text/deligne.tex
author Scott Morrison <scott@tqft.net>
Tue, 30 Mar 2010 16:48:16 -0700
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diagrams for deligne conjecture, and more work on small blobs
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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 discuss
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\newenvironment{property:deligne}{\textbf{Property \ref{property:deligne} (Higher dimensional Deligne conjecture)}\it}{}
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\begin{property:deligne}
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The singular chains of the $n$-dimensional fat graph operad act on blob cochains.
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\end{property:deligne}
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We will state this more precisely below as Proposition \ref{prop:deligne}, and just sketch a proof. First, we recall the usual Deligne conjecture, explain how to think of it as a statement about blob complexes, and begin to generalize it.
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\def\mapinf{\Maps_\infty}
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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 fat graph operad
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\nn{need ref; and need to restrict which fat graphs}, and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms
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of the blob complex of the interval.
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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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\begin{eqnarray*}
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	C_*(FG_k)\otimes \mapinf(\bc^C_*(I), \bc^C_*(I))\otimes\cdots
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	\otimes \mapinf(\bc^C_*(I), \bc^C_*(I)) & \\
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	  & \hspace{-5em} \to  \mapinf(\bc^C_*(I), \bc^C_*(I)) .
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\end{eqnarray*}
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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 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 map this topological operation to an algebraic one, we need, for each hole, element of
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$\mapinf(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$.
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So for each fixed fat graph we have a map
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\[
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	 \mapinf(\bc^C_*(I), \bc^C_*(I))\otimes\cdots
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	\otimes \mapinf(\bc^C_*(I), \bc^C_*(I))  \to  \mapinf(\bc^C_*(I), \bc^C_*(I)) .
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\]
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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 a $n$-dimensional fat graph to be a sequence of general surgeries
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on an $n$-manifold.
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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}\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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Note that the suboperad where $M_i$, $N_i$ and $R_i\cup M_i$ are all diffeomorphic to 
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the $n$-ball is equivalent to the little $n{+}1$-disks operad.
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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 \mapinf(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes 
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\mapinf(\bc_*(M_{k}), \bc_*(N_{k})) & \\
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	& \hspace{-11em}\to  \mapinf(\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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