
diff -r c2d2a8f8d70c -r a60332c29d0b text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Wed Feb 24 01:25:59 2010 +0000
+++ b/text/a_inf_blob.tex	Wed Feb 24 06:28:03 2010 +0000
@@ -295,6 +295,29 @@
 It is not hard to see that this defines a chain map from 
 $\cB^\cT(M)$ to $C_*(\Maps(M\to T))$.
 
+Next we show that $g$ induces a surjection on homology.
+Fix $k > 0$ and choose an open cover $\cU$ of $M$ fine enough so that the union 
+of any $k$ open sets of $\cU$ is contained in a disjoint union of balls in $M$.
+\nn{maybe should refer to elsewhere in this paper where we made a very similar argument}
+Let $S_*$ be the subcomplex of $C_*(\Maps(M\to T))$ generated by chains adapted to $\cU$.
+It follows from Lemma \ref{extension_lemma_b} that $C_*(\Maps(M\to T))$
+retracts onto $S_*$.
+
+Let $S_{\le k}$ denote the chains of $S_*$ of degree less than or equal to $k$.
+We claim that $S_{\le k}$ lies in the image of $g$.
+Let $c$ be a generator of $S_{\le k}$ --- that is, a $j$-parameter family of maps $M\to T$,
+$j \le k$.
+We chose $\cU$ fine enough so that the support of $c$ is contained in a disjoint union of balls
+in $M$.
+It follow that we can choose a decomposition $K$ of $M$ so that the support of $c$ is 
+disjoint from the $n{-}1$-skeleton of $K$.
+It is now easy to see that $c$ is in the image of $g$.
+
+Next we show that $g$ is injective on homology.
+
+
+
+
 \nn{...}
 
 




