185 Decide if we need a friendlier, skein-module version. |
185 Decide if we need a friendlier, skein-module version. |
186 } |
186 } |
187 \subsection{The blob complex} |
187 \subsection{The blob complex} |
188 \subsubsection{Decompositions of manifolds} |
188 \subsubsection{Decompositions of manifolds} |
189 |
189 |
190 A {\emph ball decomposition} of $W$ is a |
190 A \emph{ball decomposition} of $W$ is a |
191 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
191 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
192 $\du_a X_a$. |
192 $\du_a X_a$ and each $M_i$ is a manifold. |
193 |
193 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. |
194 If $X_a$ is some component of $M_0$, note that its image in $W$ need not be a ball; parts of $\bd X_a$ may have been glued together. |
194 A {\it permissible decomposition} of $W$ is a map |
195 Define a {\it permissible decomposition} of $W$ to be a map |
|
196 \[ |
195 \[ |
197 \coprod_a X_a \to W, |
196 \coprod_a X_a \to W, |
198 \] |
197 \] |
199 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
198 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
200 Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
199 A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls |
201 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
200 are glued up to yield $W$, and just require that there is some non-pathological way to do this. |
202 |
201 |
203 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
202 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
204 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
203 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
205 with $\du_b Y_b = M_i$ for some $i$. |
204 with $\du_b Y_b = M_i$ for some $i$. |
206 |
205 |
213 |
212 |
214 An $n$-category $\cC$ determines |
213 An $n$-category $\cC$ determines |
215 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
214 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
216 (possibly with additional structure if $k=n$). |
215 (possibly with additional structure if $k=n$). |
217 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
216 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
218 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
217 and there is a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries |
219 are splittable along this decomposition. |
218 are splittable along this decomposition. |
220 |
219 |
221 \begin{defn} |
220 \begin{defn} |
222 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
221 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
223 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
222 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
224 \begin{equation} |
223 \begin{equation*} |
225 \label{eq:psi-C} |
224 %\label{eq:psi-C} |
226 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
225 \psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl |
227 \end{equation} |
226 \end{equation*} |
228 where the restrictions to the various pieces of shared boundaries amongst the cells |
227 where the restrictions to the various pieces of shared boundaries amongst the cells |
229 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
228 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). |
230 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 $\cC$. |
229 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 $\cC$. |
231 \end{defn} |
230 \end{defn} |
232 |
231 |
244 |
243 |
245 \begin{property}[Functoriality] |
244 \begin{property}[Functoriality] |
246 \label{property:functoriality}% |
245 \label{property:functoriality}% |
247 The blob complex is functorial with respect to homeomorphisms. |
246 The blob complex is functorial with respect to homeomorphisms. |
248 That is, |
247 That is, |
249 for a fixed $n$-dimensional system of fields $\cF$, the association |
248 for a fixed $n$-category $\cC$, the association |
250 \begin{equation*} |
249 \begin{equation*} |
251 X \mapsto \bc_*(X; \cF) |
250 X \mapsto \bc_*(X; \cC) |
252 \end{equation*} |
251 \end{equation*} |
253 is a functor from $n$-manifolds and homeomorphisms between them to chain |
252 is a functor from $n$-manifolds and homeomorphisms between them to chain |
254 complexes and isomorphisms between them. |
253 complexes and isomorphisms between them. |
255 \end{property} |
254 \end{property} |
256 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cF)$; |
255 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cC)$; |
257 this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:CH} below. |
256 this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:CH} below. |
258 |
257 |
259 \begin{property}[Disjoint union] |
258 \begin{property}[Disjoint union] |
260 \label{property:disjoint-union} |
259 \label{property:disjoint-union} |
261 The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
260 The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
262 \begin{equation*} |
261 \begin{equation*} |
263 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
262 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
264 \end{equation*} |
263 \end{equation*} |
265 \end{property} |
264 \end{property} |
266 |
265 |
267 If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
266 If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ (we allow $Y = \eset$) as a codimension $0$ submanifold of its boundary, |
268 write $X_\text{gl} = X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
267 write $X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
269 Note that this includes the case of gluing two disjoint manifolds together. |
|
270 \begin{property}[Gluing map] |
268 \begin{property}[Gluing map] |
271 \label{property:gluing-map}% |
269 \label{property:gluing-map}% |
272 %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 |
270 %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 |
273 %\begin{equation*} |
271 %\begin{equation*} |
274 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
272 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
275 %\end{equation*} |
273 %\end{equation*} |
276 Given a gluing $X \to X_\mathrm{gl}$, there is |
274 Given a gluing $X \to X_\mathrm{gl}$, there is |
277 a natural map |
275 a map |
278 \[ |
276 \[ |
279 \bc_*(X) \to \bc_*(X_\mathrm{gl}) |
277 \bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow), |
280 \] |
278 \] |
281 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
279 natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
282 \end{property} |
280 \end{property} |
283 |
281 |
284 \begin{property}[Contractibility] |
282 \begin{property}[Contractibility] |
285 \label{property:contractibility}% |
283 \label{property:contractibility}% |
286 With field coefficients, the blob complex on an $n$-ball is contractible in the sense |
284 With field coefficients, the blob complex on an $n$-ball is contractible in the sense |
298 \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.} |
296 \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.} |
299 |
297 |
300 \subsection{Specializations} |
298 \subsection{Specializations} |
301 \label{sec:specializations} |
299 \label{sec:specializations} |
302 |
300 |
303 The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. |
301 The blob complex has two important special cases. |
304 |
302 |
305 \begin{thm}[Skein modules] |
303 \begin{thm}[Skein modules] |
306 \label{thm:skein-modules} |
304 \label{thm:skein-modules} |
307 The $0$-th blob homology of $X$ is the usual |
305 The $0$-th blob homology of $X$ is the usual |
308 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
306 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
319 \begin{equation*} |
317 \begin{equation*} |
320 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
318 \xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
321 \end{equation*} |
319 \end{equation*} |
322 \end{thm} |
320 \end{thm} |
323 |
321 |
324 Proposition \ref{thm:skein-modules} is immediate from the definition, and |
322 Theorem \ref{thm:skein-modules} is immediate from the definition, and |
325 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 $\bc_*(S^1; -)$. |
323 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 $\bc_*(S^1; -)$. |
326 |
324 |
327 |
325 |
328 \subsection{Structure of the blob complex} |
326 \subsection{Structure of the blob complex} |
329 \label{sec:structure} |
327 \label{sec:structure} |
382 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. |
380 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. |
383 We think of this $A_\infty$ $n$-category as a free resolution. |
381 We think of this $A_\infty$ $n$-category as a free resolution. |
384 \end{rem} |
382 \end{rem} |
385 This result is described in more detail as Example 6.2.8 of \cite{1009.5025} |
383 This result is described in more detail as Example 6.2.8 of \cite{1009.5025} |
386 |
384 |
387 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. |
385 We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above. |
388 %The notation is intended to reflect the close parallel with the definition of the TQFT skein module via a colimit. |
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389 |
|
390 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
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391 |
386 |
392 \begin{thm}[Product formula] |
387 \begin{thm}[Product formula] |
393 \label{thm:product} |
388 \label{thm:product} |
394 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
389 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
395 Let $\cC$ be an $n$-category. |
390 Let $\cC$ be an $n$-category. |
396 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Example \ref{ex:blob-complexes-of-balls}). |
391 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology. |
397 Then |
392 Then |
398 \[ |
393 \[ |
399 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
394 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
400 \] |
395 \] |
401 \end{thm} |
396 \end{thm} |
434 Then |
429 Then |
435 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
430 $$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
436 \end{thm} |
431 \end{thm} |
437 |
432 |
438 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
433 This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
439 Note that there is no restriction on the connectivity of $T$ as in \cite[Theorem 3.8.6]{0911.0018}. |
434 Note that there is no restriction on the connectivity of $T$ as there is for the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. |
440 \nn{The proof appears in \S \ref{sec:map-recon}.} |
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441 |
435 |
442 |
436 |
443 \begin{thm}[Higher dimensional Deligne conjecture] |
437 \begin{thm}[Higher dimensional Deligne conjecture] |
444 \label{thm:deligne} |
438 \label{thm:deligne} |
445 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
439 The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
446 Since the little $n{+}1$-balls operad is a suboperad of the $n$-dimensional surgery cylinder operad, |
440 Since the little $n{+}1$-balls operad is a suboperad of the $n$-dimensional surgery cylinder operad, |
447 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
441 this implies that the little $n{+}1$-balls operad acts on blob cochains of the $n$-ball. |
448 \end{thm} |
442 \end{thm} |
449 \nn{See \S \ref{sec:deligne} for a full explanation of the statement, and the proof.} |
443 \nn{Explain and sketch} |
450 |
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451 |
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452 |
444 |
453 %% == end of paper: |
445 %% == end of paper: |
454 |
446 |
455 %% Optional Materials and Methods Section |
447 %% Optional Materials and Methods Section |
456 %% The Materials and Methods section header will be added automatically. |
448 %% The Materials and Methods section header will be added automatically. |