
diff -r 50088eefeedf -r 7dc75375d376 text/a_inf_blob.tex
--- a/text/a_inf_blob.tex	Mon Jul 04 11:35:27 2011 -0600
+++ b/text/a_inf_blob.tex	Tue Jul 05 13:28:02 2011 -0600
@@ -60,8 +60,8 @@
 For simplices of dimension 1 and higher we define the map to be zero.
 It is easy to check that this is a chain map.
 
-In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;\cE)$
-and a map
+In the other direction, we will define (in the next few paragraphs) 
+a subcomplex $G_*\sub \bc_*(Y\times F;\cE)$ and a map
 \[
 	\phi: G_* \to \cl{\cC_F}(Y) .
 \]
@@ -80,8 +80,9 @@
 We will define $\phi: G_* \to \cl{\cC_F}(Y)$ using the method of acyclic models.
 Let $a$ be a generator of $G_*$.
 Let $D(a)$ denote the subcomplex of $\cl{\cC_F}(Y)$ 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).
+where $b$ is a generator appearing
+in an iterated boundary of $a$ (this includes $a$ itself)
+and $b$ splits along $K_0\times F$.
 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions;
 see \S\ref{ss:ncat_fields}.)
 By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is 
@@ -94,26 +95,39 @@
 More formally,
  
 \begin{lemma} \label{lem:d-a-acyclic}
-$D(a)$ is acyclic.
+$D(a)$ is acyclic in positive degrees.
 \end{lemma}
 
 \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 $P(a)$ denote the finite cone-product polyhedron composed of $a$ and its iterated boundaries.
+(See Remark \ref{blobsset-remark}.)
+We can think of $D(a)$ as a cell complex equipped with an obvious
+map $p: D(a) \to P(a)$ which forgets the second factor.
+For each cell $b$ of $P(a)$, let $I(b) = p\inv(b)$.
+It suffices to show that each $I(b)$ is acyclic and more generally that
+each intersection $I(b)\cap I(b')$ is acyclic.
 
-Let $K$ and $K'$ be two decompositions (0-simplices) of $Y$ compatible with $a$.
+If $I(b)\cap I(b')$ is nonempty then then as a cell complex it is isomorphic to
+$(b\cap b') \times E(b, b')$, where $E(b, b')$ consists of those simplices
+$\ol{K} = (K_0,\ldots,K_l)$ such that both $b$ and $b'$ split along $K_0\times F$.
+(Here we are thinking of $b$ and $b'$ as both blob diagrams and also faces of $P(a)$.)
+So it suffices to show that $E(b, b')$ is acyclic.
+
+Let $K$ and $K'$ be two decompositions of $Y$ (i.e.\ 0-simplices) in $E(b, b')$.
 We want to find 1-simplices which connect $K$ and $K'$.
 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 $y = e^{-1/x^2} \sin(1/x)$ in $\r^2$.)
 However, we {\it can} find another decomposition $L$ such that $L$ shares common
 refinements with both $K$ and $K'$.
+This follows from Axiom \ref{axiom:vcones}, which in turn follows from the
+splitting axiom for the system of fields $\cE$.
 Let $KL$ and $K'L$ denote these two refinements.
 Then 1-simplices associated to the four anti-refinements
 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$
 give the desired chain connecting $(a, K)$ and $(a, K')$
 (see Figure \ref{zzz4}).
+(In the language of Axiom \ref{axiom:vcones}, this is $\vcone(K \du K')$.)
 
 \begin{figure}[t] \centering
 \begin{tikzpicture}
@@ -130,9 +144,10 @@
 \label{zzz4}
 \end{figure}
 
-Consider a different choice of decomposition $L'$ in place of $L$ above.
-This leads to a cycle of 1-simplices.
-We want to find 2-simplices which fill in this cycle.
+Consider next a 1-cycle in $E(b, b')$, such as one arising from
+a different choice of decomposition $L'$ in place of $L$ above.
+%We want to find 2-simplices which fill in this cycle.
+By Axiom \ref{axiom:vcones} we can fill in this 1-cycle with 2-simplices.
 Choose a decomposition $M$ which has common refinements with each of 
 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$.
 (We also require that $KLM$ antirefines to $KM$, etc.)
@@ -175,6 +190,7 @@
 \end{figure}
 
 Continuing in this way we see that $D(a)$ is acyclic.
+By Axiom \ref{axiom:vcones} we can fill in any cycle with a V-Cone.
 \end{proof}
 
 We are now in a position to apply the method of acyclic models to get a map




