--- a/text/deligne.tex Sat May 29 15:36:14 2010 -0700
+++ b/text/deligne.tex Sat May 29 20:13:23 2010 -0700
@@ -70,7 +70,7 @@
\caption{An $n$-dimensional fat graph}\label{delfig2}
\end{figure}
-More specifically, an $n$-dimensional fat graph consists of:
+More specifically, an $n$-dimensional fat graph ($n$-FG for short) consists of:
\begin{itemize}
\item ``Upper" $n$-manifolds $M_0,\ldots,M_k$ and ``lower" $n$-manifolds $N_0,\ldots,N_k$,
with $\bd M_i = \bd N_i = E_i$ for all $i$.
@@ -95,11 +95,13 @@
cylinders.
More specifically, we impose the following two equivalence relations:
\begin{itemize}
-\item If $g:R_i\to R_i$ is a homeomorphism, we can replace
-\[
- (\ldots, f_{i-1}, f_i, \ldots) \to (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots),
-\]
-leaving the $M_i$, $N_i$ and $R_i$ fixed.
+\item If $g: R_i\to R'_i$ is a homeomorphism, we can replace
+\begin{eqnarray*}
+ (\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& (\ldots, R_{i-1}, R'_i, R_{i+1}, \ldots) \\
+ (\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots),
+\end{eqnarray*}
+leaving the $M_i$ and $N_i$ fixed.
+(Keep in mind the case $R'_i = R_i$.)
(See Figure xxxx.)
\item If $M_i = M'_i \du M''_i$ and $N_i = N'_i \du N''_i$ (and there is a
compatible disjoint union of $\bd M = \bd N$), we can replace
@@ -114,7 +116,7 @@
\end{itemize}
Note that the second equivalence increases the number of holes (or arity) by 1.
-We can make a similar identification with the rolls of $M'_i$ and $M''_i$ reversed.
+We can make a similar identification with the roles of $M'_i$ and $M''_i$ reversed.
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.
@@ -124,6 +126,8 @@
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
+are ordered differently.)
The topology comes from the spaces
\[
\Homeo(M_0\to R_1\cup M_1)\times \Homeo(R_1\cup N_1\to R_2\cup M_2)\times
@@ -132,6 +136,31 @@
and the above equivalence relations.
We will denote the typical element of $FG^n_{\ol{M}\ol{N}}$ by $\ol{f} = (f_0,\ldots,f_k)$.
+\medskip
+
+%The little $n{+}1$-ball operad injects into the $n$-FG operad.
+The $n$-FG operad contains the little $n{+}1$-ball operad.
+Roughly speaking, given a configuration of $k$ little $n{+}1$-balls in the standard
+$n{+}1$-ball, we fiber the complement of the balls by vertical intervals
+and let $M_i$ [$N_i$] be the southern [northern] hemisphere of the $i$-th ball.
+More precisely, let $x_0,\ldots,x_n$ be the coordinates of $\r^{n+1}$.
+Let $z$ be a point of the $k$-th space of the little $n{+}1$-ball operad, with
+little balls $D_1,\ldots,D_k$ inside the standard $n{+}1$-ball.
+We assume the $D_i$'s are ordered according to the $x_n$ coordinate of their centers.
+Let $\pi:\r^{n+1}\to \r^n$ be the projection corresponding to $x_n$.
+Let $B\sub\r^n$ be the standard $n$-ball.
+Let $M_i$ and $N_i$ be $B$ for all $i$.
+Identify $\pi(D_i)$ with $B$ (a.k.a.\ $M_i$ or $N_i$) via translations and dilations (no rotations).
+Let $R_i = B\setmin \pi(D_i)$.
+Let $f_i = \rm{id}$ for all $i$.
+We have now defined a map from the little $n{+}1$-ball operad to the $n$-FG operad,
+with contractible fibers.
+(The fibers correspond to moving the $D_i$'s in the $x_n$ direction without changing their ordering.)
+\nn{issue: we've described this by varying the $R_i$'s, but above we emphasize varying the $f_i$'s.
+does this need more explanation?}
+
+Another familiar subspace of the $n$-FG operad is $\Homeo(M\to N)$, which corresponds to
+case $k=0$ (no holes).
\medskip
@@ -143,41 +172,43 @@
p(\ol{f}): \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k))
\to \hom(\bc_*(M_0), \bc_*(N_0)) .
\]
-Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define $p(\ol{f}$ to be the composition
+Given $\alpha_i\in\hom(\bc_*(M_i), \bc_*(N_i))$, we define $p(\ol{f}$) to be the composition
\[
\bc_*(M_0) \stackrel{f_0}{\to} \bc_*(R_1\cup M_1)
\stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1)
- \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \to
+ \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to}
\cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k)
\stackrel{f_k}{\to} \bc_*(N_0)
\]
(Recall that the maps $\id\ot\alpha_i$ were defined in \nn{need ref}.)
It is easy to check that the above definition is compatible with the equivalence relations
and also the operad structure.
-
-\nn{little m-disks operad; }
-
-\nn{*** resume revising here}
-
-
+We can reinterpret the above as a chain map
+\[
+ p: C_0(FG^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k))
+ \to \hom(\bc_*(M_0), \bc_*(N_0)) .
+\]
+The main result of this section is that this chain map extends to the full singular
+chain complex $C_*(FG^n_{\ol{M}\ol{N}})$.
-Putting this together we get
-\begin{prop}(Precise statement of Property \ref{property:deligne})
+\begin{prop}
\label{prop:deligne}
-There is a collection of maps
-\begin{eqnarray*}
+There is a collection of chain maps
+\[
C_*(FG^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes
-\hom(\bc_*(M_{k}), \bc_*(N_{k})) & \\
- & \hspace{-11em}\to \hom(\bc_*(M_0), \bc_*(N_0))
-\end{eqnarray*}
-which satisfy an operad type compatibility condition. \nn{spell this out}
+\hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0))
+\]
+which satisfy the operad compatibility conditions.
+On $C_0(FG^n_{\ol{M}\ol{N}})$ this agrees with the chain map $p$ defined above.
+When $k=0$, this coincides with the $C_*(\Homeo(M_0\to N_0))$ action of Section \ref{sec:evaluation}.
\end{prop}
-Note that if $k=0$ then this is just the action of chains of diffeomorphisms from Section \ref{sec:evaluation}.
-And indeed, the proof is very similar \nn{...}
-
+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
+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.
+\nn{...}
-\medskip
-\hrule\medskip
-
+\nn{maybe point out that even for $n=1$ there's something new here.}