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%% \title{Almost sharp fronts for the surface quasi-geostrophic equation}

\title{The blob complex}

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%% Molecular, Murcia, Spain}, \and Franklin Sonnery\affil{2}{}}

\author{Scott Morrison\affil{1}{Miller Institute for Basic Research, UC Berkeley, CA 94704, USA} \and Kevin Walker\affil{2}{Microsoft Station Q, 2243 CNSI Building, UC Santa Barbara, CA 93106, USA}}

\contributor{Submitted to Proceedings of the National Academy of Sciences

of the United States of America}

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\begin{article}

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%\dropcap{I}n this article we study the evolution of ''almost-sharp'' fronts

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\nn{

background: TQFTs are important, historically, semisimple categories well-understood.

Many new examples arising recently which do not fit this framework, e.g. SW and OS theory.

These have more complicated gluing formulas (\cite{1003.0598,1005.1248}, etc); 

it would be nice to give generalized TQFT axioms that encompass these.

Triangulated categories are important; often calculations are via exact sequences,

and the standard TQFT constructions are quotients, which destroy exactness.

A first attempt to deal with this might be to replace all the tensor products in gluing formulas

with derived tensor products (cite Kh?).

However, in this approach it's probably difficult to prove invariance of constructions,

because they depend on explicit presentations of the manifold.

We'll give a manifestly invariant construction,

and deduce gluing formulas based on derived (actually, $A_\infty$) tensor products.}

\section{Definitions}

Decide if we need a friendlier, skein-module version.

}

\subsection{The blob complex}

\subsubsection{Decompositions of manifolds}

A \emph{ball decomposition} of $W$ is a 

sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls

$\du_a X_a$ and each $M_i$ is a manifold.

If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself.

A {\it permissible decomposition} of $W$ is a map

\[

	\coprod_a X_a \to W,

\]

which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$.

A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls

are glued up to yield $W$, and just require that there is some non-pathological way to do this.

Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement

of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$

with $\du_b Y_b = M_i$ for some $i$.

\begin{defn}

The poset $\cell(W)$ has objects the permissible decompositions of $W$, 

and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.

See Figure \ref{partofJfig} for an example.

\end{defn}

An $n$-category $\cC$ determines 

a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets 

(possibly with additional structure if $k=n$).

Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls,

and there is a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries

are splittable along this decomposition.

\begin{defn}

Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.

For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset

\begin{equation*}

%\label{eq:psi-C}

	\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl

\end{equation*}

where the restrictions to the various pieces of shared boundaries amongst the cells

If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.

\end{defn}

\subsubsection{Homotopy colimits}

\section{Properties of the blob complex}

\subsection{Formal properties}

\label{sec:properties}

The blob complex enjoys the following list of formal properties.

\begin{property}[Functoriality]

\label{property:functoriality}%

The blob complex is functorial with respect to homeomorphisms.

That is, 

for a fixed $n$-category $\cC$, the association

\begin{equation*}

X \mapsto \bc_*(X; \cC)

\end{equation*}

is a functor from $n$-manifolds and homeomorphisms between them to chain 

complexes and isomorphisms between them.

\end{property}

As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cC)$; 

this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:CH} below.

\begin{property}[Disjoint union]

\label{property:disjoint-union}

The blob complex of a disjoint union is naturally isomorphic to 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}

If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ (we allow $Y = \eset$) as a codimension $0$ submanifold of its boundary, 

write $X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$.

\begin{property}[Gluing map]

\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*}

Given a gluing $X \to X_\mathrm{gl}$, there is

a map

\[

	\bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow),

\]

natural with respect to homeomorphisms, and associative with respect to iterated gluings.

\end{property}

\begin{property}[Contractibility]

\label{property:contractibility}%

With field coefficients, the blob complex on an $n$-ball is contractible in the sense 

that it is homotopic to its $0$-th homology.

Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces 

associated by the system of fields $\cF$ to balls.

\begin{equation*}

\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)}

\end{equation*}

\end{property}

\nn{Properties \ref{property:functoriality} will be immediate from the definition given in

\S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there.

Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and 

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

\subsection{Specializations}

\label{sec:specializations}

The blob complex has two important special cases.

\begin{thm}[Skein modules]

\label{thm: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 $\cF$.

\begin{equation*}

H_0(\bc_*(X;\cF)) \iso A_{\cF}(X)

\end{equation*}

\end{thm}

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

\label{thm:hochschild}

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

quasi-isomorphic to the Hochschild complex.

\begin{equation*}

\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).}

\end{equation*}

\end{thm}

Theorem \ref{thm:skein-modules} is immediate from the definition, and

Theorem \ref{thm:hochschild} is established by extending the statement to bimodules as well as categories, then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$.

\subsection{Structure of the blob complex}

\label{sec:structure}

In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$.

\begin{thm}[$C_*(\Homeo(-))$ action]

\label{thm:CH}\label{thm:evaluation}

There is a chain map

\begin{equation*}

e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X).

\end{equation*}

such that

\begin{enumerate}

\item Restricted to $CH_0(X)$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. 

\item For

any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram

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

\begin{equation*}