% test edit #3

%%%%% excerpts from my include file of standard macros

\def\z{\mathbb{Z}}

\def\r{\mathbb{R}}

\def\c{\mathbb{C}}

\def\t{\mathbb{T}}

\def\du{\sqcup}

\def\bd{\partial}

\def\sub{\subset}

\def\sup{\supset}

%\def\setmin{\smallsetminus}

\def\setmin{\setminus}

\def\ep{\epsilon}

\def\sgl{_\mathrm{gl}}

\def\op{^\mathrm{op}}

\def\deq{\stackrel{\mathrm{def}}{=}}

\def\pd#1#2{\frac{\partial #1}{\partial #2}}

\def\nn#1{{{\it \small [#1]}}}

% equations

\newcommand{\eq}[1]{\begin{displaymath}#1\end{displaymath}}

\newcommand{\eqar}[1]{\begin{eqnarray*}#1\end{eqnarray*}}

\newcommand{\eqspl}[1]{\begin{displaymath}\begin{split}#1\end{split}\end{displaymath}}

% tricky way to iterate macros over a list

\def\semicolon{;}

\def\applytolist#1{

% \def\cA{{\cal A}} for A..Z

\applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM;

% \DeclareMathOperator{\pr}{pr} etc.

\def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}

%%%%%% end excerpt

\title{Blob Homology}

\begin{document}

\makeatletter

\@addtoreset{equation}{section}

\gdef\theequation{\thesection.\arabic{equation}}

\makeatother

\maketitle

\section{Introduction}

(motivation, summary/outline, etc.)

(1) restore exactness in pictures-mod-relations;

(1') add relations-amongst-relations etc. to pictures-mod-relations;

just go from coend to derived coend (e.g. Hochschild homology));

(3) ...

)

\section{Definitions}

\subsection{Fields}

Fix a top dimension $n$.

\nn{maybe should look for better name; but this is the name I use elsewhere}

is a collection of functors $\cC$ from manifolds of dimension $n$ or less

to sets.

These functors must satisfy various properties (see KW TQFT notes for details).

there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$;

there is a restriction map $\cC(X) \to \cC(\bd X)$;

gluing manifolds corresponds to fibered products of fields;

$c\times I \in \cC(Y\times I)$; ...

\nn{should eventually include full details of definition of fields.}

between fields and their (orientation-reversal) duals}

\nn{remark that if top dimensional fields are not already linear

then we will soon linearize them(?)}

the relevant properties of the following two examples of fields.

The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$

all maps from $X$ to $B$.

The second example will take longer to explain.

we can construct a system of fields as follows.

Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$

with codimension $i$ cells labeled by $i$-morphisms of $C$.

We'll spell this out for $n=1,2$ and then describe the general case.

If $X$ has boundary, we require that the cell decompositions are in general

position with respect to the boundary --- the boundary intersects each cell

transversely, so cells meeting the boundary are mere half-cells.

Put another way, the cell decompositions we consider are dual to standard cell

decompositions of $X$.

We will always assume that our $n$-categories have linear $n$-morphisms.

For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with

an object (0-morphism) of the 1-category $C$.

A field on a 1-manifold $S$ consists of

\begin{itemize}

of points in the interior of $S$);

by an object (0-morphism) of $C$;

``domain" and ``range" for the two adjacent 1-cells; and

domain and range determined by the transverse orientation and the labelings of the 1-cells.

\end{itemize}

If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels

interior of $S$, each transversely oriented and each labeled by an element (1-morphism)

of the algebra.

\medskip

For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories)

that are common in the literature.

We describe these carefully here.

A field on a 0-manifold $P$ is a labeling of each point of $P$ with

an object of the 2-category $C$.

A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$.

A field on a 2-manifold $Y$ consists of

\begin{itemize}

that each component of the complement is homeomorphic to a disk);

by a 0-morphism of $C$;

``domain" and ``range" for the two adjacent 2-cells;

and the labelings of the 2-cells;

of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped

to $\pm 1 \in S^1$; and

determined by the labelings of the 1-cells and the parameterizations of the previous

bullet.

\end{itemize}

\nn{need to say this better; don't try to fit everything into the bulleted list}

For general $n$, a field on a $k$-manifold $X^k$ consists of

\begin{itemize}

to the boundary of the standard $(k-j)$-dimensional bihedron; and

domain and range determined by the labelings of the link of $j$-cell.

\end{itemize}

%\nn{next definition might need some work; I think linearity relations should

%be treated differently (segregated) from other local relations, but I'm not sure

%the next definition is the best way to do it}

\medskip

in the linearized space of fields.

By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is

linear combinations of fields on $X$.

If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$.

Thus the restriction (to boundary) maps are well defined because we never

take linear combinations of fields with differing boundary conditions.

In some cases we don't linearize the default way; instead we take the

spaces $\cC_l(X; a)$ to be part of the data for the system of fields.

In particular, for fields based on linear $n$-category pictures we linearize as follows.

obvious relations on 0-cell labels.

and let $p$ be a 0-cell of $L$.

Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that

$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.

Then the subspace $K$ is generated by things of the form

$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader

to infer the meaning of $\alpha_{\lambda c + d}$.

Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms.

will do something similar below; in general, whenever a label lives in a linear

product of all the linear label spaces?  Yes:}

For top dimensional ($n$-dimensional) manifolds, we linearize as follows.

Define an ``almost-field" to be a field without labels on the 0-cells.

(Recall that 0-cells are labeled by $n$-morphisms.)

To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism

space determined by the labeling of the link of the 0-cell.

(If the 0-cell were labeled, the label would live in this space.)

We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).

above tensor products.

\subsection{Local relations}

Let $B^n$ denote the standard $n$-ball.

(for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties.

(2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and

See KW TQFT notes for details.  Need to transfer details to here.}

For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$,

where $a$ and $b$ are maps (fields) which are homotopic rel boundary.

For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map

$\cC_l(B^n; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into

domain and range.

\nn{maybe examples of local relations before general def?}

Note that the $Y$ is an $n$-manifold which is merely homeomorphic to the standard $B^n$,

then any homeomorphism $B^n \to Y$ induces the same local subspaces for $Y$.

We'll denote these by $U(Y; c) \sub \cC_l(Y; c)$, $c \in \cC(\bd Y)$.

\nn{Is this true in high (smooth) dimensions?  Self-diffeomorphisms of $B^n$

rel boundary might not be isotopic to the identity.  OK for PL and TOP?}

Given a system of fields and local relations, we define the skein space

$A(Y^n; c)$ to be the space of all finite linear combinations of fields on

the $n$-manifold $Y$ modulo local relations.

The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations

is defined to be the dual of $A(Y; c)$.

(See KW TQFT notes or xxxx for details.)

The blob complex is in some sense the derived version of $A(Y; c)$.

\subsection{The blob complex}

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

Assume a fixed system of fields.

In this section we will usually suppress boundary conditions on $X$ from the notation

(e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$).

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.

Define $\bc_0(X) = \cC_l(X)$.

(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cC_l(X; c)$.

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

\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.}

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 $u$.

just erasing the blob from the picture

(but keeping the blob label $u$).

Note that the skein space $A(X)$

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

$\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$.

2-blob diagrams (defined below), modulo the usual linear label relations.

\nn{and also modulo blob reordering relations?}

\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 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 $u_i \in U(B_i; c_i)$.

\end{itemize}

Define $\bd(B_0, B_1, r, u_0, u_1) = (B_1, ru_0, u_1) - (B_0, ru_1, u_0) \in \bc_1(X)$.

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) $B_0 \sub B_1 \sub X$.

\item A field $r \in \cC(X \setmin B_0; c_0)$

(for some $c_0 \in \cC(\bd B_0)$).

Let $r = r_1 \cup r'$, 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 $r_1u_0 \in U(B_1)$.

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 $r_1u_0$.

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

$B_i \sub B_j$ 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 A field $r \in \cC(X \setmin B^t; c^t)$,

where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$.

\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{

}

\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{

}

(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.)

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 $E_j(b)$ 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 $u_l = r_lu_j$,

where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$.

Finally, define

\eq{

}

The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel.

Thus we have a chain complex.

\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)}

relations to Chas-Sullivan string stuff}

\section{Basic properties of the blob complex}

\begin{prop} \label{disjunion}

There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.

\end{prop}

\begin{proof}

Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them

blob diagram $(b_1, b_2)$ on $X \du Y$.

Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way.

In the other direction, any blob diagram on $X\du Y$ is equal (up to sign)

to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines

a pair of blob diagrams on $X$ and $Y$.

These two maps are compatible with our sign conventions \nn{say more about this?} and

with the linear label relations.

The two maps are inverses of each other.

\nn{should probably say something about sign conventions for the differential

in a tensor product of chain complexes; ask Scott}

\end{proof}

For the next proposition we will temporarily restore $n$-manifold boundary

conditions to the notation.

of the quotient map

$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$.

\nn{always the case if we're working over $\c$}.

Then

\begin{prop} \label{bcontract}

For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$

is a chain homotopy equivalence

with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$.

Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0.

\end{prop}

\begin{proof}

By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map

$h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$.

For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding

an $(i{+}1)$-st blob equal to all of $B^n$.

In other words, add a new outermost blob which encloses all of the others.

Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to

the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$.

\nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$}

\end{proof}

for blob labels.

Also we need to blob reordering relations (?).)

(Note also that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy

equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.)

(For fields based on $n$-cats, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$.)

\medskip

As we noted above,

\begin{prop}

There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$.

\qed

\end{prop}

% oops -- duplicate

%\begin{prop} \label{functorialprop}

%The assignment $X \mapsto \bc_*(X)$ extends to a functor from the category of

%$n$-manifolds and homeomorphisms to the category of chain complexes and linear isomorphisms.

%\end{prop}

%\begin{proof}

%Obvious.

%\end{proof}