--- a/text/basic_properties.tex	Thu Mar 18 19:40:46 2010 +0000
+++ b/text/basic_properties.tex	Sat Mar 27 03:07:45 2010 +0000
@@ -3,10 +3,11 @@
 \section{Basic properties of the blob complex}
 \label{sec:basic-properties}
 
-\begin{prop} \label{disjunion}
-There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.
-\end{prop}
-\begin{proof}
+In this section we complete the proofs of Properties 1-4. Throughout the paper, where possible, we prove results using Properties 1-4, rather than the actual definition of blob homology. This allows the possibility of future improvements to or alternatives on our definition. In fact, we hope that there may be a characterisation of blob homology in terms of Properties 1-4, but at this point we are unaware of one.
+
+Recall Property \ref{property:disjoint-union}, that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.
+
+\begin{proof}[Proof of Property \ref{property:disjoint-union}]
 Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them
 (putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a
 blob diagram $(b_1, b_2)$ on $X \du Y$.
@@ -14,10 +15,8 @@
 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign)
 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines
 a pair of blob diagrams on $X$ and $Y$.
-These two maps are compatible with our sign conventions.
+These two maps are compatible with our sign conventions.  (We follow the usual convention for tensors products of complexes, as in e.g. \cite{MR1438306}: $d(a \tensor b) = da \tensor b + (-1)^{\deg(a)} a \tensor db$.)
 The two maps are inverses of each other.
-\nn{should probably say something about sign conventions for the differential
-in a tensor product of chain complexes; ask Scott}
 \end{proof}
 
 For the next proposition we will temporarily restore $n$-manifold boundary
@@ -44,8 +43,8 @@
 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to
 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$.
 \end{proof}
-
-Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy
+This proves Property \ref{property:contractibility} (the second half of the statement of this Property was immediate from the definitions).
+Note that even when there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy
 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.
 
 For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$,
@@ -56,7 +55,7 @@
 \end{cor}
 
 \begin{proof}
-This follows from \ref{disjunion} and \ref{bcontract}.
+This follows from Properties \ref{property:disjoint-union} and \ref{property:contractibility}.
 \end{proof}
 
 Define the {\it support} of a blob diagram to be the union of all the 
@@ -84,37 +83,6 @@
 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.
-But I think it's worth saying that the Diff actions will be enhanced later.
-Maybe put that in the intro too.}
-
-As we noted above,
-\begin{prop}
-There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$.
-\qed
-\end{prop}
-
-
-\begin{prop}
-For fixed fields ($n$-cat), $\bc_*$ is a functor from the category
-of $n$-manifolds and homeomorphisms to the category of chain complexes and
-(chain map) isomorphisms.
-\qed
-\end{prop}
-
-In particular,
-\begin{prop}  \label{diff0prop}
-There is an action of $\Homeo(X)$ on $\bc_*(X)$.
-\qed
-\end{prop}
-
-The above will be greatly strengthened in Section \ref{sec:evaluation}.
-
-\medskip
-
 For the next proposition we will temporarily restore $n$-manifold boundary
 conditions to the notation.
 
@@ -124,9 +92,9 @@
 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$,
 we have the blob complex $\bc_*(X; a, b, c)$.
 If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on
-$X$ to get blob diagrams on $X\sgl$:
+$X$ to get blob diagrams on $X\sgl$. This proves Property \ref{property:gluing-map}, which we restate here in more detail.
 
-\begin{prop}
+\textbf{Property \ref{property:gluing-map}.}\emph{
 There is a natural chain map
 \eq{
     \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl).
@@ -134,22 +102,7 @@
 The sum is over all fields $a$ on $Y$ compatible at their
 ($n{-}2$-dimensional) boundaries with $c$.
 `Natural' means natural with respect to the actions of diffeomorphisms.
-\qed
-\end{prop}
-
-The above map is very far from being an isomorphism, even on homology.
-This will be fixed in Section \ref{sec:gluing} below.
-
-%\nn{Next para not needed, since we already use bullet = gluing notation above(?)}
+}
 
-%An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$
-%and $X\sgl = X_1 \cup_Y X_2$.
-%(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.)
-%For $x_i \in \bc_*(X_i)$, we introduce the notation
-%\eq{
-%    x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .
-%}
-%Note that we have resumed our habit of omitting boundary labels from the notation.
-
-
-
+This map is very far from being an isomorphism, even on homology.
+We fix this deficit in Section \ref{sec:gluing} below.