +The blob complex of a disjoint union is naturally the tensor product of the blob complexes. +\begin{equation*} +\bc_*(X_1 \sqcup X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) +\end{equation*} +\end{property} + +\begin{property}[A map for gluing] +\label{property:gluing-map}% +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 $\bdy X_2$, +there is a chain map +\begin{equation*} +\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). +\end{equation*} +\end{property} + +\begin{property}[Contractibility] +\label{property:contractibility}% +\todo{Err, requires a splitting?} +The blob complex for an $n$-category on an $n$-ball is quasi-isomorphic to its $0$-th homology. +\begin{equation} +\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))} +\end{equation} +\todo{Say that this is just the original $n$-category?} +\end{property} + +\begin{property}[Skein modules] +\label{property:skein-modules}% +The $0$-th blob homology of $X$ is the usual skein module associated to $X$. (See \S \ref{sec:local-relations}.) +\begin{equation*} +H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X) +\end{equation*} +\end{property} -\subsection{Fields} +\begin{property}[Hochschild homology when $X=S^1$] +\label{property:hochschild}% +The blob complex for a $1$-category $\cC$ on the circle is +quasi-isomorphic to the Hochschild complex. +\begin{equation*} +\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & HC_*(\cC)} +\end{equation*} +\end{property} + +\begin{property}[Evaluation map] +\label{property:evaluation}% +There is an `evaluation' chain map +\begin{equation*} +\ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). +\end{equation*} +(Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.) + +Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for +any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram +(using the gluing maps described in Property \ref{property:gluing-map}) commutes. +\begin{equation*} +\xymatrix{ + \CD{X} \otimes \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) \\ + \CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2) + \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & + \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} +} +\end{equation*} +\end{property} + +\begin{property}[Gluing formula] +\label{property:gluing}% +\mbox{}% <-- gets the indenting right +\begin{itemize} +\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is +naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. + +\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an +$A_\infty$ module for $\bc_*(Y \times I)$. + +\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension +$0$-submanifold of its boundary, the blob homology of $X'$, obtained from +$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of +$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. +\begin{equation*} +\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} +\end{equation*} +\todo{How do you write self tensor product?} +\end{itemize} +\end{property} + +Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in +\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} +Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. +Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, +and Property \ref{property:gluing} in \S \ref{sec:gluing}. + +\section{Definitions} +\label{sec:definitions} + +\subsection{Systems of fields} +\label{sec:fields} Fix a top dimension $n$. @@ -245,6 +372,7 @@ \subsection{Local relations} +\label{sec:local-relations} Let $B^n$ denote the standard $n$-ball. A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ @@ -282,6 +410,7 @@ \subsection{The blob complex} +\label{sec:blob-definition} Let $X$ be an $n$-manifold. Assume a fixed system of fields. @@ -441,6 +570,7 @@ \section{Basic properties of the blob complex} +\label{sec:basic-properties} \begin{prop} \label{disjunion} There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. @@ -536,7 +666,7 @@ \qed \end{prop} -The above will be greatly strengthened in Section \ref{diffsect}. +The above will be greatly strengthened in Section \ref{sec:evaluation}. \medskip @@ -563,7 +693,7 @@ \end{prop} The above map is very far from being an isomorphism, even on homology. -This will be fixed in Section \ref{gluesect} below. +This will be fixed in Section \ref{sec:gluing} below. An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ and $X\sgl = X_1 \cup_Y X_2$. @@ -583,7 +713,8 @@ \label{sec:hochschild} \input{text/hochschild} -\section{Action of $C_*(\Diff(X))$} \label{diffsect} +\section{Action of $\CD{X}$} +\label{sec:evaluation} Let $CD_*(X)$ denote $C_*(\Diff(X))$, the singular chain complex of the space of diffeomorphisms @@ -664,7 +795,7 @@ Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. \end{lemma} -The proof will be given in Section \ref{fam_diff_sect}. +The proof will be given in Section \ref{sec:localising}. \medskip @@ -781,10 +912,102 @@ \nn{say something about associativity here} +\section{Gluing} +\label{sec:gluing}% + +\subsection{`Topological' $A_\infty$ $n$-categories} +\label{sec:topological-A-infty}% + +This section prepares the ground for establishing Property \ref{property:gluing} by defining the notion of a \emph{topological $A_\infty$ +$n$-category}. The main result of this section is + +\begin{thm} +Topological $A_\infty$ $1$-categories are equivalent to `standard' +$A_\infty$ $1$-categories. +\end{thm} -\section{Families of Diffeomorphisms} \label{fam_diff_sect} +\nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG +$n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty +easy, I think, so maybe it should be done earlier??} + +\bigskip + +Outline: +\begin{itemize} +\item recall defs of $A_\infty$ category (1-category only), modules, (self-) tensor product. +use graphical/tree point of view, rather than following Keller exactly +\item define blob complex in $A_\infty$ case; fat mapping cones? tree decoration? +\item topological $A_\infty$ cat def (maybe this should go first); also modules gluing +\item motivating example: $C_*(\maps(X, M))$ +\item maybe incorporate dual point of view (for $n=1$), where points get +object labels and intervals get 1-morphism labels +\end{itemize} + + +\subsection{$A_\infty$ action on the boundary} + +Let $Y$ be an $n{-}1$-manifold. +The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary +conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure +of an $A_\infty$ category. + +Composition of morphisms (multiplication) depends of a choice of homeomorphism +$I\cup I \cong I$. Given this choice, gluing gives a map +\eq{ + \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) + \cong \bc_*(Y\times I; a, c) +} +Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various +higher associators of the $A_\infty$ structure, more or less canonically. + +\nn{is this obvious? does more need to be said?} + +Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. + +Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism +$(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ +(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the +$A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. +Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood +of $Y$ in $X$. + +In the next section we use the above $A_\infty$ actions to state and prove +a gluing theorem for the blob complexes of $n$-manifolds. + + +\subsection{The gluing formula} + +Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy +of $Y \du -Y$ contained in its boundary. +Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. +We wish to describe the blob complex of $X\sgl$ in terms of the blob complex +of $X$. +More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, +where $c\sgl \in \cC(\bd X\sgl)$, +in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation +of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. + +\begin{thm} +$\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product +of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. +\end{thm} + +The proof will occupy the remainder of this section. + +\nn{...} + +\bigskip + +\nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} + + + + +\appendix + +\section{Families of Diffeomorphisms} \label{sec:localising} Lo, the proof of Lemma (\ref{extension_lemma}): @@ -972,121 +1195,24 @@ \input{text/explicit.tex} - -\section{$A_\infty$ action on the boundary} - -Let $Y$ be an $n{-}1$-manifold. -The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary -conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure -of an $A_\infty$ category. - -Composition of morphisms (multiplication) depends of a choice of homeomorphism -$I\cup I \cong I$. Given this choice, gluing gives a map -\eq{ - \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) - \cong \bc_*(Y\times I; a, c) -} -Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various -higher associators of the $A_\infty$ structure, more or less canonically. - -\nn{is this obvious? does more need to be said?} - -Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$. - -Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism -$(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ -(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the -$A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. -Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood -of $Y$ in $X$. - -In the next section we use the above $A_\infty$ actions to state and prove -a gluing theorem for the blob complexes of $n$-manifolds. - - - - - - - -\section{Gluing} \label{gluesect} - -Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy -of $Y \du -Y$ contained in its boundary. -Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. -We wish to describe the blob complex of $X\sgl$ in terms of the blob complex -of $X$. -More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, -where $c\sgl \in \cC(\bd X\sgl)$, -in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation -of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. - -\begin{thm} -$\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product -of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. -\end{thm} - -The proof will occupy the remainder of this section. +% ---------------------------------------------------------------- +\newcommand{\urlprefix}{} +\bibliographystyle{gtart} +%Included for winedt: +%input "bibliography/bibliography.bib" +\bibliography{bibliography/bibliography} +% ---------------------------------------------------------------- -\nn{...} - -\bigskip - -\nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} - - - - - -\section{Extension to ...} - -\nn{Need to let the input $n$-category $C$ be a graded thing -(e.g. DG $n$-category or $A_\infty$ $n$-category). -DG $n$-category case is pretty easy, I think, so maybe it should be done earlier?? -Also, $A_\infty$ stuff (this section) should go before gluing section.} - -\bigskip +This paper is available online at \arxiv{?????}, and at +\url{http://tqft.net/blobs}. -Outline: -\begin{itemize} -\item recall defs of $A_\infty$ category (1-category only), modules, (self-) tensor product. -use graphical/tree point of view, rather than following Keller exactly -\item define blob complex in $A_\infty$ case; fat mapping cones? tree decoration? -\item topological $A_\infty$ cat def (maybe this should go first); also modules gluing -\item motivating example: $C_*(\maps(X, M))$ -\item maybe incorporate dual point of view (for $n=1$), where points get -object labels and intervals get 1-morphism labels -\end{itemize} - - - - - - - - - +% A GTART necessity: +% \Addresses +% ---------------------------------------------------------------- +\end{document} +% ---------------------------------------------------------------- -\section{What else?...} - -\begin{itemize} -\item Derive Hochschild standard results from blob point of view? -\item $n=2$ examples -\item Kh -\item dimension $n+1$ (generalized Deligne conjecture?) -\item should be clear about PL vs Diff; probably PL is better -(or maybe not) -\item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category -\item something about higher derived coend things (derived 2-coend, e.g.) -\end{itemize} - - - - - -\end{document} - %Recall that for $n$-category picture fields there is an evaluation map