--- a/text/deligne.tex	Sun Jun 06 20:56:47 2010 +0200
+++ b/text/deligne.tex	Mon Jun 07 05:58:52 2010 +0200
@@ -226,9 +226,18 @@
 
 \begin{proof}
 As described above, $FG^n_{\overline{M}, \overline{N}}$ is equal to the disjoint
-union of products of homeomorphisms spaces, modulo some relations.
-By \ref{CHprop}, 
-\nn{...}
+union of products of homeomorphism spaces, modulo some relations.
+By Proposition \ref{CHprop} and the Eilenberg-Zilber theorem, we have for each such product $P$
+a chain map
+\[
+	C_*(P)\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots\otimes 
+\hom(\bc_*(M_{k}), \bc_*(N_{k})) \to  \hom(\bc_*(M_0), \bc_*(N_0)) .
+\]
+It suffices to show that the above maps are compatible with the relations whereby
+$FG^n_{\overline{M}, \overline{N}}$ is constructed from the various $P$'s.
+This in turn follows easily from the actions of $C_*(\Homeo(\cdot\to\cdot))$ are local (compatible with gluing) and associative.
+
+\nn{should add some detail to above}
 \end{proof}
 
 \nn{maybe point out that even for $n=1$ there's something new here.}