# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1313122451 21600
# Node ID c570a7a75b07da7aa123d9b94449eb2f7c5a2149
# Parent  61541264d4b3c4629447a00f777b027892a4ff43
last remaining items from referee's typo/minor list

diff -r 61541264d4b3 -r c570a7a75b07 RefereeReport.pdf
Binary file RefereeReport.pdf has changed
diff -r 61541264d4b3 -r c570a7a75b07 text/deligne.tex
--- a/text/deligne.tex	Thu Aug 11 13:54:38 2011 -0700
+++ b/text/deligne.tex	Thu Aug 11 22:14:11 2011 -0600
@@ -178,7 +178,8 @@
 	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 
+$p(\ol{f})(\alpha_1\ot\cdots\ot\alpha_k)$ 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)
@@ -201,7 +202,7 @@
 \label{thm:deligne}
 There is a collection of chain maps
 \[
-	C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes 
+	C_*(SC^n_{\ol{M}\ol{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes 
 \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to  \hom(\bc_*(M_0), \bc_*(N_0))
 \]
 which satisfy the operad compatibility conditions.
@@ -216,7 +217,7 @@
 a higher dimensional version of the Deligne conjecture for Hochschild cochains and the little 2-disks operad.
 
 \begin{proof}
-As described above, $SC^n_{\overline{M}, \overline{N}}$ is equal to the disjoint
+As described above, $SC^n_{\ol{M}\ol{N}}$ 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
@@ -225,7 +226,7 @@
 \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
-$SC^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s.
+$SC^n_{\ol{M}\ol{N}}$ 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}
diff -r 61541264d4b3 -r c570a7a75b07 text/hochschild.tex
--- a/text/hochschild.tex	Thu Aug 11 13:54:38 2011 -0700
+++ b/text/hochschild.tex	Thu Aug 11 22:14:11 2011 -0600
@@ -344,8 +344,8 @@
 $$\sum_j d_{j,1} \tensor \cdots \tensor d_{j,k_j} \tensor m_j \tensor d_{j,1}' \tensor \cdots \tensor d_{j,k'_j}' \in \ker(\DirectSum_{k,k'} C^{\tensor k} \tensor M \tensor C^{\tensor k'} \to M),$$
 and so
 \begin{align*}
-\ev(\bdy y) & = \sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j} \\
-            & = \sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k \\
+\pi\left(\ev(\bdy y)\right) & = \pi\left(\sum_j m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k d_{j,1} \cdots d_{j,k_j}\right) \\
+            & = \pi\left(\sum_j d_{j,1} \cdots d_{j,k_j} m_j d_{j,1}' \cdots d_{j,k'_j}' c_1 \cdots c_k\right) \\
             & = 0
 \end{align*}
 where this time we use the fact that we're mapping to $\coinv{M}$, not just $M$.
diff -r 61541264d4b3 -r c570a7a75b07 text/tqftreview.tex
--- a/text/tqftreview.tex	Thu Aug 11 13:54:38 2011 -0700
+++ b/text/tqftreview.tex	Thu Aug 11 22:14:11 2011 -0600
@@ -85,7 +85,7 @@
 \item The subset $\cC_n(X;c)$ of top-dimensional fields 
 with a given boundary condition is an object in our symmetric monoidal category $\cS$.
 (This condition is of course trivial when $\cS = \Set$.) 
-If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), 
+If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$ (chain complexes)), 
 then this extra structure is considered part of the definition of $\cC_n$.
 Any maps mentioned below between fields on $n$-manifolds must be morphisms in $\cS$.
 \item $\cC_k$ is compatible with the symmetric monoidal
@@ -299,7 +299,7 @@
 domain and range determined by the transverse orientation and the labelings of the 1-cells.
 \end{itemize}
 
-We want fields on 1-manifolds to be enriched over Vect, so we also allow formal linear combinations
+We want fields on 1-manifolds to be enriched over $\Vect$, so we also allow formal linear combinations
 of the above fields on a 1-manifold $X$ so long as these fields restrict to the same field on $\bd X$.
 
 In addition, we mod out by the relation which replaces
@@ -371,7 +371,7 @@
 \subsection{Local relations}
 \label{sec:local-relations}
 
-For convenience we assume that fields are enriched over Vect.
+For convenience we assume that fields are enriched over $\Vect$.
 
 Local relations are subspaces $U(B; c)\sub \cC(B; c)$ of the fields on balls which form an ideal under gluing.
 Again, we give the examples first.
@@ -400,7 +400,7 @@
 \begin{enumerate}
 \item Functoriality: 
 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$
-\item Local relations imply extended isotopy: 
+\item Local relations imply extended isotopy invariance: 
 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic 
 to $y$, then $x-y \in U(B; c)$.
 \item Ideal with respect to gluing: