%!TEX root = ../blob1.tex

\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 the 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 
(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
by $(\cF,\cU)$. (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*}
\nn{should probably say something about associativity here (or not?)}
\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}

\nn{add product formula?  $n$-dimensional fat graph operad stuff?}

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}.
