diff -r 5cf5940d1a2c -r b7bc1a931b73 blob1.tex --- a/blob1.tex Tue Jul 08 21:52:06 2008 +0000 +++ b/blob1.tex Wed Jul 09 00:10:29 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}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{sign}{supp}{maps}; +\applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{sign}{supp}{maps}; @@ -763,7 +763,13 @@ Let $f: P \times X \to X$ be a family of diffeomorphisms and $S \sub X$. We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all -$x \notin S$ and $p, q \in P$. +$x \notin S$ and $p, q \in P$. Equivalently \todo{really?}, $f$ is supported on $S$ if there is a family of diffeomorphisms $f' : P \times S \to S$ and a `background' +diffeomorphism $f_0 : X \to X$ so that +\begin{align} +\restrict{f}{P \times S}(p,s) & = f_0(f'(p,s)) \\ +\intertext{and} +\restrict{f}{P \times (X \setmin S)}(p,x) & = f_0(x). +\end{align} Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$. Let $\cU = \{U_\alpha\}$ be an open cover of $X$. @@ -779,12 +785,12 @@ } such that \begin{itemize} -\item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$; -\item the $V_i$'s are mutually disjoint; +\item each $f_i(p, \cdot): X \to X$\scott{This should just read ``each $f_i$ is supported''} is supported on some connected $V_i \sub X$; +\item the sets $V_i$ are mutually disjoint; \item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, where $k_i = \dim(P_i)$; and \item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot) \circ g$ -for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. +for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$.\scott{hmm, can we do $g$ last, instead?} \end{itemize} A chain $x \in C_k(\Diff(X))$ is (by definition) adapted to $\cU$ if it is the sum of singular cells, each of which is adapted to $\cU$. @@ -915,6 +921,34 @@ \section{Gluing} \label{sec:gluing}% +We now turn to establishing the gluing formula for blob homology, restated from Property \ref{property:gluing} in the Introduction +\begin{itemize} +%\mbox{}% <-- gets the indenting right +\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} + +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 +definitions of `topological' $A_\infty$-categories and their bimodules, explain how to translate between the `algebraic' and `topological' definitions, +and then prove the gluing formula in the topological langauge. Section \ref{sec:topological-A-infty} below explains these definitions, and establishes +the desired equivalence. This is quite involved, and in particular requires us to generalise the definition of blob homology to allow $A_\infty$ algebras +as inputs, and to re-establish many of the properties of blob homology in this generality. Many readers may prefer to read the +Definitions \ref{defn:topological-algebra} and \ref{defn:topological-module} of `topological' $A_\infty$-categories, and Definition \ref{???} of the +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 +of the gluing formula in the topological context. + \subsection{`Topological' $A_\infty$ $n$-categories} \label{sec:topological-A-infty}% @@ -922,12 +956,12 @@ The main result of this section is \begin{thm} -Topological $A_\infty$-$1$-categories are equivalent to `standard' +Topological $A_\infty$-$1$-categories are equivalent to the usual notion of $A_\infty$-$1$-categories. \end{thm} Before proving this theorem, we embark upon a long string of definitions. -For expository purposes, we begin with the $n=1$ special cases, and define +For expository purposes, we begin with the $n=1$ special cases,\scott{Why are we treating the $n>1$ cases at all?} and define first topological $A_\infty$-algebras, then topological $A_\infty$-categories, and then topological $A_\infty$-modules over these. We then turn to the general $n$ case, defining topological $A_\infty$-$n$-categories and their modules. \nn{Something about duals?} @@ -955,7 +989,7 @@ \item For each pair of intervals $J,J'$ an `evaluation' chain map $\ev_{J \to J'} : \CD{J \to J'} \tensor A(J) \to A(J')$. \item For each decomposition of intervals $J = J'\cup J''$, -a gluing map $\gl_{J,J'} : A(J') \tensor A(J'') \to A(J)$. +a gluing map $\gl_{J',J''} : A(J') \tensor A(J'') \to A(J)$. % or do it as two separate pieces of data %\item along with an `evaluation' chain map $\ev_J : \CD{J} \tensor A(J) \to A(J)$, %\item for each diffeomorphism $\phi : J \to J'$, an isomorphism $A(\phi) : A(J) \isoto A(J')$, @@ -966,21 +1000,19 @@ \item The evaluation chain map is associative, in that the diagram \begin{equation*} \xymatrix{ -\CD{J' \to J''} \tensor \CD{J \to J'} \tensor A(J) \ar[r]^{\Id \tensor \ev_{J \to J'}} \ar[d]_{\compose \tensor \Id} & -\CD{J' \to J''} \tensor A(J') \ar[d]^{\ev_{J' \to J''}} \\ -\CD{J \to J''} \tensor A(J) \ar[r]_{\ev_{J \to J''}} & -A(J'') + & \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} & \\ +\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''}} \\ + & A(J'') & } \end{equation*} -commutes. -\kevin{commutes up to homotopy? in the blob case the evaluation map is ambiguous up to homotopy} -(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.) +commutes up to homotopy. +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. %% or the version for separate pieces of data: %\item If $\phi$ is a diffeomorphism from $J$ to itself, the maps $\ev_J(\phi, -)$ and $A(\phi)$ are the same. %\item The evaluation chain map is associative, in that the diagram %\begin{equation*} %\xymatrix{ -%\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\Id \tensor \ev_J} \ar[d]_{\compose \tensor \Id} & +%\CD{J} \tensor \CD{J} \tensor A(J) \ar[r]^{\id \tensor \ev_J} \ar[d]_{\compose \tensor \id} & %\CD{J} \tensor A(J) \ar[d]^{\ev_J} \\ %\CD{J} \tensor A(J) \ar[r]_{\ev_J} & %A(J) @@ -990,7 +1022,7 @@ \item The gluing maps are \emph{strictly} associative. That is, given $J$, $J'$ and $J''$, the diagram \begin{equation*} \xymatrix{ -A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \Id} \ar[d]_{\Id \tensor \gl_{J',J''}} && +A(J) \tensor A(J') \tensor A(J'') \ar[rr]^{\gl_{J,J'} \tensor \id} \ar[d]_{\id \tensor \gl_{J',J''}} && A(J \cup J') \tensor A(J'') \ar[d]^{\gl_{J \cup J', J''}} \\ A(J) \tensor A(J' \cup J'') \ar[rr]_{\gl_{J, J' \cup J''}} && A(J \cup J' \cup J'') @@ -1026,7 +1058,7 @@ \gl : A(J, c_-, c_0) \tensor A(J', c_0, c_+) \to A(J \cup J', c_-, c_+). \end{equation*} The action of diffeomorphisms (and of $k$-parameter families of diffeomorphisms) ignores the boundary conditions. -\todo{we presumably need to say something about $\Id_c \in A(J, c, c)$.} +\todo{we presumably need to say something about $\id_c \in A(J, c, c)$.} At this point we can give two motivating examples. The first is `chains of maps to $M$' for some fixed target space $M$. \begin{defn} @@ -1037,9 +1069,7 @@ \begin{align*} \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)), \end{align*} -where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism\todo{inverse, really?!}, -\kevin{I think that's fine. If we recoil at taking inverses, -we should use smooth maps instead of diffeos} +where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism, \item $\gl_{J,J'} : A(J) \tensor A(J')$ takes the product of singular chains, then glues maps to $M$ together. \end{enumerate} The associativity conditions are trivially satisfied. @@ -1047,7 +1077,7 @@ The second example is simply the blob complex of $Y \times J$, for any $n-1$ manifold $Y$. We define $A(J) = \bc_*(Y \times J)$. 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 -\ref{property:evaluation} and \ref{property:gluing-map} respectively. +\ref{property:evaluation} and \ref{property:gluing-map} respectively. We'll often write $bc_*(Y)$ for this algebra. The definition of a module follows closely the definition of an algebra or category. \begin{defn} @@ -1055,7 +1085,7 @@ A topological $A_\infty$-(left-)module $M$ over a topological $A_\infty$ category $A$ consists of the following data. \begin{enumerate} -\item A functor $K \mapsto M(K)$ from $1$-manifolds diffeomorphic to the standard interval, with a marked point on a upper boundary, to complexes of vector spaces. +\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. \item For each pair of such marked intervals, an `evaluation' chain map $\ev_{K\to K'} : \CD{K \to K'} \tensor M(K) \to M(K')$. \item For each decomposition $K = J\cup K'$ of the marked interval @@ -1066,9 +1096,10 @@ conditions analogous to those in Definition \ref{defn:topological-algebra}. \end{defn} -Any manifold $X$ with $\bdy X = Y$ (or indeed just with $Y$ a codimension $0$-submanifold of $\bdy X$) becomes a topological $A_\infty$ module over -$\bc_*(Y)$, the topological $A_\infty$ category described above. For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. -(Here we glue $Y \times pt$ to $X$, where $pt$ is the marked point of $K$.) Again, the evaluation and gluing maps come directly from Properties +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 +a topological $A_\infty$ module over $\bc_*(Y)$, the topological $A_\infty$ category described above. +For each interval $K$, we have $M(K) = \bc_*((Y \times K) \cup_Y X)$. +(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 \ref{property:evaluation} and \ref{property:gluing-map} respectively. The definition of a bimodule is like the definition of a module, @@ -1079,20 +1110,23 @@ Let $X$ be an $n$-manifold with a copy of $Y \du -Y$ embedded as a codimension-0 submanifold of $\bdy X$. -Then the the assignment $K,L \mapsto \bc*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the +Then the the assignment $K,L \mapsto \bc_*(X \cup_Y (Y\times K) \cup_{-Y} (-Y\times L))$ has the structure of a topological $A_\infty$ bimodule over $\bc_*(Y)$. Next we define the coend (or gluing or tensor product or self tensor product, depending on the context) -$\gl(M)$ of a topological $A_\infty$ bimodule $M$. -$\gl(M)$ is defined to be the universal thing with the following structure. - +$\gl(M)$ of a topological $A_\infty$ bimodule $M$. This will be an `initial' or `universal' object satisfying various properties. +\begin{defn} +We define a category $\cG(M)$. Objects consist of the following data. \begin{itemize} \item For each interval $N$ with both endpoints marked, a complex of vector spaces C(N). \item For each pair of intervals $N,N'$ an evaluation chain map $\ev_{N \to N'} : \CD{N \to N'} \tensor C(N) \to C(N')$. \item For each decomposition of intervals $N = K\cup L$, a gluing map $\gl_{K,L} : M(K,L) \to C(N)$. +\end{itemize} +This data must satisfy the following conditions. +\begin{itemize} \item The evaluation maps are associative. \nn{up to homotopy?} \item Gluing is strictly associative. @@ -1102,14 +1136,32 @@ \item the gluing and evaluation maps are compatible. \end{itemize} -Bu universal we mean that given any other collection of chain complexes, evaluation maps -and gluing maps, they factor through the universal thing. -\nn{need to say this in more detail, in particular give the properties of the factoring map} +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, +satisfying the following conditions. +\begin{itemize} +\item For each pair of intervals $N,N'$, the diagram +\begin{equation*} +\xymatrix{ +\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} \\ +C(N) \ar[r]_{f_N} & C'(N) +} +\end{equation*} +commutes. +\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 +$$M(K,L) \xto{\gl_{K,L}} C(N) \xto{f_N} C'(N).$$ +\end{itemize} +\end{defn} -Given $X$ and $Y\du -Y \sub \bdy X$ as above, the assignment -$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y)$ clearly has the structure described -in the above bullet points. -Showing that it is the universal such thing is the content of the gluing theorem proved below. +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 $\cG{M}$, +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 $M(K,L) \to C'(N)$ +factor through the gluing maps for $\gl(M)$. + +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 +is essentially $C_*(\Maps(S^1 \to M))$. \todo{} + +For the second example, given $X$ and $Y\du -Y \sub \bdy X$, the assignment +$$N \mapsto \bc_*(X \cup_{Y\du -Y} (N\times Y))$$ clearly gives an object in $\cG{M}$. +Showing that it is an initial object is the content of the gluing theorem proved below. The definitions for a topological $A_\infty$-$n$-category are very similar to the above $n=1$ case. @@ -1282,7 +1334,7 @@ \subsection{$A_\infty$ action on the boundary} - +\label{sec:boundary-action}% 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 @@ -1313,7 +1365,7 @@ \subsection{The gluing formula} - +\label{sec: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 $X\sgl$.