
diff -r d4e6bf589ebe -r a5e863658e74 text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Tue Oct 13 21:32:06 2009 +0000
+++ b/text/a_inf_blob.tex	Thu Oct 15 23:29:45 2009 +0000
@@ -20,7 +20,7 @@
 Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball
 $X$ the old-fashioned blob complex $\bc_*(X\times F)$.
 
-\begin{thm}
+\begin{thm} \label{product_thm}
 The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the
 new-fangled blob complex $\bc_*^\cF(Y)$.
 \end{thm}
@@ -28,14 +28,20 @@
 \begin{proof}
 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.
 
-First we define a map from $\bc_*^\cF(Y)$ to $\bc_*^C(Y\times F)$.
+First we define a map 
+\[
+	\psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) .
+\]
 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
 $Y\times F$.
 In filtration degrees 1 and higher we define the map to be zero.
 It is easy to check that this is a chain map.
 
-Next we define a map from $\phi: \bc_*^C(Y\times F) \to \bc_*^\cF(Y)$.
+Next we define a map 
+\[
+	\phi: \bc_*^C(Y\times F) \to \bc_*^\cF(Y) .
+\]
 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
@@ -56,7 +62,7 @@
 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 $K\times F$.
+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) $K\times F$.
 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}).
@@ -109,7 +115,6 @@
 \nn{need to 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.)
-For example, ....
 
 \begin{figure}[!ht]
 \begin{equation*}
@@ -119,16 +124,48 @@
 \label{zzz5}
 \end{figure}
 
+Continuing in this way we see that $D(a)$ is acyclic.
 \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.
 
-\nn{....}
+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}
+
+We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity.
+
+$\psi\circ\phi$ is the identity.  $\phi$ 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...
+
+This concludes the proof of Theorem \ref{product_thm}.
+\nn{at least I think it does; it's pretty rough at this point.}
 \end{proof}
 
-
 \nn{need to say something about dim $< n$ above}
 
+\medskip
 
+\begin{cor}
+The new-fangled and old-fashioned blob complexes are homotopic.
+\end{cor}
+\begin{proof}
+Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point.
+\end{proof}
 
 \medskip
 \hrule




