161 If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
161 If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
162 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
162 of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
163 interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
163 interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
164 of the algebra. |
164 of the algebra. |
165 |
165 |
166 For $n=2$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
166 \medskip |
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167 |
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168 For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) |
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169 that are common in the literature. |
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170 We describe these carefully here. |
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171 |
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172 A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
167 an object of the 2-category $C$. |
173 an object of the 2-category $C$. |
168 A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
174 A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
169 A field on a 2-manifold $Y$ consists of |
175 A field on a 2-manifold $Y$ consists of |
170 \begin{itemize} |
176 \begin{itemize} |
171 \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
177 \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
172 that each component of the complement is homeomorphic to a disk); |
178 that each component of the complement is homeomorphic to a disk); |
173 \item a labeling of each 2-cell (and each half 2-cell adjacent to $\bd Y$) |
179 \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
174 by a 0-morphism of $C$; |
180 by a 0-morphism of $C$; |
175 \item a transverse orientation of each 1-cell, thought of as a choice of |
181 \item a transverse orientation of each 1-cell, thought of as a choice of |
176 ``domain" and ``range" for the two adjacent 2-cells; |
182 ``domain" and ``range" for the two adjacent 2-cells; |
177 \item a labeling of each 1-cell by a 1-morphism of $C$, with |
183 \item a labeling of each 1-cell by a 1-morphism of $C$, with |
178 domain and range determined by the transverse orientation of the 1-cell |
184 domain and range determined by the transverse orientation of the 1-cell |
193 to the boundary of the standard $(k-j)$-dimensional bihedron; and |
199 to the boundary of the standard $(k-j)$-dimensional bihedron; and |
194 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
200 \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
195 domain and range determined by the labelings of the link of $j$-cell. |
201 domain and range determined by the labelings of the link of $j$-cell. |
196 \end{itemize} |
202 \end{itemize} |
197 |
203 |
198 \nn{next definition might need some work; I think linearity relations should |
204 %\nn{next definition might need some work; I think linearity relations should |
199 be treated differently (segregated) from other local relations, but I'm not sure |
205 %be treated differently (segregated) from other local relations, but I'm not sure |
200 the next definition is the best way to do it} |
206 %the next definition is the best way to do it} |
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207 |
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208 \medskip |
201 |
209 |
202 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
210 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
203 in the linearized space of fields. |
211 in the linearized space of fields. |
204 By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is |
212 By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is |
205 the vector space of finite |
213 the vector space of finite |
243 |
251 |
244 Let $B^n$ denote the standard $n$-ball. |
252 Let $B^n$ denote the standard $n$-ball. |
245 A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ |
253 A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ |
246 (for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties. |
254 (for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties. |
247 |
255 |
248 \nn{implies (extended?) isotopy; stable under gluing; open covers?; ...} |
256 \nn{Roughly, these are (1) the local relations imply (extended) isotopy; |
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257 (2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and |
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258 (3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$). |
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259 See KW TQFT notes for details. Need to transfer details to here.} |
249 |
260 |
250 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$, |
261 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$, |
251 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
262 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
252 |
263 |
253 For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map |
264 For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map |
257 \nn{maybe examples of local relations before general def?} |
268 \nn{maybe examples of local relations before general def?} |
258 |
269 |
259 Note that the $Y$ is an $n$-manifold which is merely homeomorphic to the standard $B^n$, |
270 Note that the $Y$ is an $n$-manifold which is merely homeomorphic to the standard $B^n$, |
260 then any homeomorphism $B^n \to Y$ induces the same local subspaces for $Y$. |
271 then any homeomorphism $B^n \to Y$ induces the same local subspaces for $Y$. |
261 We'll denote these by $U(Y; c) \sub \cC_l(Y; c)$, $c \in \cC(\bd Y)$. |
272 We'll denote these by $U(Y; c) \sub \cC_l(Y; c)$, $c \in \cC(\bd Y)$. |
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273 \nn{Is this true in high (smooth) dimensions? Self-diffeomorphisms of $B^n$ |
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274 rel boundary might not be isotopic to the identity. OK for PL and TOP?} |
262 |
275 |
263 Given a system of fields and local relations, we define the skein space |
276 Given a system of fields and local relations, we define the skein space |
264 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
277 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
265 the $n$-manifold $Y$ modulo local relations. |
278 the $n$-manifold $Y$ modulo local relations. |
266 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
279 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
314 combination of fields on $X$ obtained by gluing $r$ to $u$. |
327 combination of fields on $X$ obtained by gluing $r$ to $u$. |
315 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
328 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
316 just erasing the blob from the picture |
329 just erasing the blob from the picture |
317 (but keeping the blob label $u$). |
330 (but keeping the blob label $u$). |
318 |
331 |
319 Note that the skein module $A(X)$ |
332 Note that the skein space $A(X)$ |
320 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
333 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
321 |
334 |
322 $\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$. |
335 $\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$. |
323 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
336 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
324 2-blob diagrams (defined below), modulo the usual linear label relations. |
337 2-blob diagrams (defined below), modulo the usual linear label relations. |
399 x = \sign(\pi) x' . |
412 x = \sign(\pi) x' . |
400 } |
413 } |
401 |
414 |
402 (Alert readers will have noticed that for $k=2$ our definition |
415 (Alert readers will have noticed that for $k=2$ our definition |
403 of $\bc_k(X)$ is slightly different from the previous definition |
416 of $\bc_k(X)$ is slightly different from the previous definition |
404 of $\bc_2(X)$. |
417 of $\bc_2(X)$ --- we did not impose the reordering relations. |
405 The general definition takes precedence; |
418 The general definition takes precedence; |
406 the earlier definition was simplified for purposes of exposition.) |
419 the earlier definition was simplified for purposes of exposition.) |
407 |
420 |
408 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
421 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
409 Let $b = (\{B_i\}, r, \{u_j\})$ be a $k$-blob diagram. |
422 Let $b = (\{B_i\}, r, \{u_j\})$ be a $k$-blob diagram. |
422 Thus we have a chain complex. |
435 Thus we have a chain complex. |
423 |
436 |
424 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
437 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
425 |
438 |
426 |
439 |
427 \nn{TO DO: ((?)) allow $n$-morphisms to be chain complex instead of just |
440 \nn{TO DO: |
428 a vector space; relations to Chas-Sullivan string stuff} |
441 expand definition to handle DGA and $A_\infty$ versions of $n$-categories; |
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442 relations to Chas-Sullivan string stuff} |
429 |
443 |
430 |
444 |
431 |
445 |
432 \section{Basic properties of the blob complex} |
446 \section{Basic properties of the blob complex} |
433 |
447 |
491 There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
505 There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
492 \qed |
506 \qed |
493 \end{prop} |
507 \end{prop} |
494 |
508 |
495 |
509 |
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510 % oops -- duplicate |
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511 |
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512 %\begin{prop} \label{functorialprop} |
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513 %The assignment $X \mapsto \bc_*(X)$ extends to a functor from the category of |
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514 %$n$-manifolds and homeomorphisms to the category of chain complexes and linear isomorphisms. |
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515 %\end{prop} |
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516 |
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517 %\begin{proof} |
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518 %Obvious. |
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519 %\end{proof} |
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520 |
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521 %\nn{need to same something about boundaries and boundary conditions above. |
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522 %maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.} |
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523 |
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524 |
496 \begin{prop} |
525 \begin{prop} |
497 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
526 For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
498 of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
527 of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
499 (chain map) isomorphisms. |
528 (chain map) isomorphisms. |
500 \qed |
529 \qed |
501 \end{prop} |
530 \end{prop} |
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531 |
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532 \nn{need to same something about boundaries and boundary conditions above. |
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533 maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.} |
502 |
534 |
503 |
535 |
504 In particular, |
536 In particular, |
505 \begin{prop} \label{diff0prop} |
537 \begin{prop} \label{diff0prop} |
506 There is an action of $\Diff(X)$ on $\bc_*(X)$. |
538 There is an action of $\Diff(X)$ on $\bc_*(X)$. |