--- a/text/blobdef.tex	Tue Feb 08 20:16:47 2011 +1100
+++ b/text/blobdef.tex	Tue Feb 08 07:14:18 2011 -0800
@@ -180,8 +180,12 @@
 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$.
 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}.
 \end{defn}
-Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is 
-splittable along it if it is the image of a field on $M_0$.
+
+Let $M_0 \to M_1 \to \cdots \to M_m = X$ be a gluing decomposition of $X$, 
+and let $M_0^0,\ldots,M_0^k$ be the connected components of $M_0$.
+We say that a field 
+$a\in \cF(X)$ is splittable along the decomposition if $a$ is the image 
+under gluing and disjoint union of a fields $a_i \in \cF(M_0^i)$, $0\le i\le k$.
 
 In the example above, note that
 \[
@@ -200,8 +204,8 @@
 \label{defn:configuration}
 A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ 
 of $X$ such that there exists a gluing decomposition $M_0  \to \cdots \to M_m = X$ of $X$ and 
-for each subset $B_i$ there is some $0 \leq r \leq m$ and some connected component $M_r'$ of 
-$M_r$ which is a ball, so $B_i$ is the image of $M_r'$ in $X$. 
+for each subset $B_i$ there is some $0 \leq l \leq m$ and some connected component $M_l'$ of 
+$M_l$ which is a ball, so $B_i$ is the image of $M_l'$ in $X$. 
 We say that such a gluing decomposition 
 is \emph{compatible} with the configuration. 
 A blob $B_i$ is a twig blob if no other blob $B_j$ is a strict subset of it. 
@@ -213,7 +217,7 @@
 Note that nested blobs may have boundaries that overlap, or indeed coincide. 
 Blobs may meet the boundary of $X$.
 Further, note that blobs need not actually be embedded balls in $X$, since parts of the 
-boundary of the ball $M_r'$ may have been glued together.
+boundary of the ball $M_l'$ may have been glued together.
 
 Note that often the gluing decomposition for a configuration of blobs may just be the trivial one: 
 if the boundaries of all the blobs cut $X$ into pieces which are all manifolds, 
@@ -235,8 +239,8 @@
 the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace 
 $U(B_i) \subset \cF(B_i)$. 
 (See Figure \ref{blobkdiagram}.) 
-More precisely, each twig blob $B_i$ is the image of some ball $M_r'$ as above, 
-and it is really the restriction to $M_r'$ that must lie in the subspace $U(M_r')$.
+More precisely, each twig blob $B_i$ is the image of some ball $M_l'$ as above, 
+and it is really the restriction to $M_l'$ that must lie in the subspace $U(M_l')$.
 \end{defn}
 \begin{figure}[t]\begin{equation*}
 \mathfig{.7}{definition/k-blobs}

--- a/text/intro.tex	Tue Feb 08 20:16:47 2011 +1100
+++ b/text/intro.tex	Tue Feb 08 07:14:18 2011 -0800
@@ -555,6 +555,7 @@
 and
 Alexander Kirillov
 for many interesting and useful conversations. 
+\nn{should add thanks to people from Teichner's reading course; Aaron Mazel-Gee, $\ldots$}
 During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. We'd like to thank the Aspen Center for Physics for the pleasant and productive 
 environment provided there during the final preparation of this manuscript.