--- a/blob to-do Mon Dec 12 19:26:44 2011 -0800
+++ b/blob to-do Mon Dec 12 19:26:57 2011 -0800
@@ -1,16 +1,13 @@
====== big ======
-* add "homeomorphism" spiel befure the first use of "homeomorphism in the intro
-* maybe also additional homeo warnings in other sections
-
-* Lemma 3.2.3 (\ref{support-shrink}) implicitly assumes embedded (non-self-intersecting) blobs. this can be fixed, of course, but it makes the argument more difficult to understand
-
-* Maybe give more details in 6.7.2
+[nothing!]
====== minor/optional ======
+* Maybe give more details in 6.7.2. Or maybe do this in some future paper.
+
[probably NO] * consider proving the gluing formula for higher codimension manifolds with
morita equivalence
--- a/text/basic_properties.tex Mon Dec 12 19:26:44 2011 -0800
+++ b/text/basic_properties.tex Mon Dec 12 19:26:57 2011 -0800
@@ -74,6 +74,9 @@
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 $\supp(y) \deq \bigcup_i \supp(b_i)$.
+%%%%% the following is true in spirit, but technically incorrect if blobs are not embedded;
+%%%%% we only use this once, so move lemma and proof to Hochschild section
+\noop{ %%%%%%%%%% begin \noop
For future use we prove the following lemma.
\begin{lemma} \label{support-shrink}
@@ -94,6 +97,7 @@
Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models, \S\ref{sec:moam}),
so $f$ and the identity map are homotopic.
\end{proof}
+} %%%%%%%%%%%%% end \noop
For the next proposition we will temporarily restore $n$-manifold boundary
conditions to the notation. Let $X$ be an $n$-manifold, with $\bd X = Y \cup Y \cup Z$.
--- a/text/hochschild.tex Mon Dec 12 19:26:44 2011 -0800
+++ b/text/hochschild.tex Mon Dec 12 19:26:57 2011 -0800
@@ -218,7 +218,10 @@
to distance $\ep$ from *.
(Move right or left so as to shrink the blob.)
Extend to get a chain map $f: F_*^\ep \to F_*^\ep$.
-By Lemma \ref{support-shrink}, $f$ is homotopic to the identity.
+By Corollary \ref{disj-union-contract},
+$f$ is homotopic to the identity.
+(Use the facts that $f$ factors though a map from a disjoint union of balls
+into $S^1$, and that $f$ is the identity in degree 0.)
Since the image of $f$ is in $J_*$, and since any blob chain is in $F_*^\ep$
for $\ep$ sufficiently small, we have that $J_*$ is homotopic to all of $\bc_*(S^1)$.
--- a/text/intro.tex Mon Dec 12 19:26:44 2011 -0800
+++ b/text/intro.tex Mon Dec 12 19:26:57 2011 -0800
@@ -43,6 +43,16 @@
with sufficient limits and colimits would do.
We could also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories).
+{\bf Note:} For simplicity, we will assume that all manifolds are unoriented and piecewise linear, unless stated otherwise.
+In fact, all the results in this paper also hold for smooth manifolds,
+as well as manifolds equipped with an orientation, spin structure, or $\mathrm{Pin}_\pm$ structure.
+We will use ``homeomorphism" as a shorthand for ``piecewise linear homeomorphism".
+The reader could also interpret ``homeomorphism" to mean an isomorphism in whatever category of manifolds we happen to
+be working in (e.g.\ spin piecewise linear, oriented smooth, etc.).
+In the smooth case there are additional technical details concerning corners and gluing
+which we have omitted, since
+most of the examples we are interested in require only a piecewise linear structure.
+
\subsection{Structure of the paper}
The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}),