376 with all the structure maps above (disjoint union, boundary restriction, etc.) |
376 with all the structure maps above (disjoint union, boundary restriction, etc.) |
377 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
377 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
378 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
378 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
379 \end{enumerate} |
379 \end{enumerate} |
380 |
380 |
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381 \bigskip |
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382 Using the functoriality and $\bullet\times I$ properties above, together |
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383 with boundary collar homeomorphisms of manifolds, we can define the notion of |
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384 {\it extended isotopy}. |
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385 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
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386 of $\bd M$. |
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387 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$. |
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388 Let $c$ be $x$ restricted to $Y$. |
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389 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
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390 Then we have the glued field $x \cup (c\times I)$ on $M \cup (Y\times I)$. |
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391 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
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392 Then we say that $x$ is {\it extended isotopic} to $f(x \cup (c\times I))$. |
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393 More generally, we define extended isotopy to be the equivalence relation on fields |
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394 on $M$ generated by isotopy plus all instance of the above construction |
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395 (for all appropriate $Y$ and $x$). |
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396 |
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397 \nn{should also say something about pseudo-isotopy} |
381 |
398 |
382 \bigskip |
399 \bigskip |
383 \hrule |
400 \hrule |
384 \bigskip |
401 \bigskip |
385 |
402 |
519 |
536 |
520 |
537 |
521 \subsection{Local relations} |
538 \subsection{Local relations} |
522 \label{sec:local-relations} |
539 \label{sec:local-relations} |
523 |
540 |
524 \nn{the following is not done yet} |
541 |
525 |
542 A {\it local relation} is a collection subspaces $U(B; c) \sub \c[\cC_l(B; c)]$ |
526 Let $B^n$ denote the standard $n$-ball. |
543 (for all $n$-manifolds $B$ which are |
527 A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ |
544 homeomorphic to the standard $n$-ball and |
528 (for all $c \in \cC(\bd B^n)$) satisfying the following two properties. |
545 all $c \in \cC(\bd B)$) satisfying the following properties. |
529 \begin{enumerate} |
546 \begin{enumerate} |
530 \item local relations imply (extended) isotopy \nn{...} |
547 \item functoriality: |
531 \item $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing \nn{...} |
548 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
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549 \item local relations imply extended isotopy: |
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550 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
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551 to $y$, then $x-y \in U(B; c)$. |
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552 \item ideal with respect to gluing: |
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553 if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\cup r \in U(B)$ |
532 \end{enumerate} |
554 \end{enumerate} |
533 See \cite{kw:tqft} for details. |
555 See \cite{kw:tqft} for details. |
534 |
556 |
535 |
557 |
536 For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$, |
558 For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \cC_l(B; c)$, |
537 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
559 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
538 |
560 |
539 For $n$-category pictures, $U(B^n; c)$ is equal to the kernel of the evaluation map |
561 For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
540 $\cC_l(B^n; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
562 $\cC_l(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
541 domain and range. |
563 domain and range. |
542 |
564 |
543 \nn{maybe examples of local relations before general def?} |
565 \nn{maybe examples of local relations before general def?} |
544 |
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545 Note that the $Y$ is an $n$-manifold which is merely homeomorphic to the standard $B^n$, |
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546 then any homeomorphism $B^n \to Y$ induces the same local subspaces for $Y$. |
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547 We'll denote these by $U(Y; c) \sub \cC_l(Y; c)$, $c \in \cC(\bd Y)$. |
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548 \nn{Is this true in high (smooth) dimensions? Self-diffeomorphisms of $B^n$ |
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549 rel boundary might not be isotopic to the identity. OK for PL and TOP?} |
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550 |
566 |
551 Given a system of fields and local relations, we define the skein space |
567 Given a system of fields and local relations, we define the skein space |
552 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
568 $A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
553 the $n$-manifold $Y$ modulo local relations. |
569 the $n$-manifold $Y$ modulo local relations. |
554 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
570 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |