blob: comparison blob1.tex

equal deleted inserted replaced

-:4093d7979c56
+:fdb1cd651fd2
 \def\ep{\epsilon}
 \def\sgl{_\mathrm{gl}}
 \def\op{^\mathrm{op}}
 \def\deq{\stackrel{\mathrm{def}}{=}}
 \def\pd#1#2{\frac{\partial #1}{\partial #2}}
+\def\lf{\overline{\cC}}
 \def\nn#1{{{\it \small [#1]}}}
 % equations
 [Outline for intro]
 \begin{itemize}
 \item Starting point: TQFTs via fields and local relations.
 This gives a satisfactory treatment for semisimple TQFTs
-(i.e. TQFTs for which the cylinder 1-category associated to an
+(i.e.\ TQFTs for which the cylinder 1-category associated to an
 $n{-}1$-manifold $Y$ is semisimple for all $Y$).
 \item For non-semiemple TQFTs, this approach is less satisfactory.
 Our main motivating example (though we will not develop it in this paper)
 is the $4{+}1$-dimensional TQFT associated to Khovanov homology.
 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together
 \[
 	\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) .
 \]
 Here $\bc_0$ is linear combinations of fields on $W$,
 $\bc_1$ is linear combinations of local relations on $W$,
-$\bc_1$ is linear combinations of relations amongst relations on $W$,
+$\bc_2$ is linear combinations of relations amongst relations on $W$,
 and so on.
 \item None of the above ideas depend on the details of the Khovanov homology example,
 so we develop the general theory in the paper and postpone specific applications
 to later papers.
 \item The blob complex enjoys the following nice properties \nn{...}
 Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association
 \begin{equation*}
 X \mapsto \bc_*^{\cF,\cU}(X)
 \end{equation*}
 is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them.
-\scott{Do we want to or need to weaken `isomorphisms' to `homotopy equivalences' or `quasi-isomorphisms'?}
 \end{property}
 \begin{property}[Disjoint union]
 \label{property:disjoint-union}
 The blob complex of a disjoint union is naturally the tensor product of the blob complexes.
 again comprise a natural transformation of functors.
 In addition, the orientation reversal maps are compatible with the boundary restriction maps.
 \item $\cC_k$ is compatible with the symmetric monoidal
 structures on $\cM_k$ and sets: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$,
 compatibly with homeomorphisms, restriction to boundary, and orientation reversal.
+We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$
+restriction maps.
 \item Gluing without corners.
 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds.
 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$.
 Using the boundary restriction, disjoint union, and (in one case) orientation reversal
 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two
 gluing surface, we say that fields in the image of the gluing map
 are transverse to $Y$ or cuttable along $Y$.
 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted
 $c \mapsto c\times I$.
 These maps comprise a natural transformation of functors, and commute appropriately
-with all the structure maps above (disjoint union, boundary restriction, etc.)
+with all the structure maps above (disjoint union, boundary restriction, etc.).
 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism
 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$.
 \end{enumerate}
+\nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$}
 \bigskip
 Using the functoriality and $\bullet\times I$ properties above, together
 with boundary collar homeomorphisms of manifolds, we can define the notion of
 {\it extended isotopy}.
 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold
 of $\bd M$.
 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$.
 Let $c$ be $x$ restricted to $Y$.
 Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$.
-Then we have the glued field $x \cup (c\times I)$ on $M \cup (Y\times I)$.
+Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$.
 Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism.
-Then we say that $x$ is {\it extended isotopic} to $f(x \cup (c\times I))$.
+Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$.
 More generally, we define extended isotopy to be the equivalence relation on fields
 on $M$ generated by isotopy plus all instance of the above construction
 (for all appropriate $Y$ and $x$).
 \nn{should also say something about pseudo-isotopy}
 \medskip
 For top dimensional ($n$-dimensional) manifolds, we're actually interested
 in the linearized space of fields.
-By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is
+By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is
 the vector space of finite
 linear combinations of fields on $X$.
-If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$.
+If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$.
 Thus the restriction (to boundary) maps are well defined because we never
 take linear combinations of fields with differing boundary conditions.
 In some cases we don't linearize the default way; instead we take the
-spaces $\cC_l(X; a)$ to be part of the data for the system of fields.
+spaces $\lf(X; a)$ to be part of the data for the system of fields.
 In particular, for fields based on linear $n$-category pictures we linearize as follows.
-Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
+Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by
 obvious relations on 0-cell labels.
 More specifically, let $L$ be a cell decomposition of $X$
 and let $p$ be a 0-cell of $L$.
 Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that
 $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$.
 (Recall that 0-cells are labeled by $n$-morphisms.)
 To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism
 space determined by the labeling of the link of the 0-cell.
 (If the 0-cell were labeled, the label would live in this space.)
 We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell).
-We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the
+We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the
 above tensor products.
 \subsection{Local relations}
 \label{sec:local-relations}
-A {\it local relation} is a collection subspaces $U(B; c) \sub \c[\cC_l(B; c)]$
+A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$,
-(for all $n$-manifolds $B$ which are
+for all $n$-manifolds $B$ which are
-homeomorphic to the standard $n$-ball and
+homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$,
-all $c \in \cC(\bd B)$) satisfying the following properties.
+satisfying the following properties.
 \begin{enumerate}
 \item functoriality:
 $f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$
 \item local relations imply extended isotopy:
 if $x, y \in \cC(B; c)$ and $x$ is extended isotopic
 to $y$, then $x-y \in U(B; c)$.
 \item ideal with respect to gluing:
-if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\cup r \in U(B)$
+if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$
 \end{enumerate}
 See \cite{kw:tqft} for details.
-For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \cC_l(B; c)$,
+For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$,
 where $a$ and $b$ are maps (fields) which are homotopic rel boundary.
 For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map
-$\cC_l(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
+$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into
 domain and range.
 \nn{maybe examples of local relations before general def?}
 Given a system of fields and local relations, we define the skein space
 the $n$-manifold $Y$ modulo local relations.
 The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations
 is defined to be the dual of $A(Y; c)$.
 (See \cite{kw:tqft} or xxxx for details.)
+\nn{should expand above paragraph}
 The blob complex is in some sense the derived version of $A(Y; c)$.
 \subsection{The blob complex}
 \label{sec:blob-definition}
 Let $X$ be an $n$-manifold.
 Assume a fixed system of fields.
 In this section we will usually suppress boundary conditions on $X$ from the notation
-(e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$).
+(e.g. write $\lf(X)$ instead of $\lf(X; c)$).
 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0
 submanifold of $X$, then $X \setmin Y$ implicitly means the closure
 $\overline{X \setmin Y}$.
 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case.
-Define $\bc_0(X) = \cC_l(X)$.
+Define $\bc_0(X) = \lf(X)$.
-(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cC_l(X; c)$.
+(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$.
 We'll omit this sort of detail in the rest of this section.)
 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$.
 $\bc_1(X)$ is the space of all local relations that can be imposed on $\bc_0(X)$.
 More specifically, define a 1-blob diagram to consist of
 A nested 2-blob diagram consists of
 \begin{itemize}
 \item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$.
 \item A field $r \in \cC(X \setmin B_0; c_0)$
 (for some $c_0 \in \cC(\bd B_0)$).
-Let $r = r_1 \cup r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
+Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$
 (for some $c_1 \in \cC(B_1)$) and
 $r' \in \cC(X \setmin B_1; c_1)$.
 \item A local relation field $u_0 \in U(B_0; c_0)$.
 \end{itemize}
 Define $\bd(B_0, B_1, r, u_0) = (B_1, r', r_1u_0) - (B_0, r, u_0)$.

changeset 62	fdb1cd651fd2
parent 61	4093d7979c56
child 63	71b4e45f47f6