
diff -r 76f423a9c787 -r 3f180943709f text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Fri Aug 21 23:17:10 2009 +0000
+++ b/text/a_inf_blob.tex	Wed Aug 26 01:21:59 2009 +0000
@@ -60,7 +60,9 @@
 Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \bar{K})$
 such that each $K_i$ has the aforementioned splittable property
 (see Subsection \ref{ss:ncat_fields}).
-(By $(a, \bar{K})$ we really mean $(a', \bar{K})$, where $a^\sharp$ is 
+\nn{need to define $D(a)$ more clearly; also includes $(b_j, \bar{K})$ where
+$\bd(a) = \sum b_j$.}
+(By $(a, \bar{K})$ we really mean $(a^\sharp, \bar{K})$, where $a^\sharp$ is 
 $a$ split according to $K_0\times F$.
 To simplify notation we will just write plain $a$ instead of $a^\sharp$.)
 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give
@@ -76,9 +78,30 @@
 \begin{proof}
 We will prove acyclicity in the first couple of degrees, and \nn{in this draft, at least}
 leave the general case to the reader.
+
 Let $K$ and $K'$ be two decompositions of $Y$ compatible with $a$.
-We want to show that $(a, K)$ and $(a, K')$ are homologous
-\nn{oops -- can't really ignore $\bd a$ like this}
+We want to show that $(a, K)$ and $(a, K')$ are homologous via filtration degree 1 stuff.
+\nn{need to say this better; these two chains don't have the same boundary.}
+We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily
+the case.
+(Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.)
+However, we {\it can} find another decomposition $L$ such that $L$ shares common
+refinements with both $K$ and $K'$.
+Let $KL$ and $K'L$ denote these two refinements.
+Then filtration degree 1 chains associated to the four anti-refinemnts
+$KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$
+give the desired chain connecting $(a, K)$ and $(a, K')$
+(see Figure xxxx).
+
+Consider a different choice of decomposition $L'$ in place of $L$ above.
+This leads to a cycle consisting of filtration degree 1 stuff.
+We want to show that this cycle bounds a chain of filtration degree 2 stuff.
+Choose a decomposition $M$ which has common refinements with each of 
+$K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$.
+Then we have a filtration degree 2 chain, as shown in Figure yyyy, which does the trick.
+For example, ....
+
+
 \end{proof}
 
 




