author | Kevin Walker <kevin@canyon23.net> |
Thu, 27 May 2010 22:29:49 -0700 | |
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%!TEX root = ../blob1.tex |
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\section{Higher-dimensional Deligne conjecture} |
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\label{sec:deligne} |
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In this section we |
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sketch |
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\nn{revisit ``sketch" after proof is done} |
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the proof of a higher dimensional version of the Deligne conjecture |
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about the action of the little disks operad on Hochschild cohomology. |
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The first several paragraphs lead up to a precise statement of the result |
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(Proposition \ref{prop:deligne} below). |
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Then we sketch the proof. |
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The usual Deligne conjecture \nn{need refs} gives a map |
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\[ |
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C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}} |
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\to Hoch^*(C, C) . |
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\] |
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Here $LD_k$ is the $k$-th space of the little disks operad, and $Hoch^*(C, C)$ denotes Hochschild |
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cochains. |
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The little disks operad is homotopy equivalent to the |
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(transversely orient) fat graph operad |
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\nn{need ref, or say more precisely what we mean}, |
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and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms |
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of the blob complex of the interval, thought of as a bimodule for itself. |
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\nn{need to make sure we prove this above}. |
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So the 1-dimensional Deligne conjecture can be restated as |
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\[ |
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C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
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\otimes \hom(\bc^C_*(I), \bc^C_*(I)) |
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\to \hom(\bc^C_*(I), \bc^C_*(I)) . |
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\] |
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See Figure \ref{delfig1}. |
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\begin{figure}[!ht] |
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$$\mathfig{.9}{deligne/intervals}$$ |
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\caption{A fat graph}\label{delfig1}\end{figure} |
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We emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module |
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for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the |
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morphisms of such modules as defined in |
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Subsection \ref{ss:module-morphisms}. |
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We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval |
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of Figure \ref{delfig1} and ending at the topmost interval. |
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The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph. |
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We remove the bottom interval of the bigon and replace it with the top interval. |
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To convert this topological operation to an algebraic one, we need, for each hole, an element of |
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$\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$. |
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So for each fixed fat graph we have a map |
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\[ |
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\hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots |
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\otimes \hom(\bc^C_*(I), \bc^C_*(I)) \to \hom(\bc^C_*(I), \bc^C_*(I)) . |
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\] |
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If we deform the fat graph, corresponding to a 1-chain in $C_*(FG_k)$, we get a homotopy |
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between the maps associated to the endpoints of the 1-chain. |
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Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies. |
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It should now be clear how to generalize this to higher dimensions. |
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In the sequence-of-surgeries description above, we never used the fact that the manifolds |
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involved were 1-dimensional. |
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Thus we can define an $n$-dimensional fat graph to be a sequence of general surgeries |
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on an $n$-manifold. |
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\nn{*** resume revising here} |
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More specifically, |
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the $n$-dimensional fat graph operad can be thought of as a sequence of general surgeries |
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$R_i \cup M_i \leadsto R_i \cup N_i$ together with mapping cylinders of diffeomorphisms |
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$f_i: R_i\cup N_i \to R_{i+1}\cup M_{i+1}$. |
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(See Figure \ref{delfig2}.) |
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\begin{figure}[!ht] |
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$$\mathfig{.9}{deligne/manifolds}$$ |
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\caption{A fat graph}\label{delfig2} |
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\end{figure} |
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The components of the $n$-dimensional fat graph operad are indexed by tuples |
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$(\overline{M}, \overline{N}) = ((M_0,\ldots,M_k), (N_0,\ldots,N_k))$. |
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\nn{not quite true: this is coarser than components} |
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Note that the suboperad where $M_i$, $N_i$ and $R_i\cup M_i$ are all homeomorphic to |
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the $n$-ball is equivalent to the little $n{+}1$-disks operad. |
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\nn{what about rotating in the horizontal directions?} |
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If $M$ and $N$ are $n$-manifolds sharing the same boundary, we define |
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the blob cochains $\bc^*(A, B)$ (analogous to Hochschild cohomology) to be |
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$A_\infty$ maps from $\bc_*(M)$ to $\bc_*(N)$, where we think of both |
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collections of complexes as modules over the $A_\infty$ category associated to $\bd A = \bd B$. |
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The ``holes" in the above |
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$n$-dimensional fat graph operad are labeled by $\bc^*(A_i, B_i)$. |
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\nn{need to make up my mind which notation I'm using for the module maps} |
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Putting this together we get |
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\begin{prop}(Precise statement of Property \ref{property:deligne}) |
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\label{prop:deligne} |
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There is a collection of maps |
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\begin{eqnarray*} |
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C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
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\hom(\bc_*(M_{k}), \bc_*(N_{k})) & \\ |
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& \hspace{-11em}\to \hom(\bc_*(M_0), \bc_*(N_0)) |
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\end{eqnarray*} |
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which satisfy an operad type compatibility condition. \nn{spell this out} |
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\end{prop} |
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Note that if $k=0$ then this is just the action of chains of diffeomorphisms from Section \ref{sec:evaluation}. |
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And indeed, the proof is very similar \nn{...} |
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\medskip |
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\hrule\medskip |
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