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\input{text/top_matter.tex}

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

\def\bc{{\mathcal 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\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{

    \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}

        \fi

        \next}

    \csname multi#1\endcsname}

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

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

\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

\textbf{Draft version, do not distribute.}

\versioninfo

\section*{Todo}

\subsection*{What else?...}

\begin{itemize}

\item Derive Hochschild standard results from blob point of view?

\item $n=2$ examples

\item Kh

\item dimension $n+1$ (generalized Deligne conjecture?)

\item should be clear about PL vs Diff; probably PL is better

(or maybe not)

\item say what we mean by $n$-category, $A_\infty$ or $E_\infty$ $n$-category

\item something about higher derived coend things (derived 2-coend, e.g.)

\end{itemize}

\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) ...

)

We then show that blob homology enjoys the following

\ref{property:gluing} properties.

\begin{property}[Functoriality]

\label{property:functoriality}%

Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association

\begin{equation*}

X \mapsto \bc_*^{\cF,\cU}(X)

\end{equation*}

is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them.

\scott{Do we want to or need to weaken `isomorphisms' to `homotopy equivalences' or `quasi-isomorphisms'?}

\end{property}

\begin{property}[Disjoint union]

\label{property:disjoint-union}

The blob complex of a disjoint union is naturally the tensor product of the blob complexes.

\begin{equation*}

\bc_*(X_1 \sqcup X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2)

\end{equation*}

\end{property}

\begin{property}[A map for gluing]

\label{property:gluing-map}%

If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$,

there is a chain map

\begin{equation*}

\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2).

\end{equation*}

\end{property}

\begin{property}[Contractibility]

\label{property:contractibility}%

\todo{Err, requires a splitting?}

The blob complex for an $n$-category on an $n$-ball is quasi-isomorphic to its $0$-th homology.

\begin{equation}

\xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))}

\end{equation}

\todo{Say that this is just the original $n$-category?}

\end{property}

\begin{property}[Skein modules]

\label{property:skein-modules}%

The $0$-th blob homology of $X$ is the usual skein module associated to $X$. (See \S \ref{sec:local-relations}.)

\begin{equation*}

H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X)

\end{equation*}

\end{property}

\begin{property}[Hochschild homology when $X=S^1$]

\label{property:hochschild}%

The blob complex for a $1$-category $\cC$ on the circle is

quasi-isomorphic to the Hochschild complex.

\begin{equation*}

\xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & HC_*(\cC)}

\end{equation*}

\end{property}

\begin{property}[Evaluation map]

\label{property:evaluation}%

There is an `evaluation' chain map

\begin{equation*}

\ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X).

\end{equation*}

(Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.)

Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for

any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram

(using the gluing maps described in Property \ref{property:gluing-map}) commutes.

\begin{equation*}

\xymatrix{

     \CD{X} \otimes \bc_*(X) \ar[r]^{\ev_X}    & \bc_*(X) \\

     \CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2)

        \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}}  \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y}  &

            \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y}

}

\end{equation*}

\end{property}

\begin{property}[Gluing formula]

\label{property:gluing}%

\mbox{}% <-- gets the indenting right

\begin{itemize}

\item For any $(n-1)$-manifold $Y$, the blob homology of $Y \times I$ is

naturally an $A_\infty$ category. % We'll write $\bc_*(Y)$ for $\bc_*(Y \times I)$ below.

\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob homology of $X$ is naturally an

$A_\infty$ module for $\bc_*(Y \times I)$.

\item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension

$0$-submanifold of its boundary, the blob homology of $X'$, obtained from

$X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of

$\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule.

\begin{equation*}

\bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}}

\end{equation*}

\todo{How do you write self tensor product?}

\end{itemize}

\end{property}

Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in

\S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.}

Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.

Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation},

and Property \ref{property:gluing} in \S \ref{sec:gluing}.

\section{Definitions}

\label{sec:definitions}

\subsection{Systems of fields}

\label{sec: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 \cite{kw:tqft} 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}

\label{sec: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 \cite{kw:tqft} 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 \cite{kw:tqft} or xxxx for details.)

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

\subsection{The blob complex}

\label{sec:blob-definition}

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.