--- a/blob to-do Thu Dec 08 09:45:16 2011 -0800
+++ b/blob to-do Thu Dec 08 12:06:43 2011 -0800
@@ -1,6 +1,10 @@
====== big ======
+* add "homeomorphism" spiel befure the first use of "homeomorphism in the intro
+* maybe also additional homeo warnings in other sections
+
+
====== minor/optional ======
[probably NO] * consider proving the gluing formula for higher codimension manifolds with
@@ -13,7 +17,6 @@
* figures
(** 13 "combining two balls" is lame) (but maybe leave it as is -- KW)
-** figures for email thread with Mike Schulman??
* better discussion of systems of fields from disk-like n-cats
--- a/text/blobdef.tex Thu Dec 08 09:45:16 2011 -0800
+++ b/text/blobdef.tex Thu Dec 08 12:06:43 2011 -0800
@@ -80,7 +80,7 @@
Next, we want the vector space $\bc_2(X)$ to capture ``the space of all relations
(redundancies, syzygies) among the
local relations encoded in $\bc_1(X)$''.
-A $2$-blob diagram, comes in one of two types, disjoint and nested.
+A $2$-blob diagram comes in one of two types, disjoint and nested.
A disjoint 2-blob diagram consists of
\begin{itemize}
\item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors.
@@ -190,7 +190,7 @@
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 fields $a_i \in \cF(M_0^i)$, $0\le i\le k$.
-Note that if $a$ is splittable in the sense then it makes sense to talk about the restriction of $a$ of any
+Note that if $a$ is splittable in this sense then it makes sense to talk about the restriction of $a$ to any
component $M'_j$ of any $M_j$ of the decomposition.
In the example above, note that
@@ -209,9 +209,10 @@
\begin{defn}
\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
+of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$
+with the property that
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 $X$.
+$M_l$ which is a ball, such that $B_i$ 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.
@@ -229,10 +230,10 @@
if the boundaries of all the blobs cut $X$ into pieces which are all manifolds,
we can just take $M_0$ to be these pieces, and $M_1 = X$.
-In the informal description above, in the definition of a $k$-blob diagram we asked for any
+In the initial informal definition of a $k$-blob diagram above, we allowed any
collection of $k$ balls which were pairwise disjoint or nested.
-We now further insist that the balls are a configuration in the sense of Definition \ref{defn:configuration}.
-Also, we asked for a local relation on each twig blob, and a field on the complement of the twig blobs;
+We now further require that the balls are a configuration in the sense of Definition \ref{defn:configuration}.
+We also specified a local relation on each twig blob, and a field on the complement of the twig blobs;
this is unsatisfactory because that complement need not be a manifold. Thus, the official definitions are
\begin{defn}
\label{defn:blob-diagram}
@@ -251,7 +252,7 @@
\begin{figure}[t]\begin{equation*}
\mathfig{.7}{definition/k-blobs}
\end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure}
-and
+
\begin{defn}
\label{defn:blobs}
The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum over all
@@ -287,7 +288,8 @@
\item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union
of two blob diagrams (equivalently, join two trees at the roots); and
\item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which
-encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root).
+encloses all the others (equivalently, add a new edge to the root, with the new vertex becoming the root
+of the new tree).
\end{itemize}
For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while
a diagram of $k$ disjoint blobs corresponds to a $k$-cube.