\documentclass[11pt,leqno]{article}

\usepackage{amsmath,amssymb,amsthm}

\usepackage[all]{xy}

% test edit #3

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

\def\bc{{\cal B}}

\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\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{

	\expandafter\def\csname multi#1\endcsname##1{

		\def\multiack{##1}\ifx\multiack\semicolon

			\def\next{\relax}

		\else

			\csname #1\endcsname{##1}

			\def\next{\csname multi#1\endcsname}

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		\next}

	\csname multi#1\endcsname}

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

\def\calc#1{\expandafter\def\csname c#1\endcsname{{\cal #1}}}

\applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM;

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

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

\applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{End}{Hom}{Mat}{Tet}{cat}{Diff}{sign};

%%%%%% end excerpt

\title{Blob Homology}

\begin{document}

\makeatletter

\@addtoreset{equation}{section}

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

\makeatother

\newtheorem{thm}[equation]{Theorem}

\newtheorem{prop}[equation]{Proposition}

\newtheorem{lemma}[equation]{Lemma}

\newtheorem{cor}[equation]{Corollary}

\newtheorem{defn}[equation]{Definition}

\maketitle

\section{Introduction}

(motivation, summary/outline, etc.)

(motivation: 

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

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

(2) want answer independent of handle decomp (i.e. don't 

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

(3) ...

)

\section{Definitions}

\subsection{Fields}

Fix a top dimension $n$.

A {\it system of fields} 

\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).

For example: 

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;

given a field $c \in \cC(Y)$ there is a ``product field" 

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

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

\nn{note: probably will suppress from notation the distinction 

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 definition of a system of fields is intended to generalize 

the relevant properties of the following two examples of fields.

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

is a manifold of dimension $n$ or less) to be the set of 

all maps from $X$ to $B$.

The second example will take longer to explain.

Given an $n$-category $C$ with the right sort of duality 

(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), 

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}

	\item A cell decomposition of $S$ (equivalently, a finite collection

of points in the interior of $S$);

	\item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$)

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

	\item a transverse orientation of each 0-cell, thought of as a choice of

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

	\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with

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

of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the 

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}

	\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such

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

	\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$)

by a 0-morphism of $C$;

	\item a transverse orientation of each 1-cell, thought of as a choice of

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

	\item a labeling of each 1-cell by a 1-morphism of $C$, with

domain and range determined by the transverse orientation of the 1-cell 

and the labelings of the 2-cells;

	\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood 

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

	\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range 

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}

	\item A cell decomposition of $X$;

	\item an explicit general position homeomorphism from the link of each $j$-cell

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

	\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with

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

For top dimensional ($n$-dimensional) manifolds, we're actually interested 

in the linearized space of fields.

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

the vector space of finite 

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.

Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by 

obvious relations on 0-cell labels.

More specifically, let $L$ be a cell decomposition of $X$ 

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.

\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; 

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

space we do something like this; ? say something about tensor 

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).

We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the 

above tensor products.

\subsection{Local relations}

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

A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ 

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

\nn{Roughly, these are (1) the local relations imply (extended) isotopy; 

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

(3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$). 

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 $\cC_l(X; c)$).

We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 

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 $u$.

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 $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$.

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

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?}

\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{

	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' .

}

(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{

	\bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b).

}

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)}

\nn{TO DO: 

expand definition to handle DGA and $A_\infty$ versions of $n$-categories; 

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

(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a 

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.

Suppose that for all $c \in \cC(\bd B^n)$ 

we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ 

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}

(Note that for the above proof to work, we need the linear label relations 

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}

%\nn{need to same something about boundaries and boundary conditions above.

%maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.}