--- 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}