# HG changeset patch
# User Kevin Walker <kevin@canyon23.net>
# Date 1275493392 25200
# Node ID 4b64f9c6313fc89d78eddbee95d612f1feae1297
# Parent  121c580d5ef73b24f799f09073982220e911bb0e
Finished dealing with main issues in product thm proof; small issues still remain

diff -r 121c580d5ef7 -r 4b64f9c6313f text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Tue Jun 01 23:07:42 2010 -0700
+++ b/text/a_inf_blob.tex	Wed Jun 02 08:43:12 2010 -0700
@@ -70,33 +70,27 @@
 In filtration degrees 1 and higher we define the map to be zero.
 It is easy to check that this is a chain map.
 
-Next we define a map 
+In the other direction, we will define a subcomplex $G_*\sub \bc_*^C(Y\times F)$
+and a map
 \[
-	\phi: \bc_*^C(Y\times F) \to \bc_*^\cF(Y) .
+	\phi: G_* \to \bc_*^\cF(Y) .
 \]
-Actually, we will define it on the homotopy equivalent subcomplex
-$\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with 
-respect to some open cover
-of $Y\times F$
-(Proposition \ref{thm:small-blobs}).
-We will have to show eventually that this is independent (up to homotopy) of the choice of cover.
-Also, for a fixed choice of cover we will only be able to define the map for blob degree less than
-some bound, but this bound goes to infinity as the cover become finer.
 
 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding
 decomposition of $Y\times F$ into the pieces $X_i\times F$.
 
-%We will define $\phi$ inductively, starting at blob degree 0.
-%Given a 0-blob diagram $x$ on $Y\times F$, we can choose a decomposition $K$ of $Y$
-%such that $x$ is splittable with respect to $K\times F$.
-%This defines a filtration degree 0 element of $\bc_*^\cF(Y)$
+Let $G_*\sub \bc_*^C(Y\times F)$ be the subcomplex generated by blob diagrams $a$ such that there
+exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$.
+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 $G_*$.)
 
-We will define $\phi$ using a variant of the method of acyclic models.
-Let $a\in \cS_m$ be a blob diagram on $Y\times F$.
-For $m$ sufficiently small there exists a decomposition $K$ of $Y$ into $k$-balls such that the
-codimension 1 cells of $K\times F$ miss the blobs of $a$, and more generally such that $a$ is splittable along (the codimension-1 part of) $K\times F$.
+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 each $K_i$ has the aforementioned splittable property.
+such that $a$ splits along each $K_i\times F$.
 (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
@@ -202,31 +196,25 @@
 \end{proof}
 
 We are now in a position to apply the method of acyclic models to get a map
-$\phi:\cS_* \to \bc_*^\cF(Y)$.
-This map is defined in sufficiently low degrees, sends a blob diagram $a$ to $D(a)$, 
-and is well-defined up to (iterated) homotopy.
-
-The subcomplex $\cS_* \subset \bc_*^C(Y\times F)$ depends on choice of cover of $Y\times F$.
-If we refine that cover, we get a complex $\cS'_* \subset \cS_*$
-and a map $\phi':\cS'_* \to \bc_*^\cF(Y)$.
-$\phi'$ is defined only on homological degrees below some bound, but this bound is higher than 
-the corresponding bound for $\phi$.
-We must show that $\phi$ and $\phi'$ agree, up to homotopy,
-on the intersection of the subcomplexes on which they are defined.
-This is clear, since the acyclic subcomplexes $D(a)$ above used in the definition of 
-$\phi$ and $\phi'$ do not depend on the choice of cover.
-
-%\nn{need to say (and justify) that we now have a map $\phi$ indep of choice of cover}
+$\phi:G_* \to \bc_*^\cF(Y)$.
+We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is in filtration degree zero
+and $r$ has filtration degree greater than zero.
 
 We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity.
 
-$\psi\circ\phi$ is the identity on the nose.  
-$\phi$ takes a blob diagram $a$ and chops it into pieces 
-according to some decomposition $K$ of $Y$.
-$\psi$ glues those pieces back together, yielding the same $a$ we started with.
-
-$\phi\circ\psi$ is the identity up to homotopy by another MoAM argument....
-
+$\psi\circ\phi$ is the identity on the nose:
+\[
+	\psi(\phi(a)) = \psi((a,K)) + \psi(r) = a + 0.
+\]
+Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and 
+$\psi$ glues those pieces back together, yielding $a$.
+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.
+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.
 
 This concludes the proof of Theorem \ref{product_thm}.
 \end{proof}