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\title{Blob Homology}

\begin{document}

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\textbf{Draft version, do not distribute.}

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\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 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.)

\item shuffle product vs gluing product (?)

\item commutative algebra results

\item $A_\infty$ blob complex

\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}

\section{Introduction}

[Outline for intro]

\begin{itemize}

\item Starting point: TQFTs via fields and local relations.

This gives a satisfactory treatment for semisimple TQFTs

(i.e.\ TQFTs for which the cylinder 1-category associated to an

$n{-}1$-manifold $Y$ is semisimple for all $Y$).

\item For non-semiemple TQFTs, this approach is less satisfactory.

Our main motivating example (though we will not develop it in this paper)

is the $4{+}1$-dimensional TQFT associated to Khovanov homology.

It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together

with a link $L \subset \bd W$.

The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$.

\item How would we go about computing $A_{Kh}(W^4, L)$?

For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence)

\nn{... $L_1, L_2, L_3$}.

Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt

to compute $A_{Kh}(S^1\times B^3, L)$.

According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$

corresponds to taking a coend (self tensor product) over the cylinder category

associated to $B^3$ (with appropriate boundary conditions).

The coend is not an exact functor, so the exactness of the triangle breaks.

\item The obvious solution to this problem is to replace the coend with its derived counterpart.

This presumably works fine for $S^1\times B^3$ (the answer being to Hochschild homology

of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired.

If we build our manifold up via a handle decomposition, the computation

would be a sequence of derived coends.

A different handle decomposition of the same manifold would yield a different

sequence of derived coends.

To show that our definition in terms of derived coends is well-defined, we

would need to show that the above two sequences of derived coends yield the same answer.

This is probably not easy to do.

\item Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$

which is manifestly invariant.

(That is, a definition that does not

involve choosing a decomposition of $W$.

After all, one of the virtues of our starting point --- TQFTs via field and local relations ---

is that it has just this sort of manifest invariance.)

\item The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient

\[

 \text{linear combinations of fields} \;\big/\; \text{local relations} ,

\]

with an appropriately free resolution (the ``blob complex")

\[

	\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) .

\]

Here $\bc_0$ is linear combinations of fields on $W$,

$\bc_1$ is linear combinations of local relations on $W$,

$\bc_2$ is linear combinations of relations amongst relations on $W$,

and so on.

\item None of the above ideas depend on the details of the Khovanov homology example,

so we develop the general theory in the paper and postpone specific applications

to later papers.

\item The blob complex enjoys the following nice properties \nn{...}

\end{itemize}

\bigskip

\hrule

\bigskip

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.

\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 \du 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)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]}

\end{equation*}

\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}

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

%\hrule

%\bigskip

%

%\input{text/fields.tex}

%

%

%\bigskip

%\hrule

%\bigskip

\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.