author | Scott Morrison <scott@tqft.net> |
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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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\nn{Does this generalisation encompass Kontsevich's proposed generalisation from \cite{MR1718044}, that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} |
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%from http://www.ams.org/mathscinet-getitem?mr=1805894 |
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%Different versions of the geometric counterpart of Deligne's conjecture have been proven by Tamarkin [``Formality of chain operad of small squares'', preprint, http://arXiv.org/abs/math.QA/9809164], the reviewer [in Confˇrence Moshˇ Flato 1999, Vol. II (Dijon), 307--331, Kluwer Acad. Publ., Dordrecht, 2000; MR1805923 (2002d:55009)], and J. E. McClure and J. H. Smith [``A solution of Deligne's conjecture'', preprint, http://arXiv.org/abs/math.QA/9910126] (see also a later simplified version [J. E. McClure and J. H. Smith, ``Multivariable cochain operations and little $n$-cubes'', preprint, http://arXiv.org/abs/math.QA/0106024]). The paper under review gives another proof of Deligne's conjecture, which, as the authors indicate, may be generalized to a proof of a higher-dimensional generalization of Deligne's conjecture, suggested in [M. Kontsevich, Lett. Math. Phys. 48 (1999), no. 1, 35--72; MR1718044 (2000j:53119)]. |
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The usual Deligne conjecture (proved variously in \cite{MR1805894, MR2064592, hep-th/9403055, MR1805923} 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 (Figure \ref{delfig2}). |
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\begin{figure}[!ht] |
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$$\mathfig{.9}{deligne/manifolds}$$ |
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\caption{An $n$-dimensional fat graph}\label{delfig2} |
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\end{figure} |
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More specifically, an $n$-dimensional fat graph consists of: |
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\begin{itemize} |
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\item ``Upper" $n$-manifolds $M_0,\ldots,M_k$ and ``lower" $n$-manifolds $N_0,\ldots,N_k$, |
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with $\bd M_i = \bd N_i = E_i$ for all $i$. |
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We call $M_0$ and $N_0$ the outer boundary and the remaining $M_i$'s and $N_i$'s the inner |
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boundaries. |
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\item Additional manifolds $R_1,\ldots,R_{k}$, with $\bd R_i = E_0\cup \bd M_i = E_0\cup \bd N_i$. |
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%(By convention, $M_i = N_i = \emptyset$ if $i <1$ or $i>k$.) |
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\item Homeomorphisms |
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\begin{eqnarray*} |
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f_0: M_0 &\to& R_1\cup M_1 \\ |
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f_i: R_i\cup N_i &\to& R_{i+1}\cup M_{i+1}\;\; \mbox{for}\, 1\le i \le k-1 \\ |
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f_k: R_k\cup N_k &\to& N_0 . |
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\end{eqnarray*} |
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Each $f_i$ should be the identity restricted to $E_0$. |
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\end{itemize} |
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We can think of the above data as encoding the union of the mapping cylinders $C(f_0),\ldots,C(f_k)$, |
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with $C(f_i)$ glued to $C(f_{i+1})$ along $R_{i+1}$ |
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(see Figure xxxx). |
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The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part. |
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We regard two such fat graphs as the same if there is a homeomorphism between them which is the |
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identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping |
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cylinders. |
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More specifically, we impose the following two equivalence relations: |
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\begin{itemize} |
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\item If $g:R_i\to R_i$ is a homeomorphism, we can replace |
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\[ |
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(\ldots, f_{i-1}, f_i, \ldots) \to (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots), |
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\] |
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leaving the $M_i$, $N_i$ and $R_i$ fixed. |
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(See Figure xxxx.) |
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\item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a |
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compatible disjoint union of $\bd M = \bd N$), we can replace |
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\begin{eqnarray*} |
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(\ldots, M_{i-1}, M_i, M_{i+1}, \ldots) &\to& (\ldots, M_{i-1}, M'_i, M''_i, M_{i+1}, \ldots) \\ |
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(\ldots, N_{i-1}, N_i, N_{i+1}, \ldots) &\to& (\ldots, N_{i-1}, N'_i, N''_i, N_{i+1}, \ldots) \\ |
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(\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& |
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(\ldots, R_{i-1}, R_i\cup M''_i, R_i\cup N'_i, R_{i+1}, \ldots) \\ |
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(\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, f_{i-1}, \rm{id}, f_i, \ldots) . |
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\end{eqnarray*} |
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(See Figure xxxx.) |
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\end{itemize} |
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Note that the second equivalence increases the number of holes (or arity) by 1. |
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We can make a similar identification with the rolls of $M'_i$ and $M''_i$ reversed. |
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In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries |
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do not overlap, we can perform them in reverse order or simultaneously. |
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There is an operad structure on $n$-dimensional fat graphs, given by gluing the outer boundary |
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of one graph into one of the inner boundaries of another graph. |
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We leave it to the reader to work out a more precise statement in terms of $M_i$'s, $f_i$'s etc. |
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For fixed $\ol{M} = (M_0,\ldots,M_k)$ and $\ol{N} = (N_0,\ldots,N_k)$, we let |
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$FG^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional fat graphs as above. |
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The topology comes from the spaces |
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\[ |
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\Homeo(M_0\to R_1\cup M_1)\times \Homeo(R_1\cup N_1\to R_2\cup M_2)\times |
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\cdots\times \Homeo(R_k\cup N_k\to N_0) |
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\] |
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and the above equivalence relations. |
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We will denote the typical element of $FG^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$. |
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\medskip |
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Let $\ol{f} \in FG^n_{\ol{M}\ol{N}}$. |
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Let $\hom(\bc_*(M_i), \bc_*(N_i))$ denote the morphisms from $\bc_*(M_i)$ to $\bc_*(N_i)$, |
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as modules of the $A_\infty$ 1-category $\bc_*(E_i)$. |
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We define a map |
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\[ |
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p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
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\to \hom(\bc_*(M_0), \bc_*(N_0)) . |
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\] |
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Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define $p(\ol{f}$ to be the composition |
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\[ |
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\bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1) |
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\stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) |
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\stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \to |
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\cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
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\stackrel{f_k}{\to} \bc_*(N_0) |
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\] |
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(Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.) |
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It is easy to check that the above definition is compatible with the equivalence relations |
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and also the operad structure. |
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\nn{little m-disks operad; } |
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\nn{*** resume revising here} |
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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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