author | scott@6e1638ff-ae45-0410-89bd-df963105f760 |
Fri, 30 Oct 2009 06:09:37 +0000 | |
changeset 150 | 24028ee41a91 |
parent 147 | db91d0a8ed75 |
child 160 | f38801a419f7 |
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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In this section we review the construction of TQFTs from ``topological fields". |
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For more details see \cite{kw:tqft}. |
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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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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 definiteness we |
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will stick with unoriented PL.) |
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%Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. |
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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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\nn{refer somewhere to my TQFT notes \cite{kw:tqft}} |
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Before finishing the definition of fields, we give two motivating examples |
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(actually, families of examples) of systems of fields. |
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The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps |
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from X to $B$. |
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The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be |
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the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
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$j$-morphisms of $C$. |
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One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
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This is described in more detail below. |
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Now for the rest of the definition of system of fields. |
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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 are 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 fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields 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, restriction to boundary, and orientation reversal. |
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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 $\pm Y$. |
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Using the boundary restriction, disjoint union, and (in one case) orientation reversal |
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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, orientation reversal, disjoint union). |
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Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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the gluing map is surjective. |
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From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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gluing surface, 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 Gluing with corners. |
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Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. |
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Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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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, gluing without corners, and (in one case) orientation reversal |
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maps, 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, orientation reversal, disjoint union). |
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Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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the gluing map is surjective. |
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From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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gluing surface, 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 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 |
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104 |
covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
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105 |
\end{enumerate} |
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106 |
|
141 | 107 |
There are two notations we commonly use for gluing. |
108 |
One is |
|
109 |
\[ |
|
110 |
x\sgl \deq \gl(x) \in \cC(X\sgl) , |
|
111 |
\] |
|
112 |
for $x\in\cC(X)$. |
|
113 |
The other is |
|
114 |
\[ |
|
115 |
x_1\bullet x_2 \deq \gl(x_1\otimes x_2) \in \cC(X\sgl) , |
|
116 |
\] |
|
117 |
in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. |
|
100
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118 |
|
141 | 119 |
\medskip |
120 |
||
121 |
Using the functoriality and $\cdot\times I$ properties above, together |
|
100
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122 |
with boundary collar homeomorphisms of manifolds, we can define the notion of |
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123 |
{\it extended isotopy}. |
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124 |
Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
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125 |
of $\bd M$. |
132 | 126 |
Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. |
100
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127 |
Let $c$ be $x$ restricted to $Y$. |
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128 |
Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
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129 |
Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
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130 |
Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
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131 |
Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
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132 |
More generally, we define extended isotopy to be the equivalence relation on fields |
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133 |
on $M$ generated by isotopy plus all instance of the above construction |
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134 |
(for all appropriate $Y$ and $x$). |
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135 |
|
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136 |
\nn{should also say something about pseudo-isotopy} |
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137 |
|
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138 |
|
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139 |
\nn{remark that if top dimensional fields are not already linear |
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140 |
then we will soon linearize them(?)} |
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141 |
|
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142 |
We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
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143 |
by $n$-category morphisms. |
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144 |
|
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145 |
Given an $n$-category $C$ with the right sort of duality |
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146 |
(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
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147 |
we can construct a system of fields as follows. |
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148 |
Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
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149 |
with codimension $i$ cells labeled by $i$-morphisms of $C$. |
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150 |
We'll spell this out for $n=1,2$ and then describe the general case. |
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151 |
|
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152 |
If $X$ has boundary, we require that the cell decompositions are in general |
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153 |
position with respect to the boundary --- the boundary intersects each cell |
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154 |
transversely, so cells meeting the boundary are mere half-cells. |
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155 |
|
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156 |
Put another way, the cell decompositions we consider are dual to standard cell |
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157 |
decompositions of $X$. |
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158 |
|
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159 |
We will always assume that our $n$-categories have linear $n$-morphisms. |
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160 |
|
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161 |
For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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162 |
an object (0-morphism) of the 1-category $C$. |
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163 |
A field on a 1-manifold $S$ consists of |
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164 |
\begin{itemize} |
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165 |
\item A cell decomposition of $S$ (equivalently, a finite collection |
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166 |
of points in the interior of $S$); |
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167 |
\item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
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168 |
by an object (0-morphism) of $C$; |
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169 |
\item a transverse orientation of each 0-cell, thought of as a choice of |
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170 |
``domain" and ``range" for the two adjacent 1-cells; and |
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171 |
\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
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172 |
domain and range determined by the transverse orientation and the labelings of the 1-cells. |
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173 |
\end{itemize} |
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174 |
|
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175 |
If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
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176 |
of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
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177 |
interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
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178 |
of the algebra. |
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179 |
|
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180 |
\medskip |
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181 |
|
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182 |
For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) |
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183 |
that are common in the literature. |
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184 |
We describe these carefully here. |
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185 |
|
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186 |
A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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187 |
an object of the 2-category $C$. |
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188 |
A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
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189 |
A field on a 2-manifold $Y$ consists of |
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190 |
\begin{itemize} |
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191 |
\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
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192 |
that each component of the complement is homeomorphic to a disk); |
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193 |
\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
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194 |
by a 0-morphism of $C$; |
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195 |
\item a transverse orientation of each 1-cell, thought of as a choice of |
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196 |
``domain" and ``range" for the two adjacent 2-cells; |
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197 |
\item a labeling of each 1-cell by a 1-morphism of $C$, with |
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198 |
domain and range determined by the transverse orientation of the 1-cell |
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199 |
and the labelings of the 2-cells; |
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200 |
\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
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201 |
of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
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202 |
to $\pm 1 \in S^1$; and |
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203 |
\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
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204 |
determined by the labelings of the 1-cells and the parameterizations of the previous |
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205 |
bullet. |
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206 |
\end{itemize} |
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207 |
\nn{need to say this better; don't try to fit everything into the bulleted list} |
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208 |
|
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209 |
For general $n$, a field on a $k$-manifold $X^k$ consists of |
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210 |
\begin{itemize} |
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\item A cell decomposition of $X$; |
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\item an explicit general position homeomorphism from the link of each $j$-cell |
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to the boundary of the standard $(k-j)$-dimensional bihedron; and |
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\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
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domain and range determined by the labelings of the link of $j$-cell. |
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\end{itemize} |
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217 |
|
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%\nn{next definition might need some work; I think linearity relations should |
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%be treated differently (segregated) from other local relations, but I'm not sure |
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%the next definition is the best way to do it} |
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|
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\medskip |
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|
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For top dimensional ($n$-dimensional) manifolds, we're actually interested |
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in the linearized space of fields. |
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By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
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the vector space of finite |
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linear combinations of fields on $X$. |
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If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
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Thus the restriction (to boundary) maps are well defined because we never |
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take linear combinations of fields with differing boundary conditions. |
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|
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In some cases we don't linearize the default way; instead we take the |
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spaces $\lf(X; a)$ to be part of the data for the system of fields. |
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In particular, for fields based on linear $n$-category pictures we linearize as follows. |
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Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
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obvious relations on 0-cell labels. |
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More specifically, let $L$ be a cell decomposition of $X$ |
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and let $p$ be a 0-cell of $L$. |
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Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
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$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
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Then the subspace $K$ is generated by things of the form |
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$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
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to infer the meaning of $\alpha_{\lambda c + d}$. |
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Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
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|
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\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
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will do something similar below; in general, whenever a label lives in a linear |
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space we do something like this; ? say something about tensor |
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product of all the linear label spaces? Yes:} |
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|
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For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
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Define an ``almost-field" to be a field without labels on the 0-cells. |
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(Recall that 0-cells are labeled by $n$-morphisms.) |
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To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
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space determined by the labeling of the link of the 0-cell. |
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(If the 0-cell were labeled, the label would live in this space.) |
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We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
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We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
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above tensor products. |
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|
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|
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|
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\subsection{Local relations} |
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\label{sec:local-relations} |
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|
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|
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A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
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for all $n$-manifolds $B$ which are |
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homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
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satisfying the following properties. |
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\begin{enumerate} |
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\item functoriality: |
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$f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
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\item local relations imply extended isotopy: |
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if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
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to $y$, then $x-y \in U(B; c)$. |
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\item ideal with respect to gluing: |
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if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ |
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\end{enumerate} |
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See \cite{kw:tqft} for details. |
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|
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|
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For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, |
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where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
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For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
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$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
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domain and range. |
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\nn{maybe examples of local relations before general def?} |
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132 | 293 |
\subsection{Constructing a TQFT} |
294 |
||
139 | 295 |
In this subsection we briefly review the construction of a TQFT from a system of fields and local relations. |
296 |
(For more details, see \cite{kw:tqft}.) |
|
297 |
||
298 |
Let $W$ be an $n{+}1$-manifold. |
|
299 |
We can think of the path integral $Z(W)$ as assigning to each |
|
300 |
boundary condition $x\in \cC(\bd W)$ a complex number $Z(W)(x)$. |
|
301 |
In other words, $Z(W)$ lies in $\c^{\lf(\bd W)}$, the vector space of linear |
|
302 |
maps $\lf(\bd W)\to \c$. |
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303 |
||
304 |
The locality of the TQFT implies that $Z(W)$ in fact lies in a subspace |
|
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$Z(\bd W) \sub \c^{\lf(\bd W)}$ defined by local projections. |
|
306 |
The linear dual to this subspace, $A(\bd W) = Z(\bd W)^*$, |
|
307 |
can be thought of as finite linear combinations of fields modulo local relations. |
|
308 |
(In other words, $A(\bd W)$ is a sort of generalized skein module.) |
|
309 |
This is the motivation behind the definition of fields and local relations above. |
|
132 | 310 |
|
139 | 311 |
In more detail, let $X$ be an $n$-manifold. |
312 |
%To harmonize notation with the next section, |
|
313 |
%let $\bc_0(X)$ be the vector space of finite linear combinations of fields on $X$, so |
|
314 |
%$\bc_0(X) = \lf(X)$. |
|
315 |
Define $U(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$; |
|
316 |
$U(X)$ is generated by things of the form $u\bullet r$, where |
|
317 |
$u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
|
318 |
Define |
|
319 |
\[ |
|
320 |
A(X) \deq \lf(X) / U(X) . |
|
321 |
\] |
|
322 |
(The blob complex, defined in the next section, |
|
323 |
is in some sense the derived version of $A(X)$.) |
|
324 |
If $X$ has boundary we can similarly define $A(X; c)$ for each |
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325 |
boundary condition $c\in\cC(\bd X)$. |
|
100
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139 | 327 |
The above construction can be extended to higher codimensions, assigning |
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a $k$-category $A(Y)$ to an $n{-}k$-manifold $Y$, for $0 \le k \le n$. |
|
329 |
These invariants fit together via actions and gluing formulas. |
|
330 |
We describe only the case $k=1$ below. |
|
331 |
(The construction of the $n{+}1$-dimensional part of the theory (the path integral) |
|
332 |
requires that the starting data (fields and local relations) satisfy additional |
|
333 |
conditions. |
|
334 |
We do not assume these conditions here, so when we say ``TQFT" we mean a decapitated TQFT |
|
335 |
that lacks its $n{+}1$-dimensional part.) |
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139 | 337 |
Let $Y$ be an $n{-}1$-manifold. |
338 |
Define a (linear) 1-category $A(Y)$ as follows. |
|
339 |
The objects of $A(Y)$ are $\cC(Y)$. |
|
340 |
The morphisms from $a$ to $b$ are $A(Y\times I; a, b)$, where $a$ and $b$ label the two boundary components of the cylinder $Y\times I$. |
|
341 |
Composition is given by gluing of cylinders. |
|
100
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342 |
|
139 | 343 |
Let $X$ be an $n$-manifold with boundary and consider the collection of vector spaces |
344 |
$A(X; \cdot) \deq \{A(X; c)\}$ where $c$ ranges through $\cC(\bd X)$. |
|
345 |
This collection of vector spaces affords a representation of the category $A(\bd X)$, where |
|
346 |
the action is given by gluing a collar $\bd X\times I$ to $X$. |
|
347 |
||
348 |
Given a splitting $X = X_1 \cup_Y X_2$ of a closed $n$-manifold $X$ along an $n{-}1$-manifold $Y$, |
|
349 |
we have left and right actions of $A(Y)$ on $A(X_1; \cdot)$ and $A(X_2; \cdot)$. |
|
350 |
The gluing theorem for $n$-manifolds states that there is a natural isomorphism |
|
351 |
\[ |
|
352 |
A(X) \cong A(X_1; \cdot) \otimes_{A(Y)} A(X_2; \cdot) . |
|
353 |
\] |
|
100
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354 |
|
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355 |
|
132 | 356 |
\section{The blob complex} |
100
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357 |
\label{sec:blob-definition} |
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358 |
|
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359 |
Let $X$ be an $n$-manifold. |
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360 |
Assume a fixed system of fields and local relations. |
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361 |
In this section we will usually suppress boundary conditions on $X$ from the notation |
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362 |
(e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
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363 |
|
140 | 364 |
We want to replace the quotient |
365 |
\[ |
|
366 |
A(X) \deq \lf(X) / U(X) |
|
367 |
\] |
|
368 |
of the previous section with a resolution |
|
369 |
\[ |
|
370 |
\cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
|
371 |
\] |
|
100
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372 |
|
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373 |
We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
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374 |
|
140 | 375 |
We of course define $\bc_0(X) = \lf(X)$. |
100
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376 |
(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
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377 |
We'll omit this sort of detail in the rest of this section.) |
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378 |
In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
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379 |
|
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380 |
$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
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381 |
Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear |
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382 |
combinations of 1-blob diagrams, where a 1-blob diagram to consists of |
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|
383 |
\begin{itemize} |
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384 |
\item An embedded closed ball (``blob") $B \sub X$. |
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385 |
\item A field $r \in \cC(X \setmin B; c)$ |
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|
386 |
(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
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387 |
\item A local relation field $u \in U(B; c)$ |
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388 |
(same $c$ as previous bullet). |
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389 |
\end{itemize} |
140 | 390 |
(See Figure \ref{blob1diagram}.) |
391 |
\begin{figure}[!ht]\begin{equation*} |
|
392 |
\mathfig{.9}{tempkw/blob1diagram} |
|
393 |
\end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
|
100
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394 |
In order to get the linear structure correct, we (officially) define |
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395 |
\[ |
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396 |
\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
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|
397 |
\] |
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398 |
The first direct sum is indexed by all blobs $B\subset X$, and the second |
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399 |
by all boundary conditions $c \in \cC(\bd B)$. |
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400 |
Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
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401 |
|
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402 |
Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
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|
403 |
\[ |
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404 |
(B, u, r) \mapsto u\bullet r, |
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405 |
\] |
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|
406 |
where $u\bullet r$ denotes the linear |
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407 |
combination of fields on $X$ obtained by gluing $u$ to $r$. |
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408 |
In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
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|
409 |
just erasing the blob from the picture |
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|
410 |
(but keeping the blob label $u$). |
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|
411 |
|
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|
412 |
Note that the skein space $A(X)$ |
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|
413 |
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
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|
414 |
|
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|
415 |
$\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the |
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|
416 |
local relations encoded in $\bc_1(X)$. |
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|
417 |
More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
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|
418 |
2-blob diagrams, of which there are two types, disjoint and nested. |
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|
419 |
|
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|
420 |
A disjoint 2-blob diagram consists of |
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|
421 |
\begin{itemize} |
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422 |
\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. |
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|
423 |
\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
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|
424 |
(where $c_i \in \cC(\bd B_i)$). |
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|
425 |
\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
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|
426 |
\end{itemize} |
140 | 427 |
(See Figure \ref{blob2ddiagram}.) |
428 |
\begin{figure}[!ht]\begin{equation*} |
|
429 |
\mathfig{.9}{tempkw/blob2ddiagram} |
|
430 |
\end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} |
|
100
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431 |
We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
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|
432 |
reversing the order of the blobs changes the sign. |
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433 |
Define $\bd(B_0, B_1, u_0, u_1, r) = |
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434 |
(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
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435 |
In other words, the boundary of a disjoint 2-blob diagram |
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|
436 |
is the sum (with alternating signs) |
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437 |
of the two ways of erasing one of the blobs. |
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438 |
It's easy to check that $\bd^2 = 0$. |
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439 |
|
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|
440 |
A nested 2-blob diagram consists of |
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|
441 |
\begin{itemize} |
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442 |
\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
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|
443 |
\item A field $r \in \cC(X \setmin B_0; c_0)$ |
132 | 444 |
(for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$. |
100
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445 |
\item A local relation field $u_0 \in U(B_0; c_0)$. |
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446 |
\end{itemize} |
140 | 447 |
(See Figure \ref{blob2ndiagram}.) |
448 |
\begin{figure}[!ht]\begin{equation*} |
|
449 |
\mathfig{.9}{tempkw/blob2ndiagram} |
|
450 |
\end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
|
100
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451 |
Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
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452 |
(for some $c_1 \in \cC(B_1)$) and |
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453 |
$r' \in \cC(X \setmin B_1; c_1)$. |
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454 |
Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
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455 |
Note that the requirement that |
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|
456 |
local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. |
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|
457 |
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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changeset
|
458 |
sum of the two ways of erasing one of the blobs. |
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|
459 |
If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
460 |
It is again easy to check that $\bd^2 = 0$. |
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|
461 |
|
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
462 |
As with the 1-blob diagrams, in order to get the linear structure correct it is better to define |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
463 |
(officially) |
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|
464 |
\begin{eqnarray*} |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
465 |
\bc_2(X) & \deq & |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
466 |
\left( |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
467 |
\bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
468 |
U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1) |
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|
469 |
\right) \\ |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
470 |
&& \bigoplus \left( |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
471 |
\bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} |
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|
472 |
U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
473 |
\right) . |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diff
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|
474 |
\end{eqnarray*} |
132 | 475 |
The final $\lf(X\setmin B_0; c_0)$ above really means fields splittable along $\bd B_1$, |
100
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
476 |
but we didn't feel like introducing a notation for that. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
477 |
For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
478 |
(rather than a new, linearly independent 2-blob diagram). |
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|
479 |
|
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
480 |
Now for the general case. |
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|
481 |
A $k$-blob diagram consists of |
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|
482 |
\begin{itemize} |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
483 |
\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
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|
484 |
For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
485 |
$B_i \sub B_j$ or $B_j \sub B_i$. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diff
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|
486 |
(The case $B_i = B_j$ is allowed. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
487 |
If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
488 |
If a blob has no other blobs strictly contained in it, we call it a twig blob. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
489 |
\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
490 |
(These are implied by the data in the next bullets, so we usually |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
491 |
suppress them from the notation.) |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
492 |
$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
493 |
if the latter space is not empty. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
494 |
\item A field $r \in \cC(X \setmin B^t; c^t)$, |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
495 |
where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
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|
496 |
is determined by the $c_i$'s. |
132 | 497 |
$r$ is required to be splittable along the boundaries of all blobs, twigs or not. |
100
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|
498 |
\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
499 |
where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
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|
500 |
If $B_i = B_j$ then $u_i = u_j$. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
501 |
\end{itemize} |
140 | 502 |
(See Figure \ref{blobkdiagram}.) |
503 |
\begin{figure}[!ht]\begin{equation*} |
|
504 |
\mathfig{.9}{tempkw/blobkdiagram} |
|
505 |
\end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
|
100
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|
506 |
|
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diff
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|
507 |
If two blob diagrams $D_1$ and $D_2$ |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
508 |
differ only by a reordering of the blobs, then we identify |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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|
509 |
$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
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parents:
diff
changeset
|
510 |
|
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
511 |
$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
512 |
As before, the official definition is in terms of direct sums |
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diff
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|
513 |
of tensor products: |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
514 |
\[ |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diff
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|
515 |
\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diff
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|
516 |
\left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
517 |
\] |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diff
changeset
|
518 |
Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
519 |
$\overline{c}$ runs over all boundary conditions, again as described above. |
132 | 520 |
$j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. |
100
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|
521 |
|
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
522 |
The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diff
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|
523 |
Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
524 |
Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
525 |
If $B_j$ is not a twig blob, this involves only decrementing |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
526 |
the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
527 |
If $B_j$ is a twig blob, we have to assign new local relation labels |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
528 |
if removing $B_j$ creates new twig blobs. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
529 |
If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
530 |
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
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|
531 |
Finally, define |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
532 |
\eq{ |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diff
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|
533 |
\bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
534 |
} |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diff
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|
535 |
The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diff
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|
536 |
Thus we have a chain complex. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
537 |
|
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diff
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|
538 |
\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
539 |
|
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
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diff
changeset
|
540 |
\nn{?? remark about dendroidal sets} |
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|
541 |
|
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|
542 |