# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1275508328 25200
# Node ID 091c36b943e704c2fb729cfc4845280276a56074
# Parent  76c301fdf0a222e3c65ca5f817bf1dc5a0534e3a
more futzing with product thm

diff -r 76c301fdf0a2 -r 091c36b943e7 text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Wed Jun 02 11:45:19 2010 -0700
+++ b/text/a_inf_blob.tex	Wed Jun 02 12:52:08 2010 -0700
@@ -64,8 +64,8 @@
 \[
 	\psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) .
 \]
-In filtration degree 0 we just glue together the various blob diagrams on $X\times F$
-(where $X$ is a component of a permissible decomposition of $Y$) to get a blob diagram on
+In filtration degree 0 we just glue together the various blob diagrams on $X_i\times F$
+(where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on
 $Y\times F$.
 In filtration degrees 1 and higher we define the map to be zero.
 It is easy to check that this is a chain map.
@@ -84,22 +84,20 @@
 It follows from Proposition \ref{thm:small-blobs} that $\bc_*^C(Y\times F)$ is homotopic to a subcomplex of $G_*$.
 (If the blobs of $a$ are small with respect to a sufficiently fine cover then their
 projections to $Y$ are contained in some disjoint union of balls.)
-Note that the image of $\psi$ is contained in $G_*$.
-(In fact, equal to $G_*$.)
+Note that the image of $\psi$ is equal to $G_*$.
 
 We will define $\phi: G_* \to \bc_*^\cF(Y)$ using the method of acyclic models.
 Let $a$ be a generator of $G_*$.
-Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(a, \ol{K})$
-such that $a$ splits along each $K_i\times F$.
+Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(b, \ol{K})$
+such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing
+in an iterated boundary of $a$ (this includes $a$ itself).
 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions;
 see Subsection \ref{ss:ncat_fields}.)
-\nn{need to define $D(a)$ more clearly; also includes $(b_j, \ol{K})$ where
-$\bd(a) = \sum b_j$.}
-(By $(a, \ol{K})$ we really mean $(a^\sharp, \ol{K})$, where $a^\sharp$ is 
-$a$ split according to $K_0\times F$.
-To simplify notation we will just write plain $a$ instead of $a^\sharp$.)
+By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is 
+$b$ split according to $K_0\times F$.
+To simplify notation we will just write plain $b$ instead of $b^\sharp$.
 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give
-$a$, filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, 
+$a$ (or one of its iterated boundaries), filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, 
 filtration degree 2 stuff which kills the homology created by the 
 filtration degree 1 stuff, and so on.
 More formally,
@@ -148,7 +146,7 @@
 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 $KL'$.
-\nn{need to also require that $KLM$ antirefines to $KM$, etc.}
+(We also also require that $KLM$ antirefines to $KM$, etc.)
 Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick.
 (Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.)
 
@@ -211,7 +209,7 @@
 We have $\psi(r) = 0$ since $\psi$ is zero in positive filtration degrees.
  
 $\phi\circ\psi$ is the identity up to homotopy by another MoAM argument.
-To each generator $(a, \ol{K})$ of we associated the acyclic subcomplex $D(a)$ defined above.
+To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above.
 Both the identity map and $\phi\circ\psi$ are compatible with this
 collection of acyclic subcomplexes, so by the usual MoAM argument these two maps
 are homotopic.