diff -r f77cb464248e -r adc0780aa5e7 text/intro.tex --- a/text/intro.tex Thu Jun 03 21:59:55 2010 -0700 +++ b/text/intro.tex Thu Jun 03 23:08:47 2010 -0700 @@ -217,6 +217,7 @@ \end{equation*} \end{property} +\todo{Somehow, the Hochschild homology thing isn't a "property". Let's move it and call it a theorem? -S} \begin{property}[Hochschild homology when $X=S^1$] \label{property:hochschild}% The blob complex for a $1$-category $\cC$ on the circle is @@ -270,63 +271,60 @@ \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] \label{property:blobs-ainfty} Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold. -There is an $A_\infty$ $k$-category $A_*(Y, \cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$A_*(Y,\cC)(D) = A^\cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}. +There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}. \end{property} \begin{rem} Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category. We think of this $A_\infty$ $n$-category as a free resolution. \end{rem} There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category -instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. +instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. \begin{property}[Product formula] \label{property:product} Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category. -Let $A_*(Y,\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). +Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). Then \[ - \bc_*(Y\times W, \cC) \simeq \bc_*(W, A_*(Y,\cC)) . + \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). \] -Note on the right hand side we have the version of the blob complex for $A_\infty$ $n$-categories. \end{property} -It seems reasonable to expect a generalization describing an arbitrary fibre bundle. See in particular \S \ref{moddecss} for the framework for such a statement. +We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps. + +Fix a topological $n$-category $\cC$, which we'll omit from the notation. Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. (See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) \begin{property}[Gluing formula] \label{property:gluing}% \mbox{}% <-- gets the indenting right \begin{itemize} -\item For any $(n-1)$-manifold $Y$, the blob complex of $Y \times I$ is -naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below. - \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an -$A_\infty$ module for $\bc_*(Y \times I)$. +$A_\infty$ module for $\bc_*(Y)$. \item For any $n$-manifold $X_\text{glued} = X_\text{cut} \bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{glued})$ is the $A_\infty$ self-tensor product of -$\bc_*(X_\text{cut})$ as an $\bc_*(Y \times I)$-bimodule: +$\bc_*(X_\text{cut})$ as an $\bc_*(Y)$-bimodule: \begin{equation*} -\bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \selfarrow +\bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow \end{equation*} \end{itemize} \end{property} -Finally, we state two more properties, which we will not prove in this paper. -\nn{revise this; expect that we will prove these in the paper} +Finally, we prove two theorems which we consider as applications. -\begin{property}[Mapping spaces] +\begin{thm}[Mapping spaces] Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps $B^n \to T$. (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) Then $$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ -\end{property} +\end{thm} This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. -\begin{property}[Higher dimensional Deligne conjecture] -\label{property:deligne} +\begin{thm}[Higher dimensional Deligne conjecture] +\label{thm:deligne} The singular chains of the $n$-dimensional fat graph operad act on blob cochains. -\end{property} -See \S \ref{sec:deligne} for an explanation of the terms appearing here. The proof will appear elsewhere. +\end{thm} +See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof. Properties \ref{property:functoriality} and \ref{property:skein-modules} will be immediate from the definition given in \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. @@ -352,11 +350,3 @@ Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{and who else?} for many interesting and useful conversations. During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. - -\medskip\hrule\medskip - -Still to do: -\begin{itemize} -\item say something about starting with semisimple n-cat (trivial?? not trivial?) -\end{itemize} -