--- a/text/appendixes/famodiff.tex	Wed Aug 10 13:11:59 2011 -0600
+++ b/text/appendixes/famodiff.tex	Wed Aug 10 13:30:17 2011 -0600
@@ -313,6 +313,18 @@
 \item[(I)] the support of $A_\beta$ is contained in $U_i^{i-1} \cup V_\beta^{N-i+1}$.
 \end{itemize}
 
+Next we define $B_\beta$.
+Theorem 5.1 of \cite{MR0283802} implies that we can choose a homotopy $B_\beta:Q_\beta\times I\to \Homeo(X)$
+such that
+\begin{itemize}
+\item[(J)] $B_\beta(\cdot, 0) = A_\beta(\cdot, 1)$;
+\item[(K)] $B_\beta(q,1) = g$ on $W_i^i$;
+\item[(L)] the support of $B_\beta(\cdot,1)$ is contained in $V_\beta^{N-i}$; and
+\item[(M)] the support of $B_\beta$ is contained in $U_i^i \cup V_\beta^{N-i}$.
+\end{itemize}
+
+All that remains is to define the ``glue" $C$ which interpolates between adjacent $\beta$ and $\beta'$.
+First consider the $k=2$ case.