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 \end{axiom}
 Note that the functoriality in the above axiom allows us to operate via
 homeomorphisms which are not the identity on the boundary of the $k$-ball.
 The action of these homeomorphisms gives the ``strong duality" structure.
+As such, we don't subdivide the boundary of a morphism
-Next we consider domains and ranges of $k$-morphisms.
+into domain and range --- the duality operations can convert between domain and range.
-Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism
-into domain and range --- the duality operations can convert domain to range and vice-versa.
-Instead, we will use a unified domain/range, which we will call a ``boundary".
 Later \todo{} we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom.
 \begin{axiom}[Boundaries]\label{nca-boundary}
 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.
 \end{axiom}
 For $c\in \cl{\cC}_{k-1}(\bd X)$ we define $\cC_k(X; c) = \bd^{-1}(c)$.
 Many of the examples we are interested in are enriched in some auxiliary category $\cS$
-(e.g. $\cS$ is vector spaces or rings, or, in the $A_\infty$ case, chain complex or topological spaces).
+(e.g. vector spaces or rings, or, in the $A_\infty$ case, chain complex or topological spaces).
-This means (by definition) that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure
+This means that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure
 of an object of $\cS$, and all of the structure maps of the category (above and below) are
 compatible with the $\cS$ structure on $\cC_n(X; c)$.
 Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to assemble them into a boundary value of the entire sphere.
 For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices.
 \begin{axiom}[Product (identity) morphisms]
 \label{axiom:product}
 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$),
 there is a map $\pi^*:\cC(X)\to \cC(E)$.
+These maps must be
+\begin{enumerate}
+\item natural with respect to maps of pinched products,
+\item functorial with respect to composition of pinched products,
+\item compatible with gluing and restriction of pinched products.
+\end{enumerate}
+%%% begin noop %%%
+% this was the original list of conditions, which I've replaced with the much terser list above -S
+\noop{
 These maps must satisfy the following conditions.
 \begin{enumerate}
 \item
 If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and
 if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram
 such that $\rho$ and $\pi$ are pinched products, then
 \[
 	\res_D\circ\pi^* = \rho^*\circ\res_Y .
 \]
 \end{enumerate}
+} %%% end \noop %%%
 \end{axiom}
 \begin{axiom}[\textup{\textbf{[plain  version]}} Extended isotopy invariance in dimension $n$.]
 \label{axiom:extended-isotopies}
 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
 to the identity on $\bd X$ and isotopic (rel boundary) to the identity.
 An $n$-dimensional surgery cylinder is a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another.
 By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module.
-\todo{Sketch proof}
+\begin{proof}
+We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, and the action of surgeries is just composition of maps of $A_\infty$-modules. We only need to check that the relations of the $n$-SC operad are satisfied. This follows immediately from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and functoriality.
+\end{proof}
 The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map
 \[
 	C_*(LD_k)\otimes \overbrace{Hoch^*(C, C)\otimes\cdots\otimes Hoch^*(C, C)}^{\text{$k$ copies}}
 			\to  Hoch^*(C, C),
 %% \appendix Appendix text...
 %% or, for appendix with title, use square brackets:
 %% \appendix[Appendix Title]
 \begin{acknowledgments}
--- text of acknowledgments here, including grant info --
+\nn{say something here}
 \end{acknowledgments}
 %% PNAS does not support submission of supporting .tex files such as BibTeX.
 %% Instead all references must be included in the article .tex document.
 %% If you currently use BibTeX, your bibliography is formed because the

changeset 595	9c708975b61b
parent 594	6945422bed13
child 597	26c4d576e155