diff -r 11532ce39ec0 -r 4f008d0a29d4 text/deligne.tex --- a/text/deligne.tex Wed Sep 22 20:42:47 2010 -0700 +++ b/text/deligne.tex Thu Sep 23 10:03:26 2010 -0700 @@ -8,9 +8,6 @@ (Theorem \ref{thm:deligne} below). Then we give the proof. -%\nn{Does this generalization encompass Kontsevich's proposed generalization from \cite[\S2.5]{MR1718044}, -%that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} - %from http://www.ams.org/mathscinet-getitem?mr=1805894 %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)]. @@ -22,53 +19,48 @@ \] Here $LD_k$ is the $k$-th space of the little disks operad and $Hoch^*(C, C)$ denotes Hochschild cochains. -The little disks operad is homotopy equivalent to the -(transversely oriented) fat graph operad -(see below), -and Hochschild cochains are homotopy equivalent to $A_\infty$ endomorphisms -of the blob complex of the interval, thought of as a bimodule for itself. -(see \S\ref{ss:module-morphisms}). -So the 1-dimensional Deligne conjecture can be restated as -\[ - C_*(FG_k)\otimes \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots - \otimes \hom(\bc^C_*(I), \bc^C_*(I)) - \to \hom(\bc^C_*(I), \bc^C_*(I)) . -\] -See Figure \ref{delfig1}. + +We now reinterpret $C_*(LD_k)$ and $Hoch^*(C, C)$ in such a way as to make the generalization to +higher dimensions clear. + +The little disks operad is homotopy equivalent to configurations of little bigons inside a big bigon, +as shown in Figure \ref{delfig1}. +We can think of such a configuration as encoding a sequence of surgeries, starting at the bottommost interval +of Figure \ref{delfig1} and ending at the topmost interval. \begin{figure}[t] $$\mathfig{.9}{deligne/intervals}$$ -\caption{A fat graph}\label{delfig1}\end{figure} +\caption{Little bigons, though of as encoding surgeries}\label{delfig1}\end{figure} +The surgeries correspond to the $k$ bigon-shaped ``holes". +We remove the bottom interval of each little bigon and replace it with the top interval. +To convert this topological operation to an algebraic one, we need, for each hole, an element of +$\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$, which is homotopy equivalent to $Hoch^*(C, C)$. +So for each fixed configuration we have a map +\[ + \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots + \otimes \hom(\bc^C_*(I), \bc^C_*(I)) \to \hom(\bc^C_*(I), \bc^C_*(I)) . +\] +If we deform the configuration, corresponding to a 1-chain in $C_*(LD_k)$, we get a homotopy +between the maps associated to the endpoints of the 1-chain. +Similarly, higher-dimensional chains in $C_*(LD_k)$ give rise to higher homotopies. + We emphasize that in $\hom(\bc^C_*(I), \bc^C_*(I))$ we are thinking of $\bc^C_*(I)$ as a module for the $A_\infty$ 1-category associated to $\bd I$, and $\hom$ means the morphisms of such modules as defined in \S\ref{ss:module-morphisms}. -We can think of a fat graph as encoding a sequence of surgeries, starting at the bottommost interval -of Figure \ref{delfig1} and ending at the topmost interval. -The surgeries correspond to the $k$ bigon-shaped ``holes" in the fat graph. -We remove the bottom interval of the bigon and replace it with the top interval. -To convert this topological operation to an algebraic one, we need, for each hole, an element of -$\hom(\bc^C_*(I_{\text{bottom}}), \bc^C_*(I_{\text{top}}))$. -So for each fixed fat graph we have a map -\[ - \hom(\bc^C_*(I), \bc^C_*(I))\otimes\cdots - \otimes \hom(\bc^C_*(I), \bc^C_*(I)) \to \hom(\bc^C_*(I), \bc^C_*(I)) . -\] -If we deform the fat graph, corresponding to a 1-chain in $C_*(FG_k)$, we get a homotopy -between the maps associated to the endpoints of the 1-chain. -Similarly, higher-dimensional chains in $C_*(FG_k)$ give rise to higher homotopies. - It should now be clear how to generalize this to higher dimensions. In the sequence-of-surgeries description above, we never used the fact that the manifolds involved were 1-dimensional. -Thus we can define an $n$-dimensional fat graph to be a sequence of general surgeries -on an $n$-manifold (Figure \ref{delfig2}). +So we will define, below, the operad of $n$-dimensional surgery cylinders, analogous to mapping +cylinders of homeomorphisms (Figure \ref{delfig2}). \begin{figure}[t] $$\mathfig{.9}{deligne/manifolds}$$ -\caption{An $n$-dimensional fat graph}\label{delfig2} +\caption{An $n$-dimensional surgery cylinder}\label{delfig2} \end{figure} +(Note that $n$ is the dimension of the manifolds we are doing surgery on; the surgery cylinders +are $n{+}1$-dimensional.) -More specifically, an $n$-dimensional fat graph ($n$-FG for short) consists of: +An $n$-dimensional surgery cylinder ($n$-SC for short) consists of: \begin{itemize} \item ``Lower" $n$-manifolds $M_0,\ldots,M_k$ and ``upper" $n$-manifolds $N_0,\ldots,N_k$, with $\bd M_i = \bd N_i = E_i$ for all $i$. @@ -89,10 +81,10 @@ (see Figure \ref{xdfig2}). \begin{figure}[t] $$\mathfig{.9}{deligne/mapping-cylinders}$$ -\caption{An $n$-dimensional fat graph constructed from mapping cylinders}\label{xdfig2} +\caption{An $n$-dimensional surgery cylinder constructed from mapping cylinders}\label{xdfig2} \end{figure} -The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part. -We regard two such fat graphs as the same if there is a homeomorphism between them which is the +%The $n$-manifolds are the ``$n$-dimensional graph" and the $I$ direction of the mapping cylinders is the ``fat" part. +We regard two such surgery cylinders as the same if there is a homeomorphism between them which is the identity on the boundary and which preserves the 1-dimensional fibers coming from the mapping cylinders. More specifically, we impose the following two equivalence relations: @@ -131,13 +123,13 @@ In terms of the ``sequence of surgeries" picture, this says that if two successive surgeries do not overlap, we can perform them in reverse order or simultaneously. -There is an operad structure on $n$-dimensional fat graphs, given by gluing the outer boundary -of one graph into one of the inner boundaries of another graph. +There is an operad structure on $n$-dimensional surgery cylinders, given by gluing the outer boundary +of one cylinder into one of the inner boundaries of another cylinder. We leave it to the reader to work out a more precise statement in terms of $M_i$'s, $f_i$'s etc. For fixed $\ol{M} = (M_0,\ldots,M_k)$ and $\ol{N} = (N_0,\ldots,N_k)$, we let -$FG^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional fat graphs as above. -(Note that in different parts of $FG^n_{\ol{M}\ol{N}}$ the $M_i$'s and $N_i$'s +$SC^n_{\ol{M}\ol{N}}$ denote the topological space of all $n$-dimensional surgery cylinders as above. +(Note that in different parts of $SC^n_{\ol{M}\ol{N}}$ the $M_i$'s and $N_i$'s are ordered differently.) The topology comes from the spaces \[ @@ -145,17 +137,17 @@ \cdots\times \Homeo(R_k\cup N_k\to N_0) \] and the above equivalence relations. -We will denote the typical element of $FG^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$. +We will denote the typical element of $SC^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$. \medskip -%The little $n{+}1$-ball operad injects into the $n$-FG operad. -The $n$-FG operad contains the little $n{+}1$-balls operad. +%The little $n{+}1$-balls operad injects into the $n$-SC operad. +The $n$-SC operad contains the little $n{+}1$-balls operad. Roughly speaking, given a configuration of $k$ little $n{+}1$-balls in the standard $n{+}1$-ball, we fiber the complement of the balls by vertical intervals and let $M_i$ [$N_i$] be the southern [northern] hemisphere of the $i$-th ball. More precisely, let $x_1,\ldots,x_{n+1}$ be the coordinates of $\r^{n+1}$. -Let $z$ be a point of the $k$-th space of the little $n{+}1$-ball operad, with +Let $z$ be a point of the $k$-th space of the little $n{+}1$-balls operad, with little balls $D_1,\ldots,D_k$ inside the standard $n{+}1$-ball. We assume the $D_i$'s are ordered according to the $x_{n+1}$ coordinate of their centers. Let $\pi:\r^{n+1}\to \r^n$ be the projection corresponding to $x_{n+1}$. @@ -164,19 +156,20 @@ Identify $\pi(D_i)$ with $B$ (a.k.a.\ $M_i$ or $N_i$) via translations and dilations (no rotations). Let $R_i = B\setmin \pi(D_i)$. Let $f_i = \rm{id}$ for all $i$. -We have now defined a map from the little $n{+}1$-ball operad to the $n$-FG operad, +We have now defined a map from the little $n{+}1$-balls operad to the $n$-SC operad, with contractible fibers. (The fibers correspond to moving the $D_i$'s in the $x_{n+1}$ direction without changing their ordering.) %\nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s. %does this need more explanation?} -Another familiar subspace of the $n$-FG operad is $\Homeo(M_0\to N_0)$, which corresponds to +Another familiar subspace of the $n$-SC operad is $\Homeo(M_0\to N_0)$, which corresponds to case $k=0$ (no holes). +In this case the surgery cylinder is just a single mapping cylinder. \medskip -Let $\ol{f} \in FG^n_{\ol{M}\ol{N}}$. +Let $\ol{f} \in SC^n_{\ol{M}\ol{N}}$. Let $\hom(\bc_*(M_i), \bc_*(N_i))$ denote the morphisms from $\bc_*(M_i)$ to $\bc_*(N_i)$, as modules of the $A_\infty$ 1-category $\bc_*(E_i)$. We define a map @@ -197,32 +190,32 @@ and also the operad structure. We can reinterpret the above as a chain map \[ - p: C_0(FG^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) + p: C_0(SC^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) \to \hom(\bc_*(M_0), \bc_*(N_0)) . \] The main result of this section is that this chain map extends to the full singular -chain complex $C_*(FG^n_{\ol{M}\ol{N}})$. +chain complex $C_*(SC^n_{\ol{M}\ol{N}})$. \begin{thm} \label{thm:deligne} There is a collection of chain maps \[ - C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes + C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) \] which satisfy the operad compatibility conditions. -On $C_0(FG^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. +On $C_0(SC^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above. When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of \S\ref{sec:evaluation}. \end{thm} If, in analogy to Hochschild cochains, we define elements of $\hom(M, N)$ -to be ``blob cochains", we can summarize the above proposition by saying that the $n$-FG operad acts on +to be ``blob cochains", we can summarize the above proposition by saying that the $n$-SC operad acts on blob cochains. -As noted above, the $n$-FG operad contains the little $n{+}1$-ball operad, so this constitutes -a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disk operad. +As noted above, the $n$-SC operad contains the little $n{+}1$-balls operad, so this constitutes +a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disks operad. \begin{proof} -As described above, $FG^n_{\overline{M}, \overline{N}}$ is equal to the disjoint +As described above, $SC^n_{\overline{M}, \overline{N}}$ is equal to the disjoint union of products of homeomorphism spaces, modulo some relations. By Theorem \ref{thm:CH} and the Eilenberg-Zilber theorem, we have for each such product $P$ a chain map @@ -231,7 +224,7 @@ \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) . \] It suffices to show that the above maps are compatible with the relations whereby -$FG^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s. +$SC^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s. This in turn follows easily from the fact that the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative. %\nn{should add some detail to above} @@ -240,4 +233,4 @@ We note that even when $n=1$, the above theorem goes beyond an action of the little disks operad. $M_i$ could be a disjoint union of intervals, and $N_i$ could connect the end points of the intervals in a different pattern from $M_i$. -The genus of the fat graph could be greater than zero. +The genus of the surface associated to the surgery cylinder could be greater than zero.