diff -r f582f921bd95 -r febbf06c3610 text/deligne.tex --- 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.}