--- a/blob1.tex Tue Mar 03 16:40:59 2009 +0000
+++ b/blob1.tex Tue Mar 03 23:26:11 2009 +0000
@@ -585,7 +585,7 @@
\label{sec:blob-definition}
Let $X$ be an $n$-manifold.
-Assume a fixed system of fields.
+Assume a fixed system of fields and local relations.
In this section we will usually suppress boundary conditions on $X$ from the notation
(e.g. write $\lf(X)$ instead of $\lf(X; c)$).
@@ -593,36 +593,37 @@
submanifold of $X$, then $X \setmin Y$ implicitly means the closure
$\overline{X \setmin Y}$.
-We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case.
+We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$.
Define $\bc_0(X) = \lf(X)$.
(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$.
We'll omit this sort of detail in the rest of this section.)
In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$.
-$\bc_1(X)$ is the space of all local relations that can be imposed on $\bc_0(X)$.
-More specifically, define a 1-blob diagram to consist of
+$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$.
+Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear
+combinations of 1-blob diagrams, where a 1-blob diagram to consists of
\begin{itemize}
\item An embedded closed ball (``blob") $B \sub X$.
-%\nn{Does $B$ need a homeo to the standard $B^n$? I don't think so.
-%(See note in previous subsection.)}
-%\item A field (boundary condition) $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$.
\item A field $r \in \cC(X \setmin B; c)$
(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$).
\item A local relation field $u \in U(B; c)$
(same $c$ as previous bullet).
\end{itemize}
-%(Note that the field $c$ is determined (implicitly) as the boundary of $u$ and/or $r$,
-%so we will omit $c$ from the notation.)
-Define $\bc_1(X)$ to be the space of all finite linear combinations of
-1-blob diagrams, modulo the simple relations relating labels of 0-cells and
-also the label ($u$ above) of the blob.
-\nn{maybe spell this out in more detail}
-(See xxxx above.)
-\nn{maybe restate this in terms of direct sums of tensor products.}
+In order to get the linear structure correct, we (officially) define
+\[
+ \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) .
+\]
+The first direct sum is indexed by all blobs $B\subset X$, and the second
+by all boundary conditions $c \in \cC(\bd B)$.
+Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$.
-There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear
-combination of fields on $X$ obtained by gluing $r$ to $u$.
+Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by
+\[
+ (B, u, r) \mapsto u\bullet r,
+\]
+where $u\bullet r$ denotes the linear
+combination of fields on $X$ obtained by gluing $u$ to $r$.
In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by
just erasing the blob from the picture
(but keeping the blob label $u$).
@@ -630,23 +631,22 @@
Note that the skein space $A(X)$
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.
-$\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$.
+$\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the
+local relations encoded in $\bc_1(X)$.
More specifically, $\bc_2(X)$ is the space of all finite linear combinations of
-2-blob diagrams (defined below), modulo the usual linear label relations.
-\nn{and also modulo blob reordering relations?}
+2-blob diagrams, of which there are two types, disjoint and nested.
-\nn{maybe include longer discussion to motivate the two sorts of 2-blob diagrams}
-
-There are two types of 2-blob diagram: disjoint and nested.
A disjoint 2-blob diagram consists of
\begin{itemize}
-\item A pair of disjoint closed balls (blobs) $B_0, B_1 \sub X$.
-%\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$.
+\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors.
\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$
(where $c_i \in \cC(\bd B_i)$).
-\item Local relation fields $u_i \in U(B_i; c_i)$.
+\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$.
\end{itemize}
-Define $\bd(B_0, B_1, r, u_0, u_1) = (B_1, ru_0, u_1) - (B_0, ru_1, u_0) \in \bc_1(X)$.
+We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$;
+reversing the order of the blobs changes the sign.
+Define $\bd(B_0, B_1, u_0, u_1, r) =
+(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$.
In other words, the boundary of a disjoint 2-blob diagram
is the sum (with alternating signs)
of the two ways of erasing one of the blobs.
@@ -656,74 +656,86 @@
\begin{itemize}
\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$.
\item A field $r \in \cC(X \setmin B_0; c_0)$
-(for some $c_0 \in \cC(\bd B_0)$).
+(for some $c_0 \in \cC(\bd B_0)$), which is cuttable along $\bd B_1$.
+\item A local relation field $u_0 \in U(B_0; c_0)$.
+\end{itemize}
Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
(for some $c_1 \in \cC(B_1)$) and
$r' \in \cC(X \setmin B_1; c_1)$.
-\item A local relation field $u_0 \in U(B_0; c_0)$.
-\end{itemize}
-Define $\bd(B_0, B_1, r, u_0) = (B_1, r', r_1u_0) - (B_0, r, u_0)$.
-Note that xxxx above guarantees that $r_1u_0 \in U(B_1)$.
+Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$.
+Note that the requirement that
+local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$.
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating
sum of the two ways of erasing one of the blobs.
-If we erase the inner blob, the outer blob inherits the label $r_1u_0$.
+If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$.
+It is again easy to check that $\bd^2 = 0$.
+
+\nn{should draw figures for 1, 2 and $k$-blob diagrams}
+
+As with the 1-blob diagrams, in order to get the linear structure correct it is better to define
+(officially)
+\begin{eqnarray*}
+ \bc_2(X) & \deq &
+ \left(
+ \bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1}
+ U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1)
+ \right) \\
+ && \bigoplus \left(
+ \bigoplus_{B_0 \subset B_1} \bigoplus_{c_0}
+ U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0)
+ \right) .
+\end{eqnarray*}
+The final $\lf(X\setmin B_0; c_0)$ above really means fields cuttable along $\bd B_1$,
+but we didn't feel like introducing a notation for that.
+For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign
+(rather than a new, linearly independent 2-blob diagram).
Now for the general case.
A $k$-blob diagram consists of
\begin{itemize}
\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$.
-For each $i$ and $j$, we require that either $B_i \cap B_j$ is empty or
+For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or
$B_i \sub B_j$ or $B_j \sub B_i$.
(The case $B_i = B_j$ is allowed.
If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.)
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.)
-%$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$
-%if the latter space is not empty.
+\item 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.)
+$c_i$ 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 $c^t \in \cC(\bd B^t)$.
+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.
+$r$ is required to be cuttable along the boundaries of all blobs, twigs or not.
\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$,
where $c_j$ is the restriction of $c^t$ to $\bd B_j$.
If $B_i = B_j$ then $u_i = u_j$.
\end{itemize}
-We define $\bc_k(X)$ to be the vector space of all finite linear combinations
-of $k$-blob diagrams, modulo the linear label relations and
-blob reordering relations defined in the remainder of this paragraph.
-Let $x$ be a blob diagram with one undetermined $n$-morphism label.
-The unlabeled entity is either a blob or a 0-cell outside of the twig blobs.
-Let $a$ and $b$ be two possible $n$-morphism labels for
-the unlabeled blob or 0-cell.
-Let $c = \lambda a + b$.
-Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly.
-Then we impose the relation
-\eq{
- x_c = \lambda x_a + x_b .
-}
-\nn{should do this in terms of direct sums of tensor products}
-Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$
-of their blob labelings.
-Then we impose the relation
-\eq{
- x = \sign(\pi) x' .
-}
+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 $D_2$.
-(Alert readers will have noticed that for $k=2$ our definition
-of $\bc_k(X)$ is slightly different from the previous definition
-of $\bc_2(X)$ --- we did not impose the reordering relations.
-The general definition takes precedence;
-the earlier definition was simplified for purposes of exposition.)
+$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams.
+As before, the official definition is in terms of direct sums
+of tensor products:
+\[
+ \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}}
+ \left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) .
+\]
+Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above.
+$\overline{c}$ runs over all boundary conditions, again as described above.
+$j$ runs over all indices of twig blobs.
The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows.
-Let $b = (\{B_i\}, r, \{u_j\})$ be a $k$-blob diagram.
+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 $B_{j+1},\ldots,B_{k-1}$.
If $B_j$ is a twig blob, we have to assign new local relation labels
if removing $B_j$ creates new twig blobs.
-If $B_l$ becomes a twig after removing $B_j$, then set $u_l = r_lu_j$,
+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{
@@ -734,10 +746,7 @@
\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)}
-
-\nn{TO DO:
-expand definition to handle DGA and $A_\infty$ versions of $n$-categories;
-relations to Chas-Sullivan string stuff}
+\nn{?? remark about dendroidal sets}