--- a/text/blobdef.tex Fri Jul 23 08:14:27 2010 -0600
+++ b/text/blobdef.tex Fri Jul 23 13:52:30 2010 -0700
@@ -137,7 +137,7 @@
\medskip
-Roughly, $\bc_k(X)$ is generated by configurations of $k$ blobs, pairwise disjoint or nested.
+Roughly, $\bc_k(X)$ is generated by configurations of $k$ blobs, pairwise disjoint or nested, along with fields on all the components that the blobs divide $X$ into. Blobs which have no other blobs inside are called `twig blobs', and the fields on the twig blobs must be local relations.
The boundary is the alternating sum of erasing one of the blobs.
In order to describe this general case in full detail, we must give a more precise description of
which configurations of balls inside $X$ we permit.
@@ -151,7 +151,7 @@
(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
-a manifold.
+a manifold. \todo{example}
Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs.
@@ -165,20 +165,10 @@
\end{defn}
Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is splittable along it if it is the image of a field on $M_0$.
-By ``a ball in $X$'' we don't literally mean a submanifold homeomorphic to a ball, but rather the image of a map from the pair $(B^n, S^{n-1})$ into $X$, which is an embedding on the interior. The boundary of a ball in $X$ is the image of a locally embedded $n{-}1$-sphere. \todo{examples, e.g. balls which actually look like an annulus, but we remember the boundary}
-\nn{not all balls in $X$ can arise via gluing, but I suppose that's OK.}
-
-\nn{do we need this next def?}
-\begin{defn}
-\label{defn:ball-decomposition}
-A \emph{ball decomposition} of an $n$-manifold $X$ is a collection of balls in $X$, such that there exists some gluing decomposition $M_0 \to \cdots \to M_m = X$ so that the balls are the images of the components of $M_0$ in $X$.
-\end{defn}
-In particular, the union of all the balls in a ball decomposition comprises all of $X$. \todo{example}
-
We'll now slightly restrict the possible configurations of blobs.
\begin{defn}
\label{defn:configuration}
-A configuration of $k$ blobs in $X$ is an ordered collection of $k$ balls in $X$ such that there is some gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and each of the balls is the image of some connected component of one of the $M_k$. Such a gluing decomposition is \emph{compatible} with the 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 for each subset $B_i$ there is some $0 \leq r \leq m$ and some connected component $M_r'$ of $M_r$ which is a ball, so $B_i$ is the image of $M_r'$ in $X$. Such a gluing decomposition is \emph{compatible} with the configuration. A blob $B_i$ is a twig blob if no other blob $B_j$ maps into the appropriate $M_r'$. \nn{that's a really clumsy way to say it, but I struggled to say it nicely and still allow boundaries to intersect -S}
\end{defn}
In particular, this implies what we said about blobs above:
that for any two blobs in a configuration of blobs in $X$,
@@ -186,114 +176,41 @@
We describe these as disjoint blobs and nested blobs.
Note that nested blobs may have boundaries that overlap, or indeed coincide.
Blobs may meet the boundary of $X$.
+Further, note that blobs need not actually be embedded balls in $X$, since parts of the boundary of the ball $M_r'$ may have been glued together.
-% (already said above)
-%Note that the boundaries of a configuration of $k$ blobs may cut up the manifold $X$ into components which are not themselves manifolds. \todo{example: the components between the boundaries of the balls may be pathological}
+\todo{Say something reassuring: that 'most of the time' all the regions are manifolds anyway, and you can take the `trivial' gluing decomposition}
In the informal description above, in the definition of a $k$-blob diagram we asked for 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; this is unsatisfactory because that complement need not be a manifold. Thus, the official definition is
\begin{defn}
\label{defn:blob-diagram}
A $k$-blob diagram on $X$ consists of
\begin{itemize}
-\item a configuration of $k$ blobs in $X$,
+\item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$,
\item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration,
\end{itemize}
such that
the restriction of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$.
\end{defn}
+\todo{Careful here: twig blobs aren't necessarily balls?}
+(See Figure \ref{blobkdiagram}. \todo{update diagram})
+\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 of 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 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 ball from the configuration, preserving the field $r$.
+The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum of 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 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 $r$:
+\begin{equation*}
+\bdy(\{B_1, \ldots B_k\}, r) = \sum_{i=1}^{k} (-1)^{i+1} (\{B_1, \ldots, \widehat{B_i}, \ldots, B_k\}, r)
+\end{equation*}
\end{defn}
We readily see that if a gluing decomposition is compatible with some configuration of blobs, then it is also compatible with any configuration obtained by forgetting some blobs, ensuring that the differential in fact lands in the space of $k{-}1$-blob diagrams.
A slight compensation to the complication of the official definition arising from attention to splitting is that the differential now just preserves the entire field $r$ without having to say anything about gluing together fields on smaller components.
-
-
-
-
-
-
-\nn{should merge this informal def with official one above}
-
-Before describing the general case, note that when we say blobs are disjoint,
-we will only mean that their interiors are disjoint.
-Nested blobs may have boundaries that overlap, or indeed may coincide.
-A $k$-blob diagram consists of
-\begin{itemize}
-\item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$.
-For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or
-$B_i \sub B_j$ or $B_j \sub B_i$.
-If a blob has no other blobs strictly contained in it, we call it a twig blob.
-\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$.
-(These are implied by the data in the next bullets, so we usually
-suppress them from the notation.)
-The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$
-if the latter space is not empty.
-\item A field $r \in \cC(X \setmin B^t; c^t)$,
-where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$
-is determined by the $c_i$'s.
-The field $r$ is required to be splittable along the boundaries of all blobs, twigs or not.
-(This is equivalent to asking for a field on of the components of $X \setmin B^t$.)
-\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$.
-If $B_i = B_j$ then $u_i = u_j$.
-\end{itemize}
-(See Figure \ref{blobkdiagram}.)
-\begin{figure}[t]\begin{equation*}
-\mathfig{.7}{definition/k-blobs}
-\end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure}
-
-If two blob diagrams $D_1$ and $D_2$
-differ only by a reordering of the blobs, then we identify
-$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$.
-
-Roughly, then, $\bc_k(X)$ is all finite linear combinations of $k$-blob diagrams.
-As before, the official definition is in terms of direct sums
-of tensor products:
-\[
- \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
- \left( \bigotimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
-\]
-Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
-The index $\overline{c}$ runs over all boundary conditions,
-again as described above and $j$ runs over all indices of twig blobs.
-The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are
-splittable along all of the blobs in $\overline{B}$.
-
-The boundary map
-\[
- \bd : \bc_k(X) \to \bc_{k-1}(X)
-\]
-is defined as follows.
-Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram.
-Let $E_j(b)$ denote the result of erasing the $j$-th blob.
-If $B_j$ is not a twig blob, this involves only decrementing
-the indices of blobs $B_{j+1},\ldots,B_{k}$.
-If $B_j$ is a twig blob, we have to assign new local relation labels
-if removing $B_j$ creates new twig blobs.
-\todo{Have to say what happens when no new twig blobs are created}
-\nn{KW: I'm confused --- why isn't it OK as written?}
-If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$,
-where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$.
-Finally, define
-\eq{
- \bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b).
-}
-The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel.
-Thus we have a chain complex.
-
Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms,
is immediately obvious from the definition.
A homeomorphism acts in an obvious way on blobs and on fields.
-
-\nn{end relocated informal def}
-
-
-
-
-
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),