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%!TEX root = ../blob1.tex
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\section{Gluing - needs to be rewritten/replaced}
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\label{sec:gluing}%
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We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction
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\begin{itemize}
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%\mbox{}% <-- gets the indenting right
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\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is
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naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below.
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\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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$A_\infty$ module for $\bc_*(Y \times I)$.
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\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension
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$0$-submanifold of its boundary, the blob homology of $X'$, obtained from
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$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of
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$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule.
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\begin{equation*}
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\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]}
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\end{equation*}
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\end{itemize}
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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
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definitions of `topological' $A_\infty$-categories and their bimodules, explain how to translate between the `algebraic' and `topological' definitions,
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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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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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as inputs, and to re-establish many of the properties of blob homology in this generality. Many readers may prefer to read the
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Definitions \ref{defn:topological-algebra} and \ref{defn:topological-module} of `topological' $A_\infty$-categories, and Definition \ref{???} of the
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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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of the gluing formula in the topological context.
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\subsection{`Topological' $A_\infty$ $n$-categories}
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\label{sec:topological-A-infty}%
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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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The main result of this section is
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\begin{thm}
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Topological $A_\infty$-$1$-categories are equivalent to the usual notion of
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$A_\infty$-$1$-categories.
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\end{thm}
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Before proving this theorem, we embark upon a long string of definitions.
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For expository purposes, we begin with the $n=1$ special cases,
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and define
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first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn
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to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules.
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\nn{Something about duals?}
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\todo{Explain that we're not making contact with any previous notions for the general $n$ case?}
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\nn{probably we should say something about the relation
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to [framed] $E_\infty$ algebras
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}
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\todo{}
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Various citations we might want to make:
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\begin{itemize}
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\item \cite{MR2061854} McClure and Smith's review article
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\item \cite{MR0420610} May, (inter alia, definition of $E_\infty$ operad)
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\item \cite{MR0236922,MR0420609} Boardman and Vogt
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\item \cite{MR1256989} definition of framed little-discs operad
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\end{itemize}
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\begin{defn}
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\label{defn:topological-algebra}%
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A ``topological $A_\infty$-algebra'' $A$ consists of the following data.
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\begin{enumerate}
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\item For each $1$-manifold $J$ diffeomorphic to the standard interval
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$I=\left[0,1\right]$, a complex of vector spaces $A(J)$.
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% either roll functoriality into the evaluation map
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\item For each pair of intervals $J,J'$ an `evaluation' chain map
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$\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$.
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\item For each decomposition of intervals $J = J'\cup J''$,
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a gluing map $\gl_{J',J''} : A(J') \tensor A(J'') \to A(J)$.
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% or do it as two separate pieces of data
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%\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$,
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%\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$,
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%\item and for each pair of intervals $J,J'$ a gluing map $\gl_{J,J'} : A(J) \tensor A(J') \to A(J \cup J')$,
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\end{enumerate}
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This data is required to satisfy the following conditions.
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\begin{itemize}
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\item The evaluation chain map is associative, in that the diagram
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\begin{equation*}
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\xymatrix{
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& \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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\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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& A(J'') &
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}
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\end{equation*}
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commutes up to homotopy.
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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.
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%% or the version for separate pieces of data:
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%\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same.
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%\item The evaluation chain map is associative, in that the diagram
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%\begin{equation*}
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%\xymatrix{
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%\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\id \tensor \ev_J} \ar[d]_{\compose \tensor \id} &
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%\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\
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%\CD{J} \tensor A(J) \ar[r]_{\ev_J} &
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%A(J)
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%}
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%\end{equation*}
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%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 $\Diff(J)$.)
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\item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram
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\begin{equation*}
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\xymatrix{
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A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \id} \ar[d]_{\id \tensor \gl_{J',J''}} &&
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A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\
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A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} &&
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A(J \cup J' \cup J'')
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}
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\end{equation*}
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commutes.
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\item The gluing and evaluation maps are compatible.
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\nn{give diagram, or just say ``in the obvious way", or refer to diagram in blob eval map section?}
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\end{itemize}
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\end{defn}
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\begin{rem}
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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
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constitute a functor from the category of intervals and diffeomorphisms between them to the category of complexes of vector spaces.
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Further, once this functor has been specified, we only need to know how the evaluation map acts when $J = J'$.
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\end{rem}
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%% if we do things separately, we should say this:
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%\begin{rem}
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%Of course, the first and third pieces of data (the complexes, and the isomorphisms) together just constitute a functor from the category of
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%intervals and diffeomorphisms between them to the category of complexes of vector spaces.
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%Further, one can combine the second and third pieces of data, asking instead for a map
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|
130 |
%\begin{equation*}
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parents:
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|
131 |
%\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J').
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parents:
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|
132 |
%\end{equation*}
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parents:
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|
133 |
%(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
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parents:
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|
134 |
%diffeomorphisms in $\CD{J'}$.)
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parents:
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|
135 |
%\end{rem}
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parents:
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|
136 |
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parents:
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|
137 |
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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parents:
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|
138 |
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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parents:
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|
139 |
\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
140 |
\gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+).
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parents:
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|
141 |
\end{equation*}
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parents:
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|
142 |
The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions.
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parents:
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|
143 |
\todo{we presumably need to say something about $\id_c \in A(J, c, c)$.}
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parents:
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|
144 |
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145 |
At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$.
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parents:
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|
146 |
\begin{defn}
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parents:
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|
147 |
Define the topological $A_\infty$ category $C_*(\Maps(\bullet \to M))$ by
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parents:
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|
148 |
\begin{enumerate}
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parents:
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|
149 |
\item $A(J) = C_*(\Maps(J \to M))$, singular chains on the space of smooth maps from $J$ to $M$,
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parents:
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|
150 |
\item $\ev_{J,J'} : \CD{J \to J'} \tensor A(J) \to A(J')$ is the composition
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
151 |
\begin{align*}
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parents:
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|
152 |
\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)),
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parents:
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|
153 |
\end{align*}
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parents:
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|
154 |
where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism,
|
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parents:
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|
155 |
\item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together.
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parents:
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|
156 |
\end{enumerate}
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parents:
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|
157 |
The associativity conditions are trivially satisfied.
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parents:
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|
158 |
\end{defn}
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parents:
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|
159 |
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parents:
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|
160 |
The second example is simply the blob complex of $Y \times J$, for any $n-1$ manifold $Y$. We define $A(J) = \bc_*(Y \times J)$.
|
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parents:
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|
161 |
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
|
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parents:
diff
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|
162 |
\ref{property:evaluation} and \ref{property:gluing-map} respectively. We'll often write $bc_*(Y)$ for this algebra.
|
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parents:
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|
163 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
164 |
The definition of a module follows closely the definition of an algebra or category.
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parents:
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|
165 |
\begin{defn}
|
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parents:
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|
166 |
\label{defn:topological-module}%
|
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parents:
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|
167 |
A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$
|
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parents:
diff
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|
168 |
consists of the following data.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
169 |
\begin{enumerate}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
170 |
\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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parents:
diff
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|
171 |
\item For each pair of such marked intervals,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
172 |
an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$.
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parents:
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changeset
|
173 |
\item For each decomposition $K = J\cup K'$ of the marked interval
|
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parents:
diff
changeset
|
174 |
$K$ into an unmarked interval $J$ and a marked interval $K'$, a gluing map
|
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parents:
diff
changeset
|
175 |
$\gl_{J,K'} : A(J) \tensor M(K') \to M(K)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
176 |
\end{enumerate}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
177 |
The above data is required to satisfy
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
178 |
conditions analogous to those in Definition \ref{defn:topological-algebra}.
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parents:
diff
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|
179 |
\end{defn}
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scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
180 |
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parents:
diff
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|
181 |
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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parents:
diff
changeset
|
182 |
a topological $A_\infty$ module over $\bc_*(Y)$, the topological $A_\infty$ category described above.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
183 |
For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
184 |
(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
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
185 |
\ref{property:evaluation} and \ref{property:gluing-map} respectively.
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parents:
diff
changeset
|
186 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
187 |
The definition of a bimodule is like the definition of a module,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
188 |
except that we have two disjoint marked intervals $K$ and $L$, one with a marked point
|
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parents:
diff
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|
189 |
on the upper boundary and the other with a marked point on the lower boundary.
|
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parents:
diff
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|
190 |
There are evaluation maps corresponding to gluing unmarked intervals
|
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parents:
diff
changeset
|
191 |
to the unmarked ends of $K$ and $L$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
192 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
193 |
Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
194 |
codimension-0 submanifold of $\bdy X$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
195 |
Then the the assignment $K,L \mapsto \bc_*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
196 |
structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$.
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scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
197 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
198 |
Next we define the coend
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
199 |
(or gluing or tensor product or self tensor product, depending on the context)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
200 |
$\gl(M)$ of a topological $A_\infty$ bimodule $M$. This will be an `initial' or `universal' object satisfying various properties.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
201 |
\begin{defn}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
202 |
We define a category $\cG(M)$. Objects consist of the following data.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
203 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
204 |
\item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N).
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
205 |
\item For each pair of intervals $N,N'$ an evaluation chain map
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
206 |
$\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
207 |
\item For each decomposition of intervals $N = K\cup L$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
208 |
a gluing map $\gl_{K,L} : M(K,L) \to C(N)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
209 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
210 |
This data must satisfy the following conditions.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
211 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
212 |
\item The evaluation maps are associative.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
213 |
\nn{up to homotopy?}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
214 |
\item Gluing is strictly associative.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
215 |
That is, given a decomposition $N = K\cup J\cup L$, the chain maps associated to
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
216 |
$K\du J\du L \to (K\cup J)\du L \to N$ and $K\du J\du L \to K\du (J\cup L) \to N$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
217 |
agree.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
218 |
\item the gluing and evaluation maps are compatible.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
219 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
220 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
221 |
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,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
222 |
satisfying the following conditions.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
223 |
\begin{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
224 |
\item For each pair of intervals $N,N'$, the diagram
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
225 |
\begin{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
226 |
\xymatrix{
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
227 |
\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} \\
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
228 |
C(N) \ar[r]_{f_N} & C'(N)
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
229 |
}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
230 |
\end{equation*}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
231 |
commutes.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
232 |
\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
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
233 |
$$M(K,L) \xto{\gl_{K,L}} C(N) \xto{f_N} C'(N).$$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
234 |
\end{itemize}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
235 |
\end{defn}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
236 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
237 |
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 $\cG{M}$,
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
238 |
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 $M(K,L) \to C'(N)$
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
239 |
factor through the gluing maps for $\gl(M)$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
240 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
241 |
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
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
242 |
is essentially $C_*(\Maps(S^1 \to M))$. \todo{}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
243 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
244 |
For the second example, given $X$ and $Y\du -Y \sub \bdy X$, the assignment
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
245 |
$$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y))$$ clearly gives an object in $\cG{M}$.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
246 |
Showing that it is an initial object is the content of the gluing theorem proved below.
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
247 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
248 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
249 |
\nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
250 |
$n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
251 |
easy, I think, so maybe it should be done earlier??}
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
252 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
253 |
\bigskip
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
254 |
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
255 |
Outline:
|
scott@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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\begin{itemize}
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\item recall defs of $A_\infty$ category (1-category only), modules, (self-) tensor product.
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use graphical/tree point of view, rather than following Keller exactly
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\item define blob complex in $A_\infty$ case; fat mapping cones? tree decoration?
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\item topological $A_\infty$ cat def (maybe this should go first); also modules gluing
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\item motivating example: $C_*(\Maps(X, M))$
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\item maybe incorporate dual point of view (for $n=1$), where points get
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object labels and intervals get 1-morphism labels
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\end{itemize}
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\subsection{$A_\infty$ action on the boundary}
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\label{sec:boundary-action}%
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Let $Y$ be an $n{-}1$-manifold.
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The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary
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conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure
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of an $A_\infty$ category.
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Composition of morphisms (multiplication) depends of a choice of homeomorphism
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$I\cup I \cong I$. Given this choice, gluing gives a map
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\eq{
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\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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\cong \bc_*(Y\times I; a, c)
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}
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Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various
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higher associators of the $A_\infty$ structure, more or less canonically.
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\nn{is this obvious? does more need to be said?}
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Let $\cA(Y)$ denote the $A_\infty$ category $\bc_*(Y\times I; \cdot, \cdot)$.
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Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism
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$(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$
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(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the
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$A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$.
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Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood
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of $Y$ in $X$.
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In the next section we use the above $A_\infty$ actions to state and prove
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a gluing theorem for the blob complexes of $n$-manifolds.
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\subsection{The gluing formula}
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\label{sec:gluing-formula}%
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Let $Y$ be an $n{-}1$-manifold and let $X$ be an $n$-manifold with a copy
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of $Y \du -Y$ contained in its boundary.
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Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$.
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We wish to describe the blob complex of $X\sgl$ in terms of the blob complex
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of $X$.
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More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$,
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where $c\sgl \in \cC(\bd X\sgl)$,
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in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation
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of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$.
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\begin{thm}
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$\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product
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of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$.
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\end{thm}
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The proof will occupy the remainder of this section.
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\nn{...}
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\bigskip
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\nn{need to define/recall def of (self) tensor product over an $A_\infty$ category}
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322 |
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