
diff -r 2313b05f4906 -r ef503460486d text/blobdef.tex
--- a/text/blobdef.tex	Tue Jan 25 14:57:07 2011 -0800
+++ b/text/blobdef.tex	Sun Feb 06 20:53:43 2011 -0800
@@ -33,9 +33,11 @@
 to define fields on these pieces.
 
 We of course define $\bc_0(X) = \cF(X)$.
-(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cF(X; c)$ for each $c \in \cF(\bdy X)$.
+In other words, $\bc_0(X)$ is just the vector space of all fields on $X$.
+
+(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cF(X; c)$ for $c \in \cF(\bdy X)$.
+The blob complex $\bc_*(X; c)$ will depend on a fixed boundary condition $c\in \cF(\bdy X)$.
 We'll omit such boundary conditions from the notation in the rest of this section.)
-In other words, $\bc_0(X)$ is just the vector space of all fields on $X$.
 
 We want the vector space $\bc_1(X)$ to capture 
 ``the space of all local relations that can be imposed on $\bc_0(X)$".
@@ -148,8 +150,8 @@
 \item For any (possibly empty) configuration of blobs on an $n$-ball $D$, we can add
 $D$ itself as an outermost blob.
 (This is used in the proof of Proposition \ref{bcontract}.)
-\item If $X'$ is obtained from $X$ by gluing, then any permissible configuration of blobs
-on $X$ gives rise to a permissible configuration on $X'$.
+\item If $X\sgl$ is obtained from $X$ by gluing, then any permissible configuration of blobs
+on $X$ gives rise to a permissible configuration on $X\sgl$.
 (This is necessary for Proposition \ref{blob-gluing}.)
 \end{itemize}
 Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not
@@ -166,8 +168,8 @@
 \end{align*}
 Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$. 
 Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, 
-and $\{C\}$ is a valid configuration of blobs in $C \cup D$, 
-so we must allow $\{A, C\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. 
+and $\{D\}$ is a valid configuration of blobs in $C \cup D$, 
+so we must allow $\{A, D\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. 
 Note however that the complement is not a manifold.
 \end{example}
 
@@ -244,7 +246,7 @@
 \label{defn:blobs}
 The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum over all 
 configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, 
-modulo identifying the vector spaces for configurations that only differ by a permutation of the balls 
+modulo identifying the vector spaces for configurations that only differ by a permutation of the blobs 
 by the sign of that permutation. 
 The differential $\bc_k(X) \to \bc_{k-1}(X)$ is, as above, the signed sum of ways of 
 forgetting one blob from the configuration, preserving the field $r$:
@@ -263,11 +265,6 @@
 is immediately obvious from the definition.
 A homeomorphism acts in an obvious way on blobs and on fields.
 
-We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, 
-to be the union of the blobs of $b$.
-For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram),
-we define $\supp(y) \deq \bigcup_i \supp(b_i)$.
-
 \begin{remark} \label{blobsset-remark} \rm
 We note that blob diagrams in $X$ have a structure similar to that of a simplicial set,
 but with simplices replaced by a more general class of combinatorial shapes.




