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\title{Blob Homology}

\begin{document}

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\textbf{Draft version, do not distribute.}

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[later than 11 June 2009]

\noop{

\section*{Todo}

\subsection*{What else?...}

\begin{itemize}

\item higher priority

\begin{itemize}

\item K\&S: learn the state of the art in A-inf categories

(tensor products, Kadeishvili result, ...)

\item K: so-called evaluation map stuff

\item K: topological fields

\item section describing intended applications

\item say something about starting with semisimple n-cat (trivial?? not trivial?)

\item T.O.C.

\end{itemize}

\item medium priority

\begin{itemize}

\item $n=2$ examples

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

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

(or maybe not)

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

\item shuffle product vs gluing product (?)

\item connection between $A_\infty$ operad and topological $A_\infty$ cat defs

\end{itemize}

\item lower priority

\begin{itemize}

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

\item Kh

\item Mention somewhere \cite{MR1624157} ``Skein homology''; it's not directly related, but has similar motivations.

\end{itemize}

\end{itemize}

} %end \noop

\input{text/intro.tex}

\section{Definitions}

\label{sec:definitions}

\subsection{Systems of fields}

\label{sec:fields}

Let $\cM_k$ denote the category (groupoid, in fact) with objects 

oriented PL manifolds of dimension

$k$ and morphisms homeomorphisms.

(We could equally well work with a different category of manifolds ---

unoriented, topological, smooth, spin, etc. --- but for definiteness we

will stick with oriented PL.)

Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$.

A $n$-dimensional {\it system of fields} in $\cS$

is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$

together with some additional data and satisfying some additional conditions, all specified below.

\nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris}

Before finishing the definition of fields, we give two motivating examples

(actually, families of examples) of systems of fields.

The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps

from X to $B$.

The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be 

the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by

$j$-morphisms of $C$.

One can think of such sub-cell-complexes as dual to pasting diagrams for $C$.

This is described in more detail below.

Now for the rest of the definition of system of fields.

\begin{enumerate}

\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, 

and these maps are a natural

transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$.

For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of 

$\cC(X)$ which restricts to $c$.

In this context, we will call $c$ a boundary condition.

\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$.

\item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps

again comprise a natural transformation of functors.

In addition, the orientation reversal maps are compatible with the boundary restriction maps.

\item $\cC_k$ is compatible with the symmetric monoidal

structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,

compatibly with homeomorphisms, restriction to boundary, and orientation reversal.

We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$

restriction maps.

\item Gluing without corners.

Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds.

Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.

Using the boundary restriction, disjoint union, and (in one case) orientation reversal

maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two

copies of $Y$ in $\bd X$.

Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps.

Then (here's the axiom/definition part) there is an injective ``gluing" map

\[

	\Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) ,

\]

and this gluing map is compatible with all of the above structure (actions

of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).

Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,

the gluing map is surjective.

From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the 

gluing surface, we say that fields in the image of the gluing map

are transverse to $Y$ or cuttable along $Y$.

\item Gluing with corners.

Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries.

Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.

Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself

(without corners) along two copies of $\bd Y$.

Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a cuttable field on $W\sgl$ and let

$c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$.

Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$.

(This restriction map uses the gluing without corners map above.)

Using the boundary restriction, gluing without corners, and (in one case) orientation reversal

maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two

copies of $Y$ in $\bd X$.

Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps.

Then (here's the axiom/definition part) there is an injective ``gluing" map

\[

	\Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) ,

\]

and this gluing map is compatible with all of the above structure (actions

of homeomorphisms, boundary restrictions, orientation reversal, disjoint union).

Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity,

the gluing map is surjective.

From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the 

gluing surface, we say that fields in the image of the gluing map

are transverse to $Y$ or cuttable along $Y$.

\item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted

$c \mapsto c\times I$.

These maps comprise a natural transformation of functors, and commute appropriately

with all the structure maps above (disjoint union, boundary restriction, etc.).

Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism

covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$.

\end{enumerate}

\nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$}

\bigskip

Using the functoriality and $\bullet\times I$ properties above, together

with boundary collar homeomorphisms of manifolds, we can define the notion of 

{\it extended isotopy}.

Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold

of $\bd M$.

Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$.

Let $c$ be $x$ restricted to $Y$.

Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$.

Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$.

Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism.

Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$.

More generally, we define extended isotopy to be the equivalence relation on fields

on $M$ generated by isotopy plus all instance of the above construction

(for all appropriate $Y$ and $x$).

\nn{should also say something about pseudo-isotopy}

%\bigskip

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\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(?)}

We now describe in more detail systems of fields coming from sub-cell-complexes labeled

by $n$-category morphisms.

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 $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is

the vector space of finite

linear combinations of fields on $X$.

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

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 $\lf(X; a)$ 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 $\lf(X; a) = \c[\cC(X; a)]/K$, 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 $\lf(X; a)$ to be the direct sum over all almost labelings of the

above tensor products.

\subsection{Local relations}

\label{sec:local-relations}

A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$,

for all $n$-manifolds $B$ which are

homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, 

satisfying the following properties.

\begin{enumerate}

\item functoriality: 

$f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$

\item local relations imply extended isotopy: 

if $x, y \in \cC(B; c)$ and $x$ is extended isotopic 

to $y$, then $x-y \in U(B; c)$.

\item ideal with respect to gluing:

if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$

\end{enumerate}

See \cite{kw:tqft} for details.

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

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

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

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

domain and range.

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

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.)

\nn{should expand above paragraph}

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 and local relations.

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 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 $\bc_k(X)$.

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 space of all linearized fields on $X$.

$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$.

Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear

combinations of 1-blob diagrams, where a 1-blob diagram to consists of

\begin{itemize}

\item An embedded closed ball (``blob") $B \sub X$.

\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}

In order to get the linear structure correct, we (officially) 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 linear

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

$\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the 

local relations encoded in $\bc_1(X)$.

More specifically, $\bc_2(X)$ is the space of all finite linear combinations of

2-blob diagrams, of which there are two types, disjoint and nested.

A disjoint 2-blob diagram consists of

\begin{itemize}

\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors.

\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 $u_i \in U(B_i; c_i)$, $i=1,2$.

\end{itemize}

We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$;

reversing the order of the blobs changes the sign.

Define $\bd(B_0, B_1, u_0, u_1, r) = 

(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\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) $B_0 \sub B_1 \sub X$.

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

(for some $c_0 \in \cC(\bd B_0)$), which is cuttable along $\bd B_1$.

\item A local relation field $u_0 \in U(B_0; c_0)$.

\end{itemize}

Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$

(for some $c_1 \in \cC(B_1)$) and

$r' \in \cC(X \setmin B_1; c_1)$.

Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$.

Note that the requirement that

local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$.

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 $u_0\bullet r_1$.

It is again easy to check that $\bd^2 = 0$.

\nn{should draw figures for 1, 2 and $k$-blob diagrams}

As with the 1-blob diagrams, in order to get the linear structure correct it is better to define

(officially)

\begin{eqnarray*}

	\bc_2(X) & \deq &

	\left( 

		\bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1}

			U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1)

	\right) \\

	&& \bigoplus \left( 

		\bigoplus_{B_0 \subset B_1} \bigoplus_{c_0}

			U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0)

	\right) .

\end{eqnarray*}

The final $\lf(X\setmin B_0; c_0)$ above really means fields cuttable along $\bd B_1$,

but we didn't feel like introducing a notation for that.

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).

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 $B_i$ 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.)

$c_i$ 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 $c^t \in \cC(\bd B^t)$