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\section{Higher-dimensional Deligne conjecture} |
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\label{sec:deligne} |
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In this section we prove a higher dimensional version of the Deligne conjecture |
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about the action of the little disks operad on Hochschild cochains. |
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The first several paragraphs lead up to a precise statement of the result |
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(Theorem \ref{thm:deligne} below). |
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Then we give the proof. |
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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, MR1328534, MR2064592, hep-th/9403055, MR1805923}) gives a map |
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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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We now reinterpret $C_*(LD_k)$ and $Hoch^*(C, C)$ in such a way as to make the generalization to |
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higher dimensions clear. |
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The little disks operad is homotopy equivalent to configurations of little bigons inside a big bigon, |
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as shown in Figure \ref{delfig1}. |
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We can think of such a configuration 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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\begin{figure}[t] |
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$$\mathfig{.9}{deligne/intervals}$$ |
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\caption{Little bigons, thought of as encoding surgeries}\label{delfig1}\end{figure} |
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The surgeries correspond to the $k$ bigon-shaped ``holes". |
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We remove the bottom interval of each little 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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$\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$, which is homotopy equivalent to $Hoch^*(C, C)$. |
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So for each fixed configuration 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 configuration, corresponding to a 1-chain in $C_*(LD_k)$, 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 $C_*(LD_k)$ give rise to higher homotopies. |
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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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\S\ref{ss:module-morphisms}. |
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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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So we will define, below, the operad of $n$-dimensional surgery cylinders, analogous to mapping |
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cylinders of homeomorphisms (Figure \ref{delfig2}). |
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\begin{figure}[t] |
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$$\mathfig{.9}{deligne/manifolds}$$ |
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\caption{An $n$-dimensional surgery cylinder}\label{delfig2} |
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\end{figure} |
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(Note that $n$ is the dimension of the manifolds we are doing surgery on; the surgery cylinders |
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are $n{+}1$-dimensional.) |
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An $n$-dimensional surgery cylinder ($n$-SC for short) consists of: |
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\begin{itemize} |
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\item ``Lower" $n$-manifolds $M_0,\ldots,M_k$ and ``upper" $n$-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 \ref{xdfig2}). |
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\begin{figure}[t] |
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$$\mathfig{.9}{deligne/mapping-cylinders}$$ |
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\caption{An $n$-dimensional surgery cylinder constructed from mapping cylinders}\label{xdfig2} |
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\end{figure} |
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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 surgery cylinders 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 which restricts to the identity on |
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$\bd R_i = \bd R'_i = E_0\cup \bd M_i$, we can replace |
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\begin{eqnarray*} |
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(\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& (\ldots, R_{i-1}, R'_i, R_{i+1}, \ldots) \\ |
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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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\end{eqnarray*} |
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leaving the $M_i$ and $N_i$ fixed. |
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(Keep in mind the case $R'_i = R_i$.) |
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(See Figure \ref{xdfig3}.) |
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\begin{figure}[t] |
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$$\mathfig{.4}{deligne/dfig3a} \to \mathfig{.4}{deligne/dfig3b} $$ |
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\caption{Conjugating by a homeomorphism.} |
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\label{xdfig3} |
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\end{figure} |
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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 \ref{xdfig1}.) |
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\begin{figure}[t] |
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$$\mathfig{.3}{deligne/dfig1a} \leftarrow \mathfig{.3}{deligne/dfig1b} \rightarrow \mathfig{.3}{deligne/dfig1c}$$ |
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\caption{Changing the order of a surgery.}\label{xdfig1} |
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\end{figure} |
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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 roles 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 a colored operad structure on $n$-dimensional surgery cylinders, given by gluing the outer boundary |
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of one cylinder into one of the inner boundaries of another cylinder. |
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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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$SC^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional surgery cylinders as above. |
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(Note that in different parts of $SC^n_{\ol{M}\ol{N}}$ the $M_i$'s and $N_i$'s |
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are ordered differently.) |
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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 $SC^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$. |
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\medskip |
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%The little $n{+}1$-balls operad injects into the $n$-SC operad. |
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The $n$-SC operad contains the little $n{+}1$-balls operad. |
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Roughly speaking, given a configuration of $k$ little $n{+}1$-balls in the standard |
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$n{+}1$-ball, we fiber the complement of the balls by vertical intervals |
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and let $M_i$ [$N_i$] be the southern [northern] hemisphere of the $i$-th ball. |
349 | 150 |
More precisely, let $x_1,\ldots,x_{n+1}$ be the coordinates of $\r^{n+1}$. |
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Let $z$ be a point of the $k$-th space of the little $n{+}1$-balls operad, with |
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little balls $D_1,\ldots,D_k$ inside the standard $n{+}1$-ball. |
349 | 153 |
We assume the $D_i$'s are ordered according to the $x_{n+1}$ coordinate of their centers. |
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Let $\pi:\r^{n+1}\to \r^n$ be the projection corresponding to $x_{n+1}$. |
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Let $B\sub\r^n$ be the standard $n$-ball. |
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Let $M_i$ and $N_i$ be $B$ for all $i$. |
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Identify $\pi(D_i)$ with $B$ (a.k.a.\ $M_i$ or $N_i$) via translations and dilations (no rotations). |
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Let $R_i = B\setmin \pi(D_i)$. |
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Let $f_i = \rm{id}$ for all $i$. |
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We have now defined a map from the little $n{+}1$-balls operad to the $n$-SC operad, |
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with contractible fibers. |
349 | 162 |
(The fibers correspond to moving the $D_i$'s in the $x_{n+1}$ |
879 | 163 |
direction while keeping them disjoint.) |
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%\nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s. |
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%does this need more explanation?} |
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Another familiar subspace of the $n$-SC operad is $\Homeo(M_0\to N_0)$, which corresponds to |
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case $k=0$ (no holes). |
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In this case the surgery cylinder is just a single mapping cylinder. |
298 | 170 |
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\medskip |
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172 |
||
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Let $\ol{f} \in SC^n_{\ol{M}\ol{N}}$. |
298 | 174 |
Let $\hom(\bc_*(M_i), \bc_*(N_i))$ denote the morphisms from $\bc_*(M_i)$ to $\bc_*(N_i)$, |
175 |
as modules of the $A_\infty$ 1-category $\bc_*(E_i)$. |
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We define a map |
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\[ |
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p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
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\to \hom(\bc_*(M_0), \bc_*(N_0)) . |
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180 |
\] |
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Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define |
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$p(\ol{f})(\alpha_1\ot\cdots\ot\alpha_k)$ to be the composition |
298 | 183 |
\[ |
184 |
\bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1) |
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\stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) |
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\stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
298 | 187 |
\cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
188 |
\stackrel{f_k}{\to} \bc_*(N_0) |
|
189 |
\] |
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450 | 190 |
(Recall that the maps $\id\ot\alpha_i$ were defined in \S\ref{ss:module-morphisms}.) |
298 | 191 |
It is easy to check that the above definition is compatible with the equivalence relations |
192 |
and also the operad structure. |
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We can reinterpret the above as a chain map |
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\[ |
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p: C_0(SC^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
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\to \hom(\bc_*(M_0), \bc_*(N_0)) . |
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\] |
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The main result of this section is that this chain map extends to the full singular |
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chain complex $C_*(SC^n_{\ol{M}\ol{N}})$. |
288 | 200 |
|
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\begin{thm} |
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\label{thm:deligne} |
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There is a collection of chain maps |
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\[ |
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C_*(SC^n_{\ol{M}\ol{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
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\hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
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\] |
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which satisfy the operad compatibility conditions, up to coherent homotopy. |
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On $C_0(SC^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. |
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When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of \S\ref{sec:evaluation}. |
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\end{thm} |
167 | 212 |
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If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ |
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to be ``blob cochains", we can summarize the above proposition by saying that the $n$-SC operad acts on |
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blob cochains. |
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As noted above, the $n$-SC operad contains the little $n{+}1$-balls operad, so this constitutes |
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a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disks operad. |
163 | 218 |
|
349 | 219 |
\begin{proof} |
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As described above, $SC^n_{\ol{M}\ol{N}}$ is equal to the disjoint |
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union of products of homeomorphism spaces, modulo some relations. |
437 | 222 |
By Theorem \ref{thm:CH} and the Eilenberg-Zilber theorem, we have for each such product $P$ |
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223 |
a chain map |
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\[ |
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C_*(P)\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes |
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\hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) . |
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227 |
\] |
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It suffices to show that the above maps are compatible with the relations whereby |
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$SC^n_{\ol{M}\ol{N}}$ is constructed from the various $P$'s. |
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230 |
This in turn follows easily from the fact that |
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the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative |
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(up to coherent homotopy). |
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%\nn{should add some detail to above} |
349 | 234 |
\end{proof} |
163 | 235 |
|
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We note that even when $n=1$, the above theorem goes beyond an action of the little disks operad. |
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$M_i$ could be a disjoint union of intervals, and $N_i$ could connect the end points of the intervals |
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in a different pattern from $M_i$. |
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The genus of the surface associated to the surgery cylinder could be greater than zero. |