1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{Introduction} |
3 \section{Introduction} |
4 |
4 |
5 We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. |
5 We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. |
6 This blob complex provides a simultaneous generalisation of several well-understood constructions: |
6 This blob complex provides a simultaneous generalization of several well-understood constructions: |
7 \begin{itemize} |
7 \begin{itemize} |
8 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. |
8 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. |
9 (See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
9 (See Theorem \ref{thm:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
10 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
10 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
11 the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
11 the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
12 (See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.) |
12 (See Theorem \ref{thm:hochschild} and \S \ref{sec:hochschild}.) |
13 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have |
13 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have |
14 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
14 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
15 on the configuration space of unlabeled points in $M$. |
15 on the configuration space of unlabeled points in $M$. |
16 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
16 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
17 \end{itemize} |
17 \end{itemize} |
21 We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. |
21 We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. |
22 The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
22 The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
23 |
23 |
24 The blob complex has good formal properties, summarized in \S \ref{sec:properties}. |
24 The blob complex has good formal properties, summarized in \S \ref{sec:properties}. |
25 These include an action of $\CH{M}$, |
25 These include an action of $\CH{M}$, |
26 extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing |
26 extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Theorem \ref{thm:evaluation}) and a gluing |
27 formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}). |
27 formula allowing calculations by cutting manifolds into smaller parts (see Theorem \ref{thm:gluing}). |
28 |
28 |
29 We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. |
29 We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. |
30 See \S \ref{sec:future} for slightly more detail. |
30 See \S \ref{sec:future} for slightly more detail. |
31 |
31 |
32 \subsubsection{Structure of the paper} |
32 \subsubsection{Structure of the paper} |
33 The three subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
33 The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
34 summarise the formal properties of the blob complex (see \S \ref{sec:properties}) |
34 summarise the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}) |
35 and outline anticipated future directions and applications (see \S \ref{sec:future}). |
35 and outline anticipated future directions (see \S \ref{sec:future}). |
36 |
36 |
37 The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, |
37 The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, |
38 and establishes some of its properties. |
38 and establishes some of its properties. |
39 There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is |
39 There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is |
40 simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
40 simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
62 (using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition |
62 (using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition |
63 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
63 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
64 Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an |
64 Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an |
65 $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
65 $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
66 We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
66 We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
67 in particular the `gluing formula' of Property \ref{property:gluing} below. |
67 in particular the `gluing formula' of Theorem \ref{thm:gluing} below. |
68 |
68 |
69 The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
69 The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
70 |
70 |
71 \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} |
71 \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} |
72 |
72 |
254 \begin{equation} |
254 \begin{equation} |
255 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)} |
255 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)} |
256 \end{equation} |
256 \end{equation} |
257 \end{property} |
257 \end{property} |
258 |
258 |
259 \begin{property}[Skein modules] |
259 Properties \ref{property:functoriality} will be immediate from the definition given in |
260 \label{property:skein-modules}% |
260 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
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261 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
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262 |
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263 \subsection{Specializations} |
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264 \label{sec:specializations} |
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265 |
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266 The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology. |
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267 |
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268 \begin{thm}[Skein modules] |
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269 \label{thm:skein-modules}% |
261 The $0$-th blob homology of $X$ is the usual |
270 The $0$-th blob homology of $X$ is the usual |
262 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
271 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
263 by $\cC$. |
272 by $\cC$. |
264 (See \S \ref{sec:local-relations}.) |
273 (See \S \ref{sec:local-relations}.) |
265 \begin{equation*} |
274 \begin{equation*} |
266 H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) |
275 H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) |
267 \end{equation*} |
276 \end{equation*} |
268 \end{property} |
277 \end{thm} |
269 |
278 |
270 \todo{Somehow, the Hochschild homology thing isn't a "property". |
279 \newtheorem*{thm:hochschild}{Theorem \ref{thm:hochschild}} |
271 Let's move it and call it a theorem? -S} |
280 |
272 \begin{property}[Hochschild homology when $X=S^1$] |
281 \begin{thm:hochschild}[Hochschild homology when $X=S^1$] |
273 \label{property:hochschild}% |
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274 The blob complex for a $1$-category $\cC$ on the circle is |
282 The blob complex for a $1$-category $\cC$ on the circle is |
275 quasi-isomorphic to the Hochschild complex. |
283 quasi-isomorphic to the Hochschild complex. |
276 \begin{equation*} |
284 \begin{equation*} |
277 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
285 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
278 \end{equation*} |
286 \end{equation*} |
279 \end{property} |
287 \end{thm:hochschild} |
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288 |
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289 Theorem \ref{thm:skein-modules} is immediate from the definition, and |
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290 Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}. |
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291 We also note Appendix \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of certain commutative algebras thought of as an $n$-category. |
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292 |
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293 |
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294 \subsection{Structure of the blob complex} |
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295 \label{sec:structure} |
280 |
296 |
281 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
297 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
282 \begin{property}[$C_*(\Homeo(-))$ action]\mbox{}\\ |
298 \begin{thm}[$C_*(\Homeo(-))$ action]\mbox{}\\ |
283 \vspace{-0.5cm} |
299 \vspace{-0.5cm} |
284 \label{property:evaluation}% |
300 \label{thm:evaluation}% |
285 \begin{enumerate} |
301 \begin{enumerate} |
286 \item There is a chain map |
302 \item There is a chain map |
287 \begin{equation*} |
303 \begin{equation*} |
288 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
304 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
289 \end{equation*} |
305 \end{equation*} |
309 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\ |
325 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\ |
310 \CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) |
326 \CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) |
311 } |
327 } |
312 \end{equation*} |
328 \end{equation*} |
313 \end{enumerate} |
329 \end{enumerate} |
314 \end{property} |
330 \end{thm} |
315 |
331 |
316 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
332 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
317 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
333 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
318 for any homeomorphic pair $X$ and $Y$, |
334 for any homeomorphic pair $X$ and $Y$, |
319 satisfying corresponding conditions. |
335 satisfying corresponding conditions. |
320 |
336 |
321 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. |
337 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. |
322 Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. |
338 Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. |
323 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
339 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
324 |
340 |
325 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
341 \begin{thm}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
326 \label{property:blobs-ainfty} |
342 \label{thm:blobs-ainfty} |
327 Let $\cC$ be a topological $n$-category. |
343 Let $\cC$ be a topological $n$-category. |
328 Let $Y$ be an $n{-}k$-manifold. |
344 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 $0 \leq m < k$, |
345 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 |
346 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
331 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
347 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
332 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
348 (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 |
349 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 Property \ref{property:evaluation}. |
350 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
335 \end{property} |
351 \end{thm} |
336 \begin{rem} |
352 \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. |
353 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. |
354 We think of this $A_\infty$ $n$-category as a free resolution. |
339 \end{rem} |
355 \end{rem} |
340 |
356 |
341 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
357 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
342 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
358 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
343 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
359 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
344 |
360 |
345 \begin{property}[Product formula] |
361 \newtheorem*{thm:product}{Theorem \ref{thm:product}} |
346 \label{property:product} |
362 |
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363 \begin{thm:product}[Product formula] |
347 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
364 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
348 Let $\cC$ be an $n$-category. |
365 Let $\cC$ be an $n$-category. |
349 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
366 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Theorem \ref{thm:blobs-ainfty}). |
350 Then |
367 Then |
351 \[ |
368 \[ |
352 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
369 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
353 \] |
370 \] |
354 \end{property} |
371 \end{thm:product} |
355 We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps. |
372 We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps. |
356 |
373 |
357 Fix a topological $n$-category $\cC$, which we'll omit from the notation. |
374 Fix a topological $n$-category $\cC$, which we'll omit from the notation. |
358 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
375 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
359 (See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
376 (See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
360 |
377 |
361 \begin{property}[Gluing formula] |
378 \newtheorem*{thm:gluing}{Theorem \ref{thm:gluing}} |
362 \label{property:gluing}% |
379 |
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380 \begin{thm:gluing}[Gluing formula] |
363 \mbox{}% <-- gets the indenting right |
381 \mbox{}% <-- gets the indenting right |
364 \begin{itemize} |
382 \begin{itemize} |
365 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
383 \item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
366 $A_\infty$ module for $\bc_*(Y)$. |
384 $A_\infty$ module for $\bc_*(Y)$. |
367 |
385 |
369 $\bc_*(X_\text{cut})$ as an $\bc_*(Y)$-bimodule: |
387 $\bc_*(X_\text{cut})$ as an $\bc_*(Y)$-bimodule: |
370 \begin{equation*} |
388 \begin{equation*} |
371 \bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
389 \bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
372 \end{equation*} |
390 \end{equation*} |
373 \end{itemize} |
391 \end{itemize} |
374 \end{property} |
392 \end{thm:gluing} |
375 |
393 |
376 Finally, we prove two theorems which we consider as applications. |
394 Theorem \ref{thm:evaluation} is proved in |
377 |
395 in \S \ref{sec:evaluation}, Theorem \ref{thm:blobs-ainfty} appears as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, |
378 \begin{thm}[Mapping spaces] |
396 and Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, with Theorem \ref{thm:gluing} then a relatively straightforward consequence of the proof, explained in \S \ref{sec:gluing}. |
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397 |
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398 \subsection{Applications} |
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399 \label{sec:applications} |
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400 Finally, we give two theorems which we consider as applications. |
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401 |
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402 \newtheorem*{thm:map-recon}{Theorem \ref{thm:map-recon}} |
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403 |
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404 \begin{thm:map-recon}[Mapping spaces] |
379 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
405 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
380 $B^n \to T$. |
406 $B^n \to T$. |
381 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
407 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
382 Then |
408 Then |
383 $$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
409 $$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
384 \end{thm} |
410 \end{thm:map-recon} |
385 |
411 |
386 This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. |
412 This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. The proof appears in \S \ref{sec:map-recon}. |
387 |
413 |
388 \begin{thm}[Higher dimensional Deligne conjecture] |
414 \newtheorem*{thm:deligne}{Theorem \ref{thm:deligne}} |
389 \label{thm:deligne} |
415 |
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416 \begin{thm:deligne}[Higher dimensional Deligne conjecture] |
390 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
417 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
391 \end{thm} |
418 \end{thm:deligne} |
392 See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof. |
419 See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof. |
393 |
420 |
394 Properties \ref{property:functoriality} and \ref{property:skein-modules} will be immediate from the definition given in |
421 |
395 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. |
422 |
396 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
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397 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} |
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398 in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, |
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399 and Properties \ref{property:product} and \ref{property:gluing} in \S \ref{sec:ainfblob} as consequences of Theorem \ref{product_thm}. |
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400 |
423 |
401 \subsection{Future directions} |
424 \subsection{Future directions} |
402 \label{sec:future} |
425 \label{sec:future} |
403 Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
426 Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
404 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
427 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
423 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} |
446 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} |
424 |
447 |
425 |
448 |
426 \subsection{Thanks and acknowledgements} |
449 \subsection{Thanks and acknowledgements} |
427 We'd like to thank David Ben-Zvi, Kevin Costello, Chris Douglas, |
450 We'd like to thank David Ben-Zvi, Kevin Costello, Chris Douglas, |
428 Michael Freedman, Vaughan Jones, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{and who else?} for many interesting and useful conversations. |
451 Michael Freedman, Vaughan Jones, Alexander Kirillov, Justin Roberts, Chris Schommer-Pries, Peter Teichner \nn{and who else?} for many interesting and useful conversations. |
429 During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. |
452 During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. |
430 |
453 |