%!TEX root = ../blob1.tex

\section{The blob complex}

\label{sec:blob-definition}

Let $X$ be an $n$-manifold.

Let $\cC$ be a fixed system of fields and local relations.

We'll assume it is enriched over \textbf{Vect}, and if it is not we can make it so by allowing finite

linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$.

In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\lf(X)$ instead of $\lf(X; c)$.

We want to replace the quotient

\[

	A(X) \deq \lf(X) / U(X)

\]

of Definition \ref{defn:TQFT-invariant} with a resolution

\[

	\cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) .

\]

We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$.   \todo{create a numbered definition for the general case}

We of course 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 vector space of all fields on $X$.

We want the vector space $\bc_1(X)$ to capture `the space of all local relations that can be imposed on $\bc_0(X)$'.

Thus we say  a $1$-blob diagram consists of:

\begin{itemize}

\item An embedded closed ball (``blob") $B \sub X$.

\item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$.

\item A field $r \in \cC(X \setmin B; c)$.

\item A local relation field $u \in U(B; c)$.

\end{itemize}

(See Figure \ref{blob1diagram}.) Since $c$ is implicitly determined by $u$ or $r$, we usually omit it from the notation.

\begin{figure}[t]\begin{equation*}

\mathfig{.6}{definition/single-blob}

\end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure}

In order to get the linear structure correct, the actual definition is

\[

	\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 $(B, u, r)$.

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 field 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 $u$).

Note that directly from the definition we have

\begin{thm}

\label{thm:skein-modules}

The skein module $A(X)$ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.

\end{thm}

This also establishes the second 

half of Property \ref{property:contractibility}.

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 disjoint 2-blob diagram consists of

\begin{itemize}

\item A pair of closed balls (blobs) $B_1, B_2 \sub X$ with disjoint interiors.

\item A field $r \in \cC(X \setmin (B_1 \cup B_2); c_1, c_2)$

(where $c_i \in \cC(\bd B_i)$).

\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$.

\end{itemize}

(See Figure \ref{blob2ddiagram}.)

\begin{figure}[t]\begin{equation*}

\mathfig{.6}{definition/disjoint-blobs}

\end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure}

We also identify $(B_1, B_2, u_1, u_2, r)$ with $-(B_2, B_1, u_2, u_1, r)$;

reversing the order of the blobs changes the sign.

Define $\bd(B_1, B_2, u_1, u_2, r) = 

(B_2, u_2, u_1\bullet r) - (B_1, u_1, u_2\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.

It's easy to check that $\bd^2 = 0$.

A nested 2-blob diagram consists of

\begin{itemize}

\item A pair of nested balls (blobs) $B_1 \subseteq B_2 \subseteq X$.

\item A field $r' \in \cC(B_2 \setminus B_1; c_1, c_2)$ 

(for some $c_1 \in \cC(\bdy B_1)$ and $c_2 \in \cC(\bdy B_2)$).

\item A field $r \in \cC(X \setminus B_2; c_2)$.

\item A local relation field $u \in U(B_1; c_1)$.

\end{itemize}

(See Figure \ref{blob2ndiagram}.)

\begin{figure}[t]\begin{equation*}

\mathfig{.6}{definition/nested-blobs}

\end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure}

Define $\bd(B_1, B_2, u, r', r) = (B_2, u\bullet r', r) - (B_1, u, r' \bullet r)$.

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.

When  we erase the inner blob, the outer blob inherits the label $u\bullet r'$.

It is again easy to check that $\bd^2 = 0$. Note that the requirement that

local relations are an ideal with respect to gluing guarantees that $u\bullet r' \in U(B_2)$.

As with the $1$-blob diagrams, in order to get the linear structure correct the actual definition is 

\begin{eqnarray*}

	\bc_2(X) & \deq &

	\left( 

		\bigoplus_{B_1, B_2\; \text{disjoint}} \bigoplus_{c_1, c_2}

			U(B_1; c_1) \otimes U(B_2; c_2) \otimes \lf(X\setmin (B_1\cup B_2); c_1, c_2)

	\right)  \bigoplus \\

	&& \quad\quad  \left( 

		\bigoplus_{B_1 \subset B_2} \bigoplus_{c_1, c_2}

			U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2)

	\right) .

\end{eqnarray*}

For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign

(rather than a new, linearly independent, 2-blob diagram). 

\noop{

\nn{Hmm, I think we should be doing this for nested blobs too -- 

we shouldn't force the linear indexing of the blobs to have anything to do with 

the partial ordering by inclusion -- this is what happens below}

\nn{KW: I think adding that detail would only add distracting clutter, and the statement is true as written (in the sense that it yields a vector space isomorphic to the general def below}

}

\begin{defn}

An \emph{$n$-ball decomposition} of a topological space $X$ is 

finite collection of triples $\{(B_i, X_i, Y_i)\}_{i=1,\ldots, k}$ where $B_i$ is an $n$-ball, $X_i$ is some topological space, and $Y_i$ is pair of disjoint homeomorphic $n-1$-manifolds in the boundary of $X_{i-1}$ (for convenience, $X_0 = Y_1 = \eset$), such that $X_{i+1} = X_i \cup_{Y_i} B_i$ and $X = X_k \cup_{Y_k} B_k$.

Equivalently, we can define a ball decomposition inductively. A ball decomposition of $X$ is a topological space $X'$ along with a pair of disjoint homeomorphic $n-1$-manifolds $Y \subset \bdy X$, so $X = X' \bigcup_Y \selfarrow$, and $X'$ is either a disjoint union of balls, or a topological space equipped with a ball decomposition.

\end{defn}

Before describing the general case we should say more precisely what we mean by 

disjoint and nested blobs.

However, we require of any collection of blobs $B_1,\ldots,B_k \subseteq X$ that

$X$ is decomposable along the union of the boundaries of the blobs.

\nn{need to say more here.  we want to be able to glue blob diagrams, but avoid pathological

behavior}

\nn{need to allow the case where $B\to X$ is not an embedding

on $\bd B$.  this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$

and blobs are allowed to meet $\bd X$.

Also, the complement of the blobs (and regions between nested blobs) might not be manifolds.}

Now for the general case.

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 $B_j \sub B_i$.

(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.)

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.

$r$ 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 $u_j \in U(B_j; c_j)$,

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}

(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}

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.

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),

we define $\supp(y) \deq \bigcup_i \supp(b_i)$.

We note that blob diagrams in $X$ have a structure similar to that of a simplicial set,

but with simplices replaced by a more general class of combinatorial shapes.

Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products

and cones, and which contains the point.

We can associate an element $p(b)$ of $P$ to each blob diagram $b$ 

(equivalently, to each rooted tree) according to the following rules:

\begin{itemize}

\item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree;

\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).

\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.

(This correspondence works best if we think of each twig label $u_i$ as having the form

$x - s(e(x))$, where $x$ is an arbitrary field on $B_i$, $e: \cC(B_i) \to C$ is the evaluation map, 

and $s:C \to \cC(B_i)$ is some fixed section of $e$. \todo{This parenthetical remark mysteriously specialises to the category case})