164 \input{../text/intro}

   163 \section{}

   164 

   165 \nn{

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

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

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

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

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

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

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

   173 with derived tensor products (cite Kh?).

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

   175 because they depend on explicit presentations of the manifold.

   176 We'll give a manifestly invariant construction,

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

   178 

   179 \section{Definitions}

   180 \subsection{$n$-categories}

   181 \nn{

   182 Axioms for $n$-categories, examples (maps, string diagrams)

   183 }

   184 \nn{

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

   186 }

   187 \subsection{The blob complex}

   188 \subsubsection{Decompositions of manifolds}

   189 \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls}

   190 \subsubsection{Homotopy colimits}

   191 \nn{How can we extend an $n$-category from balls to arbitrary manifolds?}

   192 

   193 \nn{In practice, this gives the old definition}

   194 \subsubsection{}

   195 \section{Properties of the blob complex}

   196 \subsection{Formal properties}

   197 \label{sec:properties}

   198 The blob complex enjoys the following list of formal properties.

   199 

   200 \begin{property}[Functoriality]

   201 \label{property:functoriality}%

   202 The blob complex is functorial with respect to homeomorphisms.

   203 That is, 

   204 for a fixed $n$-dimensional system of fields $\cF$, the association

   205 \begin{equation*}

   206 X \mapsto \bc_*(X; \cF)

   207 \end{equation*}

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

   209 complexes and isomorphisms between them.

   210 \end{property}

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

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

   213 

   214 \begin{property}[Disjoint union]

   215 \label{property:disjoint-union}

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

   217 \begin{equation*}

   218 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2)

   219 \end{equation*}

   220 \end{property}

   221 

   222 If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, 

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

   224 Note that this includes the case of gluing two disjoint manifolds together.

   225 \begin{property}[Gluing map]

   226 \label{property:gluing-map}%

   227 %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

   228 %\begin{equation*}

   229 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2).

   230 %\end{equation*}

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

   232 a natural map

   233 \[

   234 	\bc_*(X) \to \bc_*(X_\mathrm{gl}) 

   235 \]

   236 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings).

   237 \end{property}

   238 

   239 \begin{property}[Contractibility]

   240 \label{property:contractibility}%

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

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

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

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

   245 \begin{equation*}

   246 \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)}

   247 \end{equation*}

   248 \end{property}

   249 

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

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

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

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

   254 

   255 \subsection{Specializations}

   256 \label{sec:specializations}

   257 

   258 The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology.

   259 

   260 \begin{thm}[Skein modules]

   261 \label{thm:skein-modules}

   262 The $0$-th blob homology of $X$ is the usual 

   263 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$

   264 by $\cF$.

   265 \begin{equation*}

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

   267 \end{equation*}

   268 \end{thm}

   269 

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

   271 \label{thm:hochschild}

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

   273 quasi-isomorphic to the Hochschild complex.

   274 \begin{equation*}

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

   276 \end{equation*}

   277 \end{thm}

   278 

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

   280 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 $\bc_*(S^1; -)$.

   281 

   282 

   283 \subsection{Structure of the blob complex}

   284 \label{sec:structure}

   285 

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

   287 

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

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

   290 There is a chain map

   291 \begin{equation*}

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

   293 \end{equation*}

   294 such that

   295 \begin{enumerate}

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

   297 

   298 \item For

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

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

   301 \begin{equation*}

   302 \xymatrix@C+0.3cm{

   303      \CH{X} \otimes \bc_*(X)

   304         \ar[r]_{e_{X}}  \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y}  &

   305             \bc_*(X) \ar[d]_{\gl_Y} \\

   306      \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}}    & \bc_*(X \bigcup_Y \selfarrow)

   307 }

   308 \end{equation*}

   309 \end{enumerate}

   310 

   311 Futher, this map is associative, in the sense that the following diagram commutes (up to homotopy).

   312 \begin{equation*}

   313 \xymatrix{

   314 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\

   315 \CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X)

   316 }

   317 \end{equation*}

   318 \end{thm}

   319 

   320 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps

   321 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$

   322 for any homeomorphic pair $X$ and $Y$, 

   323 satisfying corresponding conditions.

   324 

   325 \begin{thm}[Blob complexes of products with balls form an $A_\infty$ $n$-category]

   326 \label{thm:blobs-ainfty}

   327 Let $\cC$ be  a topological $n$-category.

   328 Let $Y$ be an $n{-}k$-manifold. 

   329 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, 

   330 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set 

   331 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ 

   332 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) 

   333 These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in 

   334 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}.

   335 \end{thm}

   336 \begin{rem}

   337 Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category.

   338 We think of this $A_\infty$ $n$-category as a free resolution.

   339 \end{rem}

   340 This result is described in more detail as Example 6.2.8 of \cite{1009.5025}

   341 

   342 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. The next theorem describes the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as in the previous example.

   343 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit.

   344 

   345 \newtheorem*{thm:product}{Theorem \ref{thm:product}}

   346 

   347 \begin{thm}[Product formula]

   348 \label{thm:product}

   349 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold.

   350 Let $\cC$ be an $n$-category.

   351 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}).

   352 Then

   353 \[

   354 	\bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W).

   355 \]

   356 \end{thm}

   357 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps

   358 (see \cite[\S7.1]{1009.5025}).

   359 

   360 Fix a topological $n$-category $\cC$, which we'll omit from the notation.

   361 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category.

   362 

   363 \begin{thm}[Gluing formula]

   364 \label{thm:gluing}

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

   366 \begin{itemize}

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

   368 $A_\infty$ module for $\bc_*(Y)$.

   369 

   370 \item For any $n$-manifold $X_\text{gl} = X\bigcup_Y \selfarrow$, the blob complex $\bc_*(X_\text{gl})$ is the $A_\infty$ self-tensor product of

   371 $\bc_*(X)$ as an $\bc_*(Y)$-bimodule:

   372 \begin{equation*}

   373 \bc_*(X_\text{gl}) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow

   374 \end{equation*}

   375 \end{itemize}

   376 \end{thm}

   377 

   378 \nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.}

   379 

   380 \section{Applications}

   381 \label{sec:applications}

   382 Finally, we give two applications of the above machinery.

   383 

   384 \begin{thm}[Mapping spaces]

   385 \label{thm:map-recon}

   386 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps 

   387 $B^n \to T$.

   388 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.)

   389 Then 

   390 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$

   391 \end{thm}

   392 

   393 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. 

   394 Note that there is no restriction on the connectivity of $T$ as in \cite[Theorem 3.8.6]{0911.0018}.

   395 \nn{The proof appears in \S \ref{sec:map-recon}.}

   396 

   397 

   398 \begin{thm}[Higher dimensional Deligne conjecture]

   399 \label{thm:deligne}

   400 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains.

   401 Since the little $n{+}1$-balls operad is a suboperad of the $n$-dimensional surgery cylinder operad,

   402 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball.

   403 \end{thm}

   404 \nn{See \S \ref{sec:deligne} for a full explanation of the statement, and the proof.}

   405 

   212 \begin{thebibliography}{}

   453 %%%% BIBTEX

   213 

   454 \bibliographystyle{alpha}

   214 \end{thebibliography}

   455 \bibliography{../bibliography/bibliography}

   456 

   457 %%%% non-BIBTEX

   458 %\begin{thebibliography}{}

   459 %

   460 %\end{thebibliography}

   461