%!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 (enriched over Vect) and local relations.

(If $\cC$ is not enriched over Vect, 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. write $\lf(X)$ instead of $\lf(X; c)$).

We want to replace the quotient

\[

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

\]

of the previous section 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 $\bc_k(X)$.

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 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}.)

\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 the skein space $A(X)$

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

This is Theorem \ref{thm:skein-modules}, and also used in 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 $\bc_1(X)$'.

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

Note that the requirement that

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

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 $\bd^2 = 0$.

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 $B_0$ and $B_1$ 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}

}

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

disjoint and nested blobs.

Disjoint will mean disjoint interiors.

Nested blobs are allowed to coincide, or to have overlapping boundaries.

Blob are allowed to intersect $\bd X$.

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 $\bd X$.}

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.

\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 $D_2$.

$\bc_k(X)$ 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.

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 $B_{j+1},\ldots,B_{k-1}$.

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 $u_l = u_j\bullet r_l$,

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 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 thing 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 $e$.)