Binary file RefereeReport.pdf has changed

--- a/text/evmap.tex	Sun Jul 10 14:52:33 2011 -0600
+++ b/text/evmap.tex	Fri Jul 15 14:45:59 2011 -0700
@@ -8,9 +8,9 @@
 That is, for each pair of homeomorphic manifolds $X$ and $Y$
 we define a chain map
 \[
-    e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) ,
+    e_{XY} : \CH{X, Y} \otimes \bc_*(X) \to \bc_*(Y) ,
 \]
-where $CH_*(X, Y) = C_*(\Homeo(X, Y))$, the singular chains on the space
+where $C_*(\Homeo(X, Y))$ is the singular chains on the space
 of homeomorphisms from $X$ to $Y$.
 (If $X$ and $Y$ have non-empty boundary, these families of homeomorphisms
 are required to restrict to a fixed homeomorphism on the boundaries.)
@@ -406,32 +406,32 @@
 \subsection{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}}
 \label{ss:emap-def}
 
-Let $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, the singular chain complex of
+Let  $C_*(\Homeo(X \to Y))$ denote the singular chain complex of
 the space of homeomorphisms
 between the $n$-manifolds $X$ and $Y$ 
 (any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$).
-We also will use the abbreviated notation $CH_*(X) \deq CH_*(X, X)$.
-(For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general
+We also will use the abbreviated notation $\CH{X} \deq \CH{X \to X}$.
+(For convenience, we will permit the singular cells generating $\CH{X \to Y}$ to be more general
 than simplices --- they can be based on any cone-product polyhedron (see Remark \ref{blobsset-remark}).)
 
 \begin{thm}  \label{thm:CH} \label{thm:evaluation}%
 For $n$-manifolds $X$ and $Y$ there is a chain map
 \eq{
-    e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) ,
+    e_{XY} : \CH{X \to Y} \otimes \bc_*(X) \to \bc_*(Y) ,
 }
 well-defined up to homotopy,
 such that
 \begin{enumerate}
-\item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of 
+\item on $C_0(\Homeo(X \to Y)) \otimes \bc_*(X)$ it agrees with the obvious action of 
 $\Homeo(X, Y)$ on $\bc_*(X)$  described in Property \ref{property:functoriality}, and
 \item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, 
 the following diagram commutes up to homotopy
 \begin{equation*}
 \xymatrix@C+2cm{
-      CH_*(X, Y) \otimes \bc_*(X)
+      \CH{X \to Y} \otimes \bc_*(X)
         \ar[r]_(.6){e_{XY}}  \ar[d]^{\gl \otimes \gl}   &
             \bc_*(Y)\ar[d]^{\gl} \\
-     CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}}   & 	\bc_*(Y\sgl)  
+     \CH{X\sgl, Y\sgl} \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}}   & 	\bc_*(Y\sgl)  
 }
 \end{equation*}
 \end{enumerate}
@@ -443,14 +443,14 @@
 In fact, for $\btc_*$ we get a sharper result: we can omit
 the ``up to homotopy" qualifiers.
 
-Let $f\in CH_k(X, Y)$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$, 
+Let $f\in C_k(\Homeo(X \to Y))$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$, 
 $a:Q^j \to \BD_i(X)$.
 Define $e_{XY}(f\ot a)\in \btc_{i,j+k}(Y)$ by
 \begin{align*}
 	e_{XY}(f\ot a) : P\times Q &\to \BD_i(Y) \\
 	(p,q) &\mapsto f(p)(a(q))  .
 \end{align*}
-It is clear that this agrees with the previously defined $CH_0(X, Y)$ action on $\btc_*$,
+It is clear that this agrees with the previously defined $C_0(\Homeo(X \to Y))$ action on $\btc_*$,
 and it is also easy to see that the diagram in item 2 of the statement of the theorem
 commutes on the nose.
 \end{proof}
@@ -458,14 +458,14 @@
 
 \begin{thm}
 \label{thm:CH-associativity}
-The $CH_*(X, Y)$ actions defined above are associative.
+The $\CH{X \to Y}$ actions defined above are associative.
 That is, the following diagram commutes up to homotopy:
 \[ \xymatrix{
-& CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\
-CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\
-& CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} &
+& \CH{Y\to Z} \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\
+\CH{X \to Y} \ot \CH{Y \to Z} \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & \bc_*(Z) \\
+& \CH{X \to Z} \ot \bc_*(X) \ar[ur]_{e_{XZ}} &
 } \]
-Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition
+Here $\mu:\CH{X\to Y} \ot \CH{Y \to Z}\to \CH{X \to Z}$ is the map induced by composition
 of homeomorphisms.
 \end{thm}
 \begin{proof}

--- a/text/intro.tex	Sun Jul 10 14:52:33 2011 -0600
+++ b/text/intro.tex	Fri Jul 15 14:45:59 2011 -0700
@@ -38,7 +38,7 @@
 %See \S \ref{sec:future} for slightly more detail.
 
 Throughout, we have resisted the temptation to work in the greatest possible generality.
-(Don't worry, it wasn't that hard.)
+%(Don't worry, it wasn't that hard.)
 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category 
 with sufficient limits and colimits would do.
 We could also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories).