119 |
120 |
120 [Outline for intro] |
121 [Outline for intro] |
121 \begin{itemize} |
122 \begin{itemize} |
122 \item Starting point: TQFTs via fields and local relations. |
123 \item Starting point: TQFTs via fields and local relations. |
123 This gives a satisfactory treatment for semisimple TQFTs |
124 This gives a satisfactory treatment for semisimple TQFTs |
124 (i.e. TQFTs for which the cylinder 1-category associated to an |
125 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
125 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
126 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
126 \item For non-semiemple TQFTs, this approach is less satisfactory. |
127 \item For non-semiemple TQFTs, this approach is less satisfactory. |
127 Our main motivating example (though we will not develop it in this paper) |
128 Our main motivating example (though we will not develop it in this paper) |
128 is the $4{+}1$-dimensional TQFT associated to Khovanov homology. |
129 is the $4{+}1$-dimensional TQFT associated to Khovanov homology. |
129 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
130 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
162 \[ |
163 \[ |
163 \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
164 \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
164 \] |
165 \] |
165 Here $\bc_0$ is linear combinations of fields on $W$, |
166 Here $\bc_0$ is linear combinations of fields on $W$, |
166 $\bc_1$ is linear combinations of local relations on $W$, |
167 $\bc_1$ is linear combinations of local relations on $W$, |
167 $\bc_1$ is linear combinations of relations amongst relations on $W$, |
168 $\bc_2$ is linear combinations of relations amongst relations on $W$, |
168 and so on. |
169 and so on. |
169 \item None of the above ideas depend on the details of the Khovanov homology example, |
170 \item None of the above ideas depend on the details of the Khovanov homology example, |
170 so we develop the general theory in the paper and postpone specific applications |
171 so we develop the general theory in the paper and postpone specific applications |
171 to later papers. |
172 to later papers. |
172 \item The blob complex enjoys the following nice properties \nn{...} |
173 \item The blob complex enjoys the following nice properties \nn{...} |
184 Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association |
185 Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association |
185 \begin{equation*} |
186 \begin{equation*} |
186 X \mapsto \bc_*^{\cF,\cU}(X) |
187 X \mapsto \bc_*^{\cF,\cU}(X) |
187 \end{equation*} |
188 \end{equation*} |
188 is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them. |
189 is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them. |
189 \scott{Do we want to or need to weaken `isomorphisms' to `homotopy equivalences' or `quasi-isomorphisms'?} |
|
190 \end{property} |
190 \end{property} |
191 |
191 |
192 \begin{property}[Disjoint union] |
192 \begin{property}[Disjoint union] |
193 \label{property:disjoint-union} |
193 \label{property:disjoint-union} |
194 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
194 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
326 again comprise a natural transformation of functors. |
326 again comprise a natural transformation of functors. |
327 In addition, the orientation reversal maps are compatible with the boundary restriction maps. |
327 In addition, the orientation reversal maps are compatible with the boundary restriction maps. |
328 \item $\cC_k$ is compatible with the symmetric monoidal |
328 \item $\cC_k$ is compatible with the symmetric monoidal |
329 structures on $\cM_k$ and sets: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
329 structures on $\cM_k$ and sets: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
330 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
330 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
|
331 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
|
332 restriction maps. |
331 \item Gluing without corners. |
333 \item Gluing without corners. |
332 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
334 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
333 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
335 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
334 Using the boundary restriction, disjoint union, and (in one case) orientation reversal |
336 Using the boundary restriction, disjoint union, and (in one case) orientation reversal |
335 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
337 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
371 gluing surface, we say that fields in the image of the gluing map |
373 gluing surface, we say that fields in the image of the gluing map |
372 are transverse to $Y$ or cuttable along $Y$. |
374 are transverse to $Y$ or cuttable along $Y$. |
373 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
375 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
374 $c \mapsto c\times I$. |
376 $c \mapsto c\times I$. |
375 These maps comprise a natural transformation of functors, and commute appropriately |
377 These maps comprise a natural transformation of functors, and commute appropriately |
376 with all the structure maps above (disjoint union, boundary restriction, etc.) |
378 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 |
379 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
378 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
380 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
379 \end{enumerate} |
381 \end{enumerate} |
|
382 |
|
383 \nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$} |
380 |
384 |
381 \bigskip |
385 \bigskip |
382 Using the functoriality and $\bullet\times I$ properties above, together |
386 Using the functoriality and $\bullet\times I$ properties above, together |
383 with boundary collar homeomorphisms of manifolds, we can define the notion of |
387 with boundary collar homeomorphisms of manifolds, we can define the notion of |
384 {\it extended isotopy}. |
388 {\it extended isotopy}. |
385 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
389 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
386 of $\bd M$. |
390 of $\bd M$. |
387 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$. |
391 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$. |
388 Let $c$ be $x$ restricted to $Y$. |
392 Let $c$ be $x$ restricted to $Y$. |
389 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
393 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
390 Then we have the glued field $x \cup (c\times I)$ on $M \cup (Y\times I)$. |
394 Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
391 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
395 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
392 Then we say that $x$ is {\it extended isotopic} to $f(x \cup (c\times I))$. |
396 Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
393 More generally, we define extended isotopy to be the equivalence relation on fields |
397 More generally, we define extended isotopy to be the equivalence relation on fields |
394 on $M$ generated by isotopy plus all instance of the above construction |
398 on $M$ generated by isotopy plus all instance of the above construction |
395 (for all appropriate $Y$ and $x$). |
399 (for all appropriate $Y$ and $x$). |
396 |
400 |
397 \nn{should also say something about pseudo-isotopy} |
401 \nn{should also say something about pseudo-isotopy} |
495 |
499 |
496 \medskip |
500 \medskip |
497 |
501 |
498 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
502 For top dimensional ($n$-dimensional) manifolds, we're actually interested |
499 in the linearized space of fields. |
503 in the linearized space of fields. |
500 By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is |
504 By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
501 the vector space of finite |
505 the vector space of finite |
502 linear combinations of fields on $X$. |
506 linear combinations of fields on $X$. |
503 If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$. |
507 If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
504 Thus the restriction (to boundary) maps are well defined because we never |
508 Thus the restriction (to boundary) maps are well defined because we never |
505 take linear combinations of fields with differing boundary conditions. |
509 take linear combinations of fields with differing boundary conditions. |
506 |
510 |
507 In some cases we don't linearize the default way; instead we take the |
511 In some cases we don't linearize the default way; instead we take the |
508 spaces $\cC_l(X; a)$ to be part of the data for the system of fields. |
512 spaces $\lf(X; a)$ to be part of the data for the system of fields. |
509 In particular, for fields based on linear $n$-category pictures we linearize as follows. |
513 In particular, for fields based on linear $n$-category pictures we linearize as follows. |
510 Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
514 Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
511 obvious relations on 0-cell labels. |
515 obvious relations on 0-cell labels. |
512 More specifically, let $L$ be a cell decomposition of $X$ |
516 More specifically, let $L$ be a cell decomposition of $X$ |
513 and let $p$ be a 0-cell of $L$. |
517 and let $p$ be a 0-cell of $L$. |
514 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
518 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
515 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
519 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
528 (Recall that 0-cells are labeled by $n$-morphisms.) |
532 (Recall that 0-cells are labeled by $n$-morphisms.) |
529 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
533 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
530 space determined by the labeling of the link of the 0-cell. |
534 space determined by the labeling of the link of the 0-cell. |
531 (If the 0-cell were labeled, the label would live in this space.) |
535 (If the 0-cell were labeled, the label would live in this space.) |
532 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
536 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
533 We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the |
537 We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
534 above tensor products. |
538 above tensor products. |
535 |
539 |
536 |
540 |
537 |
541 |
538 \subsection{Local relations} |
542 \subsection{Local relations} |
539 \label{sec:local-relations} |
543 \label{sec:local-relations} |
540 |
544 |
541 |
545 |
542 A {\it local relation} is a collection subspaces $U(B; c) \sub \c[\cC_l(B; c)]$ |
546 A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
543 (for all $n$-manifolds $B$ which are |
547 for all $n$-manifolds $B$ which are |
544 homeomorphic to the standard $n$-ball and |
548 homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
545 all $c \in \cC(\bd B)$) satisfying the following properties. |
549 satisfying the following properties. |
546 \begin{enumerate} |
550 \begin{enumerate} |
547 \item functoriality: |
551 \item functoriality: |
548 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
552 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
549 \item local relations imply extended isotopy: |
553 \item local relations imply extended isotopy: |
550 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
554 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
551 to $y$, then $x-y \in U(B; c)$. |
555 to $y$, then $x-y \in U(B; c)$. |
552 \item ideal with respect to gluing: |
556 \item ideal with respect to gluing: |
553 if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\cup r \in U(B)$ |
557 if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ |
554 \end{enumerate} |
558 \end{enumerate} |
555 See \cite{kw:tqft} for details. |
559 See \cite{kw:tqft} for details. |
556 |
560 |
557 |
561 |
558 For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \cC_l(B; c)$, |
562 For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, |
559 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
563 where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
560 |
564 |
561 For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
565 For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
562 $\cC_l(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
566 $\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
563 domain and range. |
567 domain and range. |
564 |
568 |
565 \nn{maybe examples of local relations before general def?} |
569 \nn{maybe examples of local relations before general def?} |
566 |
570 |
567 Given a system of fields and local relations, we define the skein space |
571 Given a system of fields and local relations, we define the skein space |
569 the $n$-manifold $Y$ modulo local relations. |
573 the $n$-manifold $Y$ modulo local relations. |
570 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
574 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
571 is defined to be the dual of $A(Y; c)$. |
575 is defined to be the dual of $A(Y; c)$. |
572 (See \cite{kw:tqft} or xxxx for details.) |
576 (See \cite{kw:tqft} or xxxx for details.) |
573 |
577 |
|
578 \nn{should expand above paragraph} |
|
579 |
574 The blob complex is in some sense the derived version of $A(Y; c)$. |
580 The blob complex is in some sense the derived version of $A(Y; c)$. |
575 |
581 |
576 |
582 |
577 |
583 |
578 \subsection{The blob complex} |
584 \subsection{The blob complex} |
579 \label{sec:blob-definition} |
585 \label{sec:blob-definition} |
580 |
586 |
581 Let $X$ be an $n$-manifold. |
587 Let $X$ be an $n$-manifold. |
582 Assume a fixed system of fields. |
588 Assume a fixed system of fields. |
583 In this section we will usually suppress boundary conditions on $X$ from the notation |
589 In this section we will usually suppress boundary conditions on $X$ from the notation |
584 (e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$). |
590 (e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
585 |
591 |
586 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
592 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
587 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
593 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
588 $\overline{X \setmin Y}$. |
594 $\overline{X \setmin Y}$. |
589 |
595 |
590 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case. |
596 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case. |
591 |
597 |
592 Define $\bc_0(X) = \cC_l(X)$. |
598 Define $\bc_0(X) = \lf(X)$. |
593 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cC_l(X; c)$. |
599 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
594 We'll omit this sort of detail in the rest of this section.) |
600 We'll omit this sort of detail in the rest of this section.) |
595 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
601 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
596 |
602 |
597 $\bc_1(X)$ is the space of all local relations that can be imposed on $\bc_0(X)$. |
603 $\bc_1(X)$ is the space of all local relations that can be imposed on $\bc_0(X)$. |
598 More specifically, define a 1-blob diagram to consist of |
604 More specifically, define a 1-blob diagram to consist of |
649 A nested 2-blob diagram consists of |
655 A nested 2-blob diagram consists of |
650 \begin{itemize} |
656 \begin{itemize} |
651 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
657 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
652 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
658 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
653 (for some $c_0 \in \cC(\bd B_0)$). |
659 (for some $c_0 \in \cC(\bd B_0)$). |
654 Let $r = r_1 \cup r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
660 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
655 (for some $c_1 \in \cC(B_1)$) and |
661 (for some $c_1 \in \cC(B_1)$) and |
656 $r' \in \cC(X \setmin B_1; c_1)$. |
662 $r' \in \cC(X \setmin B_1; c_1)$. |
657 \item A local relation field $u_0 \in U(B_0; c_0)$. |
663 \item A local relation field $u_0 \in U(B_0; c_0)$. |
658 \end{itemize} |
664 \end{itemize} |
659 Define $\bd(B_0, B_1, r, u_0) = (B_1, r', r_1u_0) - (B_0, r, u_0)$. |
665 Define $\bd(B_0, B_1, r, u_0) = (B_1, r', r_1u_0) - (B_0, r, u_0)$. |