--- a/text/evmap.tex Mon Jul 12 17:29:25 2010 -0600
+++ b/text/evmap.tex Wed Jul 14 11:06:11 2010 -0600
@@ -13,7 +13,7 @@
than simplices --- they can be based on any linear polyhedron.
\nn{be more restrictive here? does more need to be said?})
-\begin{prop} \label{CHprop}
+\begin{thm} \label{thm:CH}
For $n$-manifolds $X$ and $Y$ there is a chain map
\eq{
e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y)
@@ -21,7 +21,7 @@
such that
\begin{enumerate}
\item on $CH_0(X, Y) \otimes \bc_*(X)$ it agrees with the obvious action of
-$\Homeo(X, Y)$ on $\bc_*(X)$ (Property (\ref{property:functoriality})), and
+$\Homeo(X, Y)$ 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*}
@@ -35,7 +35,7 @@
\end{enumerate}
Up to (iterated) homotopy, there is a unique family $\{e_{XY}\}$ of chain maps
satisfying the above two conditions.
-\end{prop}
+\end{thm}
Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma},
and then give an outline of the method of proof.
@@ -75,7 +75,7 @@
\medskip
-Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}.
+Before diving into the details, we outline our strategy for the proof of Theorem \ref{thm:CH}.
Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$.
We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that
\begin{itemize}
@@ -147,9 +147,7 @@
$\supp(p)\cup\supp(b)$, and so on.
-\medskip
-
-\begin{proof}[Proof of Proposition \ref{CHprop}.]
+\begin{proof}[Proof of Theorem \ref{thm:CH}.]
We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$.
Choose a metric on $X$.
@@ -594,7 +592,7 @@
\gl: R_*\ot CH_*(X, X) \otimes \bc_*(X) \to R_*\ot CH_*(X\sgl, X \sgl) \otimes \bc_*(X \sgl) ,
\]
and it is easy to see that $\gl(G^m_*)\sub \ol{G}^m_*$.
-From this it follows that the diagram in the statement of Proposition \ref{CHprop} commutes.
+From this it follows that the diagram in the statement of Theorem \ref{thm:CH} commutes.
\todo{this paragraph isn't very convincing, or at least I don't see what's going on}
Finally we show that the action maps defined above are independent of
@@ -613,7 +611,7 @@
Similar arguments show that this homotopy from $e$ to $e'$ is well-defined
up to second order homotopy, and so on.
-This completes the proof of Proposition \ref{CHprop}.
+This completes the proof of Theorem \ref{thm:CH}.
\end{proof}
@@ -629,7 +627,8 @@
\end{rem*}
-\begin{prop}
+\begin{thm}
+\label{thm:CH-associativity}
The $CH_*(X, Y)$ actions defined above are associative.
That is, the following diagram commutes up to homotopy:
\[ \xymatrix{
@@ -639,10 +638,10 @@
} \]
Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition
of homeomorphisms.
-\end{prop}
+\end{thm}
\begin{proof}
-The strategy of the proof is similar to that of Proposition \ref{CHprop}.
+The strategy of the proof is similar to that of Theorem \ref{thm:CH}.
We will identify a subcomplex
\[
G_* \sub CH_*(X, Y) \ot CH_*(Y, Z) \ot \bc_*(X)
@@ -656,7 +655,7 @@
(If $p:P\times X\to Y$, then $p\inv(|q|)$ means the union over all $x\in P$ of
$p(x, \cdot)\inv(|q|)$.)
-As in the proof of Proposition \ref{CHprop}, we can construct a homotopy
+As in the proof of Theorem \ref{thm:CH}, we can construct a homotopy
between the upper and lower maps restricted to $G_*$.
This uses the facts that the maps agree on $CH_0(X, Y) \ot CH_0(Y, Z) \ot \bc_*(X)$,
that they are compatible with gluing, and the contractibility of $\bc_*(X)$.