--- a/text/a_inf_blob.tex Wed Apr 13 12:14:18 2011 -0700
+++ b/text/a_inf_blob.tex Mon Apr 18 22:28:40 2011 -0700
@@ -192,7 +192,7 @@
We have $\psi(r) = 0$ since $\psi$ is zero on $(\ge 1)$-simplices.
Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models.
-To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above.
+To each generator $(b, \ol{K})$ of $\cl{\cC_F}(Y)$ 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 method of acyclic models argument these two maps
are homotopic.
@@ -248,12 +248,12 @@
A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$:
assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, if $\dim(D) = k$,
or the fields $\cE(p^*(E))$, if $\dim(D) < k$.
-($p^*(E)$ denotes the pull-back bundle over $D$.)
+(Here $p^*(E)$ denotes the pull-back bundle over $D$.)
Let $\cF_E$ denote this $k$-category over $Y$.
We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to
get a chain complex $\cl{\cF_E}(Y)$.
The proof of Theorem \ref{thm:product} goes through essentially unchanged
-to show that
+to show the following result.
\begin{thm}
Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above.
Then
@@ -270,7 +270,7 @@
$D\widetilde{\times} M$ is a manifold of dimension $n-k+j$ with a collar structure along the boundary of $D$.
(If $D\to Y$ is an embedding then $D\widetilde{\times} M$ is just the part of $M$
lying above $D$.)
-We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which a re good with respect to $M$.
+We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$.
We can again adapt the homotopy colimit construction to
get a chain complex $\cl{\cF_M}(Y)$.
The proof of Theorem \ref{thm:product} again goes through essentially unchanged