# HG changeset patch # User scott@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1256609496 0 # Node ID 77a311b5e2df87b3350b2369775773f4ee0dad89 # Parent b15dafe85ee1a341ee7e0ba3653a8005d5562f55 ... diff -r b15dafe85ee1 -r 77a311b5e2df preamble.tex --- a/preamble.tex Mon Oct 26 17:14:35 2009 +0000 +++ b/preamble.tex Tue Oct 27 02:11:36 2009 +0000 @@ -119,6 +119,11 @@ \newcommand{\lmod}[1]{\leftidx{_{#1}}{\operatorname{mod}}{}} +\newcommand{\HC}{\operatorname{Hoch}} +\newcommand{\HH}{\operatorname{HH}} + +\newcommand{\selfarrow}{\ensuremath{\!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]}\phantom{++.}}} + \def\bc{{\mathcal B}} \newcommand{\into}{\hookrightarrow} diff -r b15dafe85ee1 -r 77a311b5e2df text/hochschild.tex --- a/text/hochschild.tex Mon Oct 26 17:14:35 2009 +0000 +++ b/text/hochschild.tex Tue Oct 27 02:11:36 2009 +0000 @@ -66,7 +66,7 @@ Next, we show that for any $C$-$C$-bimodule $M$, \begin{prop} \label{prop:hoch} -The complex $K_*(M)$ is quasi-isomorphic to $HC_*(M)$, the usual +The complex $K_*(M)$ is quasi-isomorphic to $\HC_*(M)$, the usual Hochschild complex of $M$. \end{prop} \begin{proof} @@ -74,19 +74,19 @@ up to quasi-isomorphism, by the following properties: \begin{enumerate} \item \label{item:hochschild-additive}% -$HC_*(M_1 \oplus M_2) \cong HC_*(M_1) \oplus HC_*(M_2)$. +$\HC_*(M_1 \oplus M_2) \cong \HC_*(M_1) \oplus \HC_*(M_2)$. \item \label{item:hochschild-exact}% An exact sequence $0 \to M_1 \into M_2 \onto M_3 \to 0$ gives rise to an -exact sequence $0 \to HC_*(M_1) \into HC_*(M_2) \onto HC_*(M_3) \to 0$. +exact sequence $0 \to \HC_*(M_1) \into \HC_*(M_2) \onto \HC_*(M_3) \to 0$. \item \label{item:hochschild-coinvariants}% -$HH_0(M)$ is isomorphic to the coinvariants of $M$, $\coinv(M) = +$\HH_0(M)$ is isomorphic to the coinvariants of $M$, $\coinv(M) = M/\langle cm-mc \rangle$. \item \label{item:hochschild-free}% -$HC_*(C\otimes C)$ is contractible. +$\HC_*(C\otimes C)$ is contractible. (Here $C\otimes C$ denotes the free $C$-$C$-bimodule with one generator.) -That is, $HC_*(C\otimes C)$ is -quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants}, is just $C$) via the quotient map $HC_0 \onto HH_0$. +That is, $\HC_*(C\otimes C)$ is +quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants}, is just $C$) via the quotient map $\HC_0 \onto \HH_0$. \end{enumerate} (Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) We'll first recall why these properties are characteristic. @@ -110,32 +110,32 @@ \intertext{and} \cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j). \end{align*} -The cone of each chain map is acyclic. In the first case, this is because the `rows' indexed by $i$ are acyclic since $HC_i$ is exact. +The cone of each chain map is acyclic. In the first case, this is because the `rows' indexed by $i$ are acyclic since $\HC_i$ is exact. In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free. Because the cones are acyclic, the chain maps are quasi-isomorphisms. Composing one with the inverse of the other, we obtain the desired quasi-isomorphism $$\cP_*(M) \quismto \coinv(F_*).$$ %If $M$ is free, that is, a direct sum of copies of %$C \tensor C$, then properties \ref{item:hochschild-additive} and -%\ref{item:hochschild-free} determine $HC_*(M)$. Otherwise, choose some +%\ref{item:hochschild-free} determine $\HC_*(M)$. Otherwise, choose some %free cover $F \onto M$, and define $K$ to be this map's kernel. Thus we %have a short exact sequence $0 \to K \into F \onto M \to 0$, and hence a -%short exact sequence of complexes $0 \to HC_*(K) \into HC_*(F) \onto HC_*(M) +%short exact sequence of complexes $0 \to \HC_*(K) \into \HC_*(F) \onto \HC_*(M) %\to 0$. Such a sequence gives a long exact sequence on homology %\begin{equation*} %%\begin{split} -%\cdots \to HH_{i+1}(F) \to HH_{i+1}(M) \to HH_i(K) \to HH_i(F) \to \cdots % \\ -%%\cdots \to HH_1(F) \to HH_1(M) \to HH_0(K) \to HH_0(F) \to HH_0(M). +%\cdots \to \HH_{i+1}(F) \to \HH_{i+1}(M) \to \HH_i(K) \to \HH_i(F) \to \cdots % \\ +%%\cdots \to \HH_1(F) \to \HH_1(M) \to \HH_0(K) \to \HH_0(F) \to \HH_0(M). %%\end{split} %\end{equation*} -%For any $i \geq 1$, $HH_{i+1}(F) = HH_i(F) = 0$, by properties +%For any $i \geq 1$, $\HH_{i+1}(F) = \HH_i(F) = 0$, by properties %\ref{item:hochschild-additive} and \ref{item:hochschild-free}, and so -%$HH_{i+1}(M) \iso HH_i(F)$. For $i=0$, \todo{}. +%$\HH_{i+1}(M) \iso \HH_i(F)$. For $i=0$, \todo{}. % %This tells us how to -%compute every homology group of $HC_*(M)$; we already know $HH_0(M)$ +%compute every homology group of $\HC_*(M)$; we already know $\HH_0(M)$ %(it's just coinvariants, by property \ref{item:hochschild-coinvariants}), -%and higher homology groups are determined by lower ones in $HC_*(K)$, and +%and higher homology groups are determined by lower ones in $\HC_*(K)$, and %hence recursively as coinvariants of some other bimodule. Proposition \ref{prop:hoch} then follows from the following lemmas, establishing that $K_*$ has precisely these required properties. @@ -390,7 +390,7 @@ \medskip For purposes of illustration, we describe an explicit chain map -$HC_*(M) \to K_*(M)$ +$\HC_*(M) \to K_*(M)$ between the Hochschild complex and the blob complex (with bimodule point) for degree $\le 2$. This map can be completed to a homotopy equivalence, though we will not prove that here. @@ -398,7 +398,7 @@ Describing the extension to higher degrees is straightforward but tedious. \nn{but probably we should include the general case in a future version of this paper} -Recall that in low degrees $HC_*(M)$ is +Recall that in low degrees $\HC_*(M)$ is \[ \cdots \stackrel{\bd}{\to} M \otimes C\otimes C \stackrel{\bd}{\to} M \otimes C \stackrel{\bd}{\to} M diff -r b15dafe85ee1 -r 77a311b5e2df text/intro.tex --- a/text/intro.tex Mon Oct 26 17:14:35 2009 +0000 +++ b/text/intro.tex Tue Oct 27 02:11:36 2009 +0000 @@ -88,7 +88,7 @@ \begin{property}[Functoriality] \label{property:functoriality}% -Blob homology is functorial with respect to homeomorphisms. That is, +The blob complex is functorial with respect to homeomorphisms. That is, for fixed $n$-category / fields $\cC$, the association \begin{equation*} X \mapsto \bc_*^{\cC}(X) @@ -96,7 +96,7 @@ is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them. \end{property} -\nn{should probably also say something about being functorial in $\cC$} +The blob complex is also functorial with respect to $\cC$, although we will not address this in detail here. \begin{property}[Disjoint union] \label{property:disjoint-union} @@ -106,17 +106,17 @@ \end{equation*} \end{property} +If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$-submanifold of its boundary, write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. Note that this includes the case of gluing two disjoint manifolds together. \begin{property}[Gluing map] \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*} -\nn{alternate version:}Given a gluing $X_\mathrm{cut} \to X_\mathrm{gl}$, there is +%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*} +Given a gluing $X_\mathrm{cut} \to X_\mathrm{glued}$, there is a natural map \[ - \bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{gl}) . + \bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{glued}) . \] (Natural with respect to homeomorphisms, and also associative with respect to iterated gluings.) \end{property} @@ -124,11 +124,10 @@ \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. +The blob complex on an $n$-ball is contractible in the sense that it is quasi-isomorphic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the original $n$-category $\cC$. \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] @@ -146,13 +145,12 @@ 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)} +\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & \HC_*(\cC)} \end{equation*} \end{property} -\nn{$HC_*$ or $\rm{Hoch}_*$?} -\begin{property}[$C_*(\Diff(\cdot))$ action] +\begin{property}[$C_*(\Diff(-))$ action] \label{property:evaluation}% There is a chain map \begin{equation*} @@ -175,6 +173,19 @@ \nn{maybe do self-gluing instead of 2 pieces case} \end{property} +There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category +instead of a garden variety $n$-category. + +\begin{property}[Product formula] +Let $M^n = Y^{n-k}\times W^k$ and let $\cC$ be an $n$-category. +Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology, which associates to each $k$-ball $D$ the complex $A_*(Y)(D) = \bc_*(Y \times D, \cC)$. +Then +\[ + \bc_*(Y^{n-k}\times W^k, \cC) \simeq \bc_*(W, A_*(Y)) . +\] +Note on the right here we have the version of the blob complex for $A_\infty$ $n$-categories. +\nn{say something about general fiber bundles?} +\end{property} \begin{property}[Gluing formula] \label{property:gluing}% @@ -186,12 +197,10 @@ \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. +\item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of +$\bc_*(X_\text{cut})$ 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>[]} +\bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \selfarrow \end{equation*} \end{itemize} \end{property} @@ -199,50 +208,30 @@ \begin{property}[Relation to mapping spaces] -There is a version of the blob complex for $C$ an $A_\infty$ $n$-category -instead of a garden variety $n$-category. - Let $\pi^\infty_{\le n}(W)$ denote the $A_\infty$ $n$-category based on maps $B^n \to W$. (The case $n=1$ is the usual $A_\infty$ category of paths in $W$.) -Then $\bc_*(M, \pi^\infty_{\le n}(W))$ is -homotopy equivalent to $C_*(\{\text{maps}\; M \to W\})$. +Then +$$\bc_*(M, \pi^\infty_{\le n}(W) \simeq \CM{M}{W}.$$ \end{property} - - - -\begin{property}[Product formula] -Let $M^n = Y^{n-k}\times W^k$ and let $C$ be an $n$-category. -Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology. -Then -\[ - \bc_*(Y^{n-k}\times W^k, C) \simeq \bc_*(W, A_*(Y)) . -\] -\nn{say something about general fiber bundles?} -\end{property} - - - - \begin{property}[Higher dimensional Deligne conjecture] The singular chains of the $n$-dimensional fat graph operad act on blob cochains. - +\end{property} +\begin{rem} The $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries of $n$-manifolds $R_i \cup A_i \leadsto R_i \cup B_i$ together with mapping cylinders of diffeomorphisms $f_i: R_i\cup B_i \to R_{i+1}\cup A_{i+1}$. (Note that the suboperad where $A_i$, $B_i$ and $R_i\cup A_i$ are all diffeomorphic to the $n$-ball is equivalent to the little $n{+}1$-disks operad.) - -If $A$ and $B$ are $n$-manifolds sharing the same boundary, define +If $A$ and $B$ are $n$-manifolds sharing the same boundary, we define the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be $A_\infty$ maps from $\bc_*(A)$ to $\bc_*(B)$, where we think of both -(collections of) complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. +collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. The ``holes" in the above $n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$. -\end{property} - +\end{rem}