diff -r e924dd389d6e -r ec3af8dfcb3c blob1.tex --- 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}