--- a/blob1.tex Tue Jul 21 15:55:06 2009 +0000
+++ b/blob1.tex Tue Jul 21 16:21:20 2009 +0000
@@ -24,7 +24,7 @@
\textbf{Draft version, do not distribute.}
%\versioninfo
-[11 June 2009]
+[later than 11 June 2009]
\noop{
@@ -63,175 +63,10 @@
} %end \noop
-\section{Introduction}
-
-[Outline for intro]
-\begin{itemize}
-\item Starting point: TQFTs via fields and local relations.
-This gives a satisfactory treatment for semisimple TQFTs
-(i.e.\ TQFTs for which the cylinder 1-category associated to an
-$n{-}1$-manifold $Y$ is semisimple for all $Y$).
-\item For non-semiemple TQFTs, this approach is less satisfactory.
-Our main motivating example (though we will not develop it in this paper)
-is the $4{+}1$-dimensional TQFT associated to Khovanov homology.
-It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together
-with a link $L \subset \bd W$.
-The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$.
-\item How would we go about computing $A_{Kh}(W^4, L)$?
-For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence)
-\nn{... $L_1, L_2, L_3$}.
-Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt
-to compute $A_{Kh}(S^1\times B^3, L)$.
-According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$
-corresponds to taking a coend (self tensor product) over the cylinder category
-associated to $B^3$ (with appropriate boundary conditions).
-The coend is not an exact functor, so the exactness of the triangle breaks.
-\item The obvious solution to this problem is to replace the coend with its derived counterpart.
-This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology
-of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired.
-If we build our manifold up via a handle decomposition, the computation
-would be a sequence of derived coends.
-A different handle decomposition of the same manifold would yield a different
-sequence of derived coends.
-To show that our definition in terms of derived coends is well-defined, we
-would need to show that the above two sequences of derived coends yield the same answer.
-This is probably not easy to do.
-\item Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$
-which is manifestly invariant.
-(That is, a definition that does not
-involve choosing a decomposition of $W$.
-After all, one of the virtues of our starting point --- TQFTs via field and local relations ---
-is that it has just this sort of manifest invariance.)
-\item The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient
-\[
- \text{linear combinations of fields} \;\big/\; \text{local relations} ,
-\]
-with an appropriately free resolution (the ``blob complex")
-\[
- \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) .
-\]
-Here $\bc_0$ is linear combinations of fields on $W$,
-$\bc_1$ is linear combinations of local relations on $W$,
-$\bc_2$ is linear combinations of relations amongst relations on $W$,
-and so on.
-\item None of the above ideas depend on the details of the Khovanov homology example,
-so we develop the general theory in the paper and postpone specific applications
-to later papers.
-\item The blob complex enjoys the following nice properties \nn{...}
-\end{itemize}
-
-\bigskip
-\hrule
-\bigskip
-
-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.
-\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 \du 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
-(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
-by $(\cF,\cU)$. (See \S \ref{sec:local-relations}.)
-\begin{equation*}
-H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X)
-\end{equation*}
-\end{property}
-
-\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*}
-\nn{should probably say something about associativity here (or not?)}
-\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 $\bc_*(Y \times I)$.
+\input{text/intro.tex}
-\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)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]}
-\end{equation*}
-\end{itemize}
-\end{property}
-
-\nn{add product formula? $n$-dimensional fat graph operad stuff?}
-
-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}
@@ -1019,193 +854,7 @@
\appendix
-\section{Families of Diffeomorphisms} \label{sec:localising}
-
-
-Lo, the proof of Lemma (\ref{extension_lemma}):
-
-\nn{should this be an appendix instead?}
-
-\nn{for pedagogical reasons, should do $k=1,2$ cases first; probably do this in
-later draft}
-
-\nn{not sure what the best way to deal with boundary is; for now just give main argument, worry
-about boundary later}
-
-Recall that we are given
-an open cover $\cU = \{U_\alpha\}$ and an
-$x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$.
-We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$.
-
-Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$.
-
-As a first approximation to the argument we will eventually make, let's replace $x$
-with a single singular cell
-\eq{
- f: P \times X \to X .
-}
-Also, we'll ignore for now issues around $\bd P$.
-
-Our homotopy will have the form
-\eqar{
- F: I \times P \times X &\to& X \\
- (t, p, x) &\mapsto& f(u(t, p, x), x)
-}
-for some function
-\eq{
- u : I \times P \times X \to P .
-}
-First we describe $u$, then we argue that it does what we want it to do.
-
-For each cover index $\alpha$ choose a cell decomposition $K_\alpha$ of $P$.
-The various $K_\alpha$ should be in general position with respect to each other.
-We will see below that the $K_\alpha$'s need to be sufficiently fine in order
-to insure that $F$ above is a homotopy through diffeomorphisms of $X$ and not
-merely a homotopy through maps $X\to X$.
-
-Let $L$ be the union of all the $K_\alpha$'s.
-$L$ is itself a cell decomposition of $P$.
-\nn{next two sentences not needed?}
-To each cell $a$ of $L$ we associate the tuple $(c_\alpha)$,
-where $c_\alpha$ is the codimension of the cell of $K_\alpha$ which contains $c$.
-Since the $K_\alpha$'s are in general position, we have $\sum c_\alpha \le k$.
-
-Let $J$ denote the handle decomposition of $P$ corresponding to $L$.
-Each $i$-handle $C$ of $J$ has an $i$-dimensional tangential coordinate and,
-more importantly, a $k{-}i$-dimensional normal coordinate.
-
-For each (top-dimensional) $k$-cell $c$ of each $K_\alpha$, choose a point $p_c \in c \sub P$.
-Let $D$ be a $k$-handle of $J$, and let $D$ also denote the corresponding
-$k$-cell of $L$.
-To $D$ we associate the tuple $(c_\alpha)$ of $k$-cells of the $K_\alpha$'s
-which contain $d$, and also the corresponding tuple $(p_{c_\alpha})$ of points in $P$.
-
-For $p \in D$ we define
-\eq{
- u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} .
-}
-(Recall that $P$ is a single linear cell, so the weighted average of points of $P$
-makes sense.)
-
-So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$.
-For handles of $J$ of index less than $k$, we will define $u$ to
-interpolate between the values on $k$-handles defined above.
-
-If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate
-of $E$.
-In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$
-with a $k$-handle.
-Let $\beta$ be the index of the $K_\beta$ containing the $k{-}1$-cell
-corresponding to $E$.
-Let $q_0, q_1 \in P$ be the points associated to the two $k$-cells of $K_\beta$
-adjacent to the $k{-}1$-cell corresponding to $E$.
-For $p \in E$, define
-\eq{
- u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha}
- + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) .
-}
-
-In general, for $E$ a $k{-}j$-handle, there is a normal coordinate
-$\eta: E \to R$, where $R$ is some $j$-dimensional polyhedron.
-The vertices of $R$ are associated to $k$-cells of the $K_\alpha$, and thence to points of $P$.
-If we triangulate $R$ (without introducing new vertices), we can linearly extend
-a map from the vertices of $R$ into $P$ to a map of all of $R$ into $P$.
-Let $\cN$ be the set of all $\beta$ for which $K_\beta$ has a $k$-cell whose boundary meets
-the $k{-}j$-cell corresponding to $E$.
-For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells.
-Now define, for $p \in E$,
-\eq{
- u(t, p, x) = (1-t)p + t \left(
- \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha}
- + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right)
- \right) .
-}
-Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension
-mentioned above.
-
-This completes the definition of $u: I \times P \times X \to P$.
-
-\medskip
-
-Next we verify that $u$ has the desired properties.
-
-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 $x$.
-Therefore $F$ is a homotopy from $f$ to something.
-
-Next we show that if the $K_\alpha$'s are sufficiently fine cell decompositions,
-then $F$ is a homotopy through diffeomorphisms.
-We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$.
-We have
-\eq{
-% \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) .
- \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} .
-}
-Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and
-\nn{bounded away from zero, or something like that}.
-(Recall that $X$ and $P$ are compact.)
-Also, $\pd{f}{p}$ is bounded.
-So if we can insure that $\pd{u}{x}$ is sufficiently small, we are done.
-It follows from Equation xxxx above that $\pd{u}{x}$ depends on $\pd{r_\alpha}{x}$
-(which is bounded)
-and the differences amongst the various $p_{c_\alpha}$'s and $q_{\beta i}$'s.
-These differences are small if the cell decompositions $K_\alpha$ are sufficiently fine.
-This completes the proof that $F$ is a homotopy through diffeomorphisms.
-
-\medskip
-
-Next we show that for each handle $D \sub P$, $F(1, \cdot, \cdot) : D\times X \to X$
-is a singular cell adapted to $\cU$.
-This will complete the proof of the lemma.
-\nn{except for boundary issues and the `$P$ is a cell' assumption}
-
-Let $j$ be the codimension of $D$.
-(Or rather, the codimension of its corresponding cell. From now on we will not make a distinction
-between handle and corresponding cell.)
-Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$,
-where the $j_i$'s are the codimensions of the $K_\alpha$
-cells of codimension greater than 0 which intersect to form $D$.
-We will show that
-if the relevant $U_\alpha$'s are disjoint, then
-$F(1, \cdot, \cdot) : D\times X \to X$
-is a product of singular cells of dimensions $j_1, \ldots, j_m$.
-If some of the relevant $U_\alpha$'s intersect, then we will get a product of singular
-cells whose dimensions correspond to a partition of the $j_i$'s.
-We will consider some simple special cases first, then do the general case.
-
-First consider the case $j=0$ (and $m=0$).
-A quick look at Equation xxxx above shows that $u(1, p, x)$, and hence $F(1, p, x)$,
-is independent of $p \in P$.
-So the corresponding map $D \to \Diff(X)$ is constant.
-
-Next consider the case $j = 1$ (and $m=1$, $j_1=1$).
-Now Equation yyyy applies.
-We can write $D = D'\times I$, where the normal coordinate $\eta$ is constant on $D'$.
-It follows that the singular cell $D \to \Diff(X)$ can be written as a product
-of a constant map $D' \to \Diff(X)$ and a singular 1-cell $I \to \Diff(X)$.
-The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set.
-
-Next case: $j=2$, $m=1$, $j_1 = 2$.
-This is similar to the previous case, except that the normal bundle is 2-dimensional instead of
-1-dimensional.
-We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell
-and a 2-cell with support $U_\beta$.
-
-Next case: $j=2$, $m=2$, $j_1 = j_2 = 1$.
-In this case the codimension 2 cell $D$ is the intersection of two
-codimension 1 cells, from $K_\beta$ and $K_\gamma$.
-We can write $D = D' \times I \times I$, where the normal coordinates are constant
-on $D'$, and the two $I$ factors correspond to $\beta$ and $\gamma$.
-If $U_\beta$ and $U_\gamma$ are disjoint, then we can factor $D$ into a constant $k{-}2$-cell and
-two 1-cells, supported on $U_\beta$ and $U_\gamma$ respectively.
-If $U_\beta$ and $U_\gamma$ intersect, then we can factor $D$ into a constant $k{-}2$-cell and
-a 2-cell supported on $U_\beta \cup U_\gamma$.
-\nn{need to check that this is true}
-
-\nn{finally, general case...}
-
-\nn{this completes proof}
-
-\input{text/explicit.tex}
+\input{text/famodiff.tex}
\section{Comparing definitions of $A_\infty$ algebras}
\label{sec:comparing-A-infty}