# HG changeset patch # User Scott Morrison # Date 1289843344 28800 # Node ID bf613e5af5a340a6e5d2441bcd5e54c7680185dc # Parent b0ed73b141d807704083a8246702f96c5133b248# Parent f699e8381c4322d0edbac4e1b4e9a02f54ea2f99 Automated merge with https://tqft.net/hg/blob/ diff -r f699e8381c43 -r bf613e5af5a3 pnas/diagrams/deligne/mapping-cylinders.pdf Binary file pnas/diagrams/deligne/mapping-cylinders.pdf has changed diff -r f699e8381c43 -r bf613e5af5a3 pnas/pnas.tex --- a/pnas/pnas.tex Sun Nov 14 19:43:47 2010 -0800 +++ b/pnas/pnas.tex Mon Nov 15 09:49:04 2010 -0800 @@ -219,6 +219,9 @@ \nn{In many places we omit details; they can be found in MW. (Blanket statement in order to avoid too many citations to MW.)} +\nn{perhaps say something explicit about the relationship of this paper to big blob paper. +like: in this paper we try to give a clear view of the big picture without getting bogged down in details} + \section{Definitions} \subsection{$n$-categories} \mbox{} @@ -621,7 +624,7 @@ \xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)} \end{equation*} \end{property} -\nn{maybe should say something about the $A_\infty$ case} +%\nn{maybe should say something about the $A_\infty$ case} \begin{proof}(Sketch) For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram @@ -630,6 +633,9 @@ $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. \end{proof} +If $\cC$ is an $A-\infty$ $n$-category then $\bc_*(B^n;\cC)$ is still homotopy equivalent to $\cC(B^n)$, +but this is no longer concentrated in degree zero. + \subsection{Specializations} \label{sec:specializations} @@ -808,18 +814,47 @@ \section{Deligne conjecture for $n$-categories} \label{sec:applications} +Let $M$ and $N$ be $n$-manifolds with common boundary $E$. +Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$ +acts on $\bc_*(M)$ and $\bc_*(N)$. +Let $\hom_A(\bc_*(M), \bc_*(N))$ denote the chain complex of $A_\infty$ module maps +from $\bc_*(M)$ to $\bc_*(N)$. +Let $R$ be another $n$-manifold with boundary $-E$. +There is a chain map +\[ + \hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) . +\] +We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and +replaces it with $N$, yielding $N\cup_E R$. +(This is a more general notion of surgery that usual --- $M$ and $N$ can be any manifolds +which share a common boundary.) +In analogy to Hochschild cochains, we will call elements of $\hom_A(\bc_*(M), \bc_*(N))$ ``blob cochains". + +Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the +blob complex. +An $n$-dimensional surgery cylinder is +defined to be a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), +modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. +One can associated to this data an $(n{+}1)$-manifold with a foliation by intervals, +and the relations we impose correspond to homeomorphisms of the $(n{+}1)$-manifolds +which preserve the foliation. + +Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. + \begin{thm}[Higher dimensional Deligne conjecture] \label{thm:deligne} The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. -Since the little $n{+}1$-balls operad is a suboperad of the $n$-SC operad, -this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. \end{thm} -An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), -modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. -Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. - -By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module. +More specifically, let $M_0, N_0, \ldots, M_k, N_k$ be $n$-manifolds and let $SC^n_{\overline{M}, \overline{N}}$ +denote the component of the operad with outer boundary $M_0\cup N_0$ and inner boundaries +$M_1\cup N_1,\ldots, M_k\cup N_k$. +Then there is a collection of chain maps +\begin{multline*} + C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots \\ + \otimes \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) +\end{multline*} +which satisfy the operad compatibility conditions. \begin{proof} We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, @@ -828,15 +863,22 @@ This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. \end{proof} -The little disks operad $LD$ is homotopy equivalent to -\nn{suboperad of} -the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cochains $Hoch^*(C, C)$. -The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map -\[ - C_*(LD_k)\tensor \overbrace{Hoch^*(C, C)\tensor\cdots\tensor Hoch^*(C, C)}^{\text{$k$ copies}} - \to Hoch^*(C, C), -\] -which we now see to be a specialization of Theorem \ref{thm:deligne}. +Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are 1-balls. +We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little +disks operad and $\hom(\bc_*(M_i), \bc_*(N_i))$ is homotopy equivalent to Hochschild cochains. +This special case is just the usual Deligne conjecture +(see \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923} +\nn{should check that this is the optimal list of references; what about Gerstenhaber-Voronov?; +if we revise this list, should propagate change back to main paper} +). + +The general case when $n=1$ goes beyond the original Deligne conjecture, as the $M_i$'s and $N_i$'s +could be disjoint unions of 1-balls and circles, and the surgery cylinders could be high genus surfaces. + +If all of the $M_i$'s and $N_i$'s are $n$-balls, then $SC^n_{\overline{M}, \overline{N}}$ +contains a copy of the little $(n{+}1)$-balls operad. +Thus the little $(n{+}1)$-balls operad acts on blob cochains of the $n$-ball. + %% == end of paper: @@ -858,7 +900,7 @@ \begin{acknowledgments} It is a pleasure to acknowledge helpful conversations with Kevin Costello, -Mike Freedman, +Michael Freedman, Justin Roberts, and Peter Teichner. @@ -952,7 +994,8 @@ \end{figure} \begin{figure} -$$\mathfig{.4}{deligne/manifolds}$$ +%$$\mathfig{.4}{deligne/manifolds}$$ +$$\mathfig{.4}{deligne/mapping-cylinders}$$ \caption{An $n$-dimensional surgery cylinder.}\label{delfig2} \end{figure}