--- 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.