%!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}; 

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 $\bc_k(X)$. 

In fact, on the first pass we will intentionally describe the definition in a misleadingly simple way, 

then explain the technical difficulties, and finally give a cumbersome but complete definition in 

Definition \ref{defn:blobs}. 

If (we don't recommend it) you want to keep track of the ways in which 

this initial description is misleading, or you're reading through a second time to understand the 

technical difficulties, keep note that later we will give precise meanings to ``a ball in $X$'', 

``nested'' and ``disjoint'', that are not quite the intuitive ones. 

Moreover some of the pieces 

into which we cut manifolds below are not themselves manifolds, and it requires special attention 

to define fields on these pieces.

We of course define $\bc_0(X) = \lf(X)$.

(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$ for each $c \in \lf(\bdy X)$.

We'll omit such boundary conditions from the notation 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 closed ball in $X$ (``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, we 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 $(B, u, r)$.

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 $u$).

\nn{it seems rather strange to make this a theorem}

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, c_2) \tensor \cC(X \setminus B_2; c_2)

	\right) .

\end{eqnarray*}

% __ (already said this above)

%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). 

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.

\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 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 $B_l$ 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.

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})