
diff -r d31a9c505f29 -r 77b0cdeb0fcd text/basic_properties.tex
--- a/text/basic_properties.tex	Tue Mar 16 14:11:07 2010 +0000
+++ b/text/basic_properties.tex	Thu Mar 18 19:40:46 2010 +0000
@@ -51,6 +51,40 @@
 For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$,
 where $(c', c'')$ is some (any) splitting of $c$ into domain and range.
 
+\begin{cor} \label{disj-union-contract}
+If $X$ is a disjoint union of $n$-balls, then $\bc_*(X; c)$ is contractible.
+\end{cor}
+
+\begin{proof}
+This follows from \ref{disjunion} and \ref{bcontract}.
+\end{proof}
+
+Define the {\it support} of a blob diagram to be the union of all the 
+blobs of the diagram.
+Define the support of a linear combination of blob diagrams to be the union of the 
+supports of the constituent diagrams.
+For future use we prove the following lemma.
+
+\begin{lemma} \label{support-shrink}
+Let $L_* \sub \bc_*(X)$ be a subcomplex generated by some
+subset of the blob diagrams on $X$, and let $f: L_* \to L_*$
+be a chain map which does not increase supports and which induces an isomorphism on
+$H_0(L_*)$.
+Then $f$ is homotopic (in $\bc_*(X)$) to the identity $L_*\to L_*$.
+\end{lemma}
+
+\begin{proof}
+We will use the method of acyclic models.
+Let $b$ be a blob diagram of $L_*$, let $S\sub X$ be the support of $b$, and let
+$r$ be the restriction of $b$ to $X\setminus S$.
+Note that $S$ is a disjoint union of balls.
+Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$.
+note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$.
+Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), 
+so $f$ and the identity map are homotopic.
+\end{proof}
+
+
 \medskip
 
 \nn{Maybe there is no longer a need to repeat the next couple of props here, since we also state them in the introduction.




