--- a/text/a_inf_blob.tex Wed Jun 29 10:44:13 2011 -0700
+++ b/text/a_inf_blob.tex Wed Jun 29 11:51:35 2011 -0700
@@ -8,10 +8,10 @@
We will show below
in Corollary \ref{cor:new-old}
-that when $\cC$ is obtained from a system of fields $\cD$
+that when $\cC$ is obtained from a system of fields $\cE$
as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}),
$\cl{\cC}(M)$ is homotopy equivalent to
-our original definition of the blob complex $\bc_*(M;\cD)$.
+our original definition of the blob complex $\bc_*(M;\cE)$.
%\medskip
@@ -51,7 +51,7 @@
First we define a map
\[
- \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;C) .
+ \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;\cE) .
\]
On 0-simplices of the hocolimit
we just glue together the various blob diagrams on $X_i\times F$
@@ -60,7 +60,7 @@
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 $G_*\sub \bc_*(Y\times F;C)$
+In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;\cE)$
and a map
\[
\phi: G_* \to \cl{\cC_F}(Y) .
@@ -69,9 +69,9 @@
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 $X_i\times F$.
-Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there
+Let $G_*\sub \bc_*(Y\times F;\cE)$ 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 Lemma \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$
+It follows from Lemma \ref{thm:small-blobs} that $\bc_*(Y\times F; \cE)$
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.)