author | Scott Morrison <scott@tqft.net> |
Mon, 22 Nov 2010 13:40:40 -0800 | |
changeset 661 | 6345c3679795 |
parent 522 | a60c035e53bd |
child 697 | 7843262cd782 |
permissions | -rw-r--r-- |
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%!TEX root = ../blob1.tex |
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\section{TQFTs via fields} |
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\label{sec:fields} |
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\label{sec:tqftsviafields} |
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In this section we review the construction of TQFTs from fields and local relations. |
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For more details see \cite{kw:tqft}. |
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For our purposes, a TQFT is {\it defined} to be something which arises |
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from this construction. |
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This is an alternative to the more common definition of a TQFT |
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as a functor on cobordism categories satisfying various conditions. |
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A fully local (``down to points") version of the cobordism-functor TQFT definition |
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should be equivalent to the fields-and-local-relations definition. |
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A system of fields is very closely related to an $n$-category. |
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In one direction, Example \ref{ex:traditional-n-categories(fields)} |
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shows how to construct a system of fields from a (traditional) $n$-category. |
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We do this in detail for $n=1,2$ (\S\ref{sec:example:traditional-n-categories(fields)}) |
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and more informally for general $n$. |
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In the other direction, |
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our preferred definition of an $n$-category in \S\ref{sec:ncats} is essentially |
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just a system of fields restricted to balls of dimensions 0 through $n$; |
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one could call this the ``local" part of a system of fields. |
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Since this section is intended primarily to motivate |
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the blob complex construction of \S\ref{sec:blob-definition}, |
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we suppress some technical details. |
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In \S\ref{sec:ncats} the analogous details are treated more carefully. |
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\medskip |
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We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
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submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
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$\overline{X \setmin Y}$. |
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\subsection{Systems of fields} |
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\label{ss:syst-o-fields} |
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Let $\cM_k$ denote the category with objects |
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unoriented PL manifolds of dimension |
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$k$ and morphisms homeomorphisms. |
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(We could equally well work with a different category of manifolds --- |
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oriented, topological, smooth, spin, etc. --- but for simplicity we |
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will stick with unoriented PL.) |
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Fix a symmetric monoidal category $\cS$. |
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Fields on $n$-manifolds will be enriched over $\cS$. |
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Good examples to keep in mind are $\cS = \Set$ or $\cS = \Vect$. |
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The presentation here requires that the objects of $\cS$ have an underlying set, |
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but this could probably be avoided if desired. |
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A $n$-dimensional {\it system of fields} in $\cS$ |
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is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
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together with some additional data and satisfying some additional conditions, all specified below. |
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Before finishing the definition of fields, we give two motivating examples of systems of fields. |
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\begin{example} |
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\label{ex:maps-to-a-space(fields)} |
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Fix a target space $T$, and let $\cC(X)$ be the set of continuous maps |
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from $X$ to $T$. |
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\end{example} |
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\begin{example} |
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\label{ex:traditional-n-categories(fields)} |
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Fix an $n$-category $C$, and let $\cC(X)$ be |
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the set of embedded cell complexes in $X$ with codimension-$j$ cells labeled by |
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$j$-morphisms of $C$. |
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One can think of such embedded cell complexes as dual to pasting diagrams for $C$. |
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This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}. |
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\end{example} |
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Now for the rest of the definition of system of fields. |
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(Readers desiring a more precise definition should refer to \S\ref{ss:n-cat-def} |
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and replace $k$-balls with $k$-manifolds.) |
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\begin{enumerate} |
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\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
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and these maps comprise a natural |
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transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
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For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
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$\cC(X)$ which restricts to $c$. |
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In this context, we will call $c$ a boundary condition. |
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\item The subset $\cC_n(X;c)$ of top-dimensional fields |
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with a given boundary condition is an object in our symmetric monoidal category $\cS$. |
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(This condition is of course trivial when $\cS = \Set$.) |
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If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), |
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then this extra structure is considered part of the definition of $\cC_n$. |
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Any maps mentioned below between fields on $n$-manifolds must be morphisms in $\cS$. |
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\item $\cC_k$ is compatible with the symmetric monoidal |
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structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
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compatibly with homeomorphisms and restriction to boundary. |
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We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
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restriction maps. |
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\item Gluing without corners. |
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Let $\bd X = Y \du Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
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Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$. |
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Using the boundary restriction and disjoint union |
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maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
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copies of $Y$ in $\bd X$. |
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Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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Then (here's the axiom/definition part) there is an injective ``gluing" map |
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\[ |
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\Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
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\] |
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and this gluing map is compatible with all of the above structure (actions |
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of homeomorphisms, boundary restrictions, disjoint union). |
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Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity |
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and collaring maps, |
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the gluing map is surjective. |
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We say that fields on $X\sgl$ in the image of the gluing map |
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are transverse to $Y$ or splittable along $Y$. |
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\item Gluing with corners. |
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Let $\bd X = (Y \du Y) \cup W$, where the two copies of $Y$ |
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are disjoint from each other and $\bd(Y\du Y) = \bd W$. |
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Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$ |
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(Figure \ref{fig:gluing-with-corners}). |
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\begin{figure}[t] |
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\begin{center} |
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\begin{tikzpicture} |
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\node(A) at (-4,0) { |
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\begin{tikzpicture}[scale=.8, fill=blue!15!white] |
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\filldraw[line width=1.5pt] (-.4,1) .. controls +(-1,-.1) and +(-1,0) .. (0,-1) |
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.. controls +(1,0) and +(1,-.1) .. (.4,1) -- (.4,3) |
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.. controls +(3,-.4) and +(3,0) .. (0,-3) |
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.. controls +(-3,0) and +(-3,-.1) .. (-.4,3) -- cycle; |
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\node at (0,-2) {$X$}; |
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\node (W) at (-2.7,-2) {$W$}; |
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\node (Y1) at (-1.2,3.5) {$Y$}; |
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\node (Y2) at (1.4,3.5) {$Y$}; |
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\node[outer sep=2.3] (y1e) at (-.4,2) {}; |
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\node[outer sep=2.3] (y2e) at (.4,2) {}; |
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\node (we1) at (-2.2,-1.1) {}; |
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\node (we2) at (-.6,-.7) {}; |
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\draw[->] (Y1) -- (y1e); |
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\draw[->] (Y2) -- (y2e); |
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\draw[->] (W) .. controls +(0,.5) and +(-.5,-.2) .. (we1); |
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\draw[->] (W) .. controls +(.5,0) and +(-.2,-.5) .. (we2); |
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\end{tikzpicture} |
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}; |
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\node(B) at (4,0) { |
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\begin{tikzpicture}[scale=.8, fill=blue!15!white] |
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\fill (0,1) .. controls +(-1,0) and +(-1,0) .. (0,-1) |
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.. controls +(1,0) and +(1,0) .. (0,1) -- (0,3) |
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.. controls +(3,0) and +(3,0) .. (0,-3) |
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.. controls +(-3,0) and +(-3,0) .. (0,3) -- cycle; |
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\draw[line width=1.5pt] (0,1) .. controls +(-1,0) and +(-1,0) .. (0,-1) |
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.. controls +(1,0) and +(1,0) .. (0,1); |
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\draw[line width=1.5pt] (0,3) .. controls +(3,0) and +(3,0) .. (0,-3) |
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.. controls +(-3,0) and +(-3,0) .. (0,3); |
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\draw[line width=.5pt, black!65!white] (0,1) -- (0,3); |
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\node at (0,-2) {$X\sgl$}; |
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\node (W) at (2.7,-2) {$W\sgl$}; |
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\node (we1) at (2.2,-1.1) {}; |
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\node (we2) at (.6,-.7) {}; |
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\draw[->] (W) .. controls +(0,.5) and +(.5,-.2) .. (we1); |
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\draw[->] (W) .. controls +(-.5,0) and +(.2,-.5) .. (we2); |
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\end{tikzpicture} |
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}; |
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\draw[->, red!50!green, line width=2pt] (A) -- node[above, black] {glue} (B); |
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\end{tikzpicture} |
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\end{center} |
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\caption{Gluing with corners} |
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\label{fig:gluing-with-corners} |
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\end{figure} |
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Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
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(without corners) along two copies of $\bd Y$. |
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Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let |
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$c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
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Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
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(This restriction map uses the gluing without corners map above.) |
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Using the boundary restriction and gluing without corners maps, |
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we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
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copies of $Y$ in $\bd X$. |
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Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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Then (here's the axiom/definition part) there is an injective ``gluing" map |
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\[ |
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\Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , |
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\] |
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and this gluing map is compatible with all of the above structure (actions |
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of homeomorphisms, boundary restrictions, disjoint union). |
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Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity |
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and collaring maps, |
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the gluing map is surjective. |
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We say that fields in the image of the gluing map |
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are transverse to $Y$ or splittable along $Y$. |
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\item Product fields. |
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There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
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$c \mapsto c\times I$. |
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These maps comprise a natural transformation of functors, and commute appropriately |
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with all the structure maps above (disjoint union, boundary restriction, etc.). |
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Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
|
199 |
covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
|
200 |
\end{enumerate} |
|
201 |
||
202 |
There are two notations we commonly use for gluing. |
|
203 |
One is |
|
204 |
\[ |
|
205 |
x\sgl \deq \gl(x) \in \cC(X\sgl) , |
|
206 |
\] |
|
207 |
for $x\in\cC(X)$. |
|
208 |
The other is |
|
209 |
\[ |
|
210 |
x_1\bullet x_2 \deq \gl(x_1\otimes x_2) \in \cC(X\sgl) , |
|
211 |
\] |
|
212 |
in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. |
|
213 |
||
214 |
\medskip |
|
215 |
||
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98b8559b0b7a
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Kevin Walker <kevin@canyon23.net>
parents:
409
diff
changeset
|
216 |
Using the functoriality and product field properties above, together |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
217 |
with boundary collar homeomorphisms of manifolds, we can define |
98b8559b0b7a
starting to work on tqdftreview.tex
Kevin Walker <kevin@canyon23.net>
parents:
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diff
changeset
|
218 |
{\it collar maps} $\cC(M)\to \cC(M)$. |
215 | 219 |
Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
220 |
of $\bd M$. |
|
221 |
Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. |
|
222 |
Let $c$ be $x$ restricted to $Y$. |
|