6 Let $X$ be an $n$-manifold. |
6 Let $X$ be an $n$-manifold. |
7 Let $\cC$ be a fixed system of fields and local relations. |
7 Let $\cC$ be a fixed system of fields and local relations. |
8 We'll assume it is enriched over \textbf{Vect}, and if it is not we can make it so by allowing finite |
8 We'll assume it is enriched over \textbf{Vect}, and if it is not we can make it so by allowing finite |
9 linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$. |
9 linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$. |
10 |
10 |
11 In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\lf(X)$ instead of $\lf(X; c)$. |
11 %In this section we will usually suppress boundary conditions on $X$ from the notation, e.g. by writing $\lf(X)$ instead of $\lf(X; c)$. |
12 |
12 |
13 We want to replace the quotient |
13 We want to replace the quotient |
14 \[ |
14 \[ |
15 A(X) \deq \lf(X) / U(X) |
15 A(X) \deq \lf(X) / U(X) |
16 \] |
16 \] |
17 of Definition \ref{defn:TQFT-invariant} with a resolution |
17 of Definition \ref{defn:TQFT-invariant} with a resolution |
18 \[ |
18 \[ |
19 \cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
19 \cdots \to \bc_2(X) \to \bc_1(X) \to \bc_0(X) . |
20 \] |
20 \] |
21 |
21 |
22 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. \todo{create a numbered definition for the general case} |
22 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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23 In fact, on the first pass we will intentionally describe the definition in a misleadingly simple way, then explain the technical difficulties, and finally give a cumbersome but complete definition in Definition \ref{defn:blob-definition}. If (we don't recommend it) you want to keep track of the ways in which this initial description is misleading, or you're reading through a second time to understand the technical difficulties, keep note that later we will give precise meanings to ``a ball in $X$'', ``nested'' and ``disjoint'', that are not quite the intuitive ones. Moreover some of the pieces into which we cut manifolds below are not themselves manifolds, and it requires special attention to define fields on these pieces. |
23 |
24 |
24 We of course define $\bc_0(X) = \lf(X)$. |
25 We of course define $\bc_0(X) = \lf(X)$. |
25 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
26 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$ for each $c \in \lf{\bdy X}$. |
26 We'll omit this sort of detail in the rest of this section.) |
27 We'll omit this sort of detail in the rest of this section.) |
27 In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. |
28 In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. |
28 |
29 |
29 We want the vector space $\bc_1(X)$ to capture `the space of all local relations that can be imposed on $\bc_0(X)$'. |
30 We want the vector space $\bc_1(X)$ to capture `the space of all local relations that can be imposed on $\bc_0(X)$'. |
30 Thus we say a $1$-blob diagram consists of: |
31 Thus we say a $1$-blob diagram consists of: |
31 \begin{itemize} |
32 \begin{itemize} |
32 \item An embedded closed ball (``blob") $B \sub X$. |
33 \item An closed ball in $X$ (``blob") $B \sub X$. |
33 \item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$. |
34 \item A boundary condition $c \in \cC(\bdy B) = \cC(\bd(X \setmin B))$. |
34 \item A field $r \in \cC(X \setmin B; c)$. |
35 \item A field $r \in \cC(X \setmin B; c)$. |
35 \item A local relation field $u \in U(B; c)$. |
36 \item A local relation field $u \in U(B; c)$. |
36 \end{itemize} |
37 \end{itemize} |
37 (See Figure \ref{blob1diagram}.) Since $c$ is implicitly determined by $u$ or $r$, we usually omit it from the notation. |
38 (See Figure \ref{blob1diagram}.) Since $c$ is implicitly determined by $u$ or $r$, we usually omit it from the notation. |
118 U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) |
119 U(B_1; c_1) \otimes \lf(B_2 \setmin B_1; c_1) \tensor \cC(X \setminus B_2; c_2) |
119 \right) . |
120 \right) . |
120 \end{eqnarray*} |
121 \end{eqnarray*} |
121 For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign |
122 For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign |
122 (rather than a new, linearly independent, 2-blob diagram). |
123 (rather than a new, linearly independent, 2-blob diagram). |
123 \noop{ |
124 |
124 \nn{Hmm, I think we should be doing this for nested blobs too -- |
125 Before describing the general case, note that when we say blobs are disjoint, we will only mean that their interiors are disjoint. Nested blobs may have boundaries that overlap, or indeed may coincide. |
125 we shouldn't force the linear indexing of the blobs to have anything to do with |
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126 the partial ordering by inclusion -- this is what happens below} |
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127 \nn{KW: I think adding that detail would only add distracting clutter, and the statement is true as written (in the sense that it yields a vector space isomorphic to the general def below} |
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128 } |
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129 |
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130 In order to precisely state the general definition, we'll need a suitable notion of cutting up a manifold into balls. |
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131 \begin{defn} |
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132 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds $M_0 \to M_1 \to \cdots \to M_m = X$ such that $M_0$ is a disjoint union of balls, and each $M_k$ is obtained from $M_{k-1}$ by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
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133 \end{defn} |
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134 |
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135 By `a ball in $X$' we don't literally mean a submanifold homeomorphic to a ball, but rather the image of a map from the pair $(B^n, S^{n-1})$ into $X$, which is an embedding on the interior. The boundary of a ball in $X$ is the image of a locally embedded $n{-}1$-sphere. \todo{examples, e.g. balls which actually look like an annulus, but we remember the boundary} |
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136 |
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137 \begin{defn} |
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138 A \emph{ball decomposition} of an $n$-manifold $X$ is a collection of balls in $X$, such that there exists some gluing decomposition $M_0 \to \cdots \to M_m = X$ so that the balls are the images of the components of $M_0$ in $X$. |
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139 \end{defn} |
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140 In particular, the union of all the balls in a ball decomposition comprises all of $X$. \todo{example} |
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141 |
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142 We'll now slightly restrict the possible configurations of blobs. |
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143 \begin{defn} |
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144 A configuration of $k$ blobs in $X$ is a collection of $k$ balls in $X$ such that there is some gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and each of the balls is the image of some connected component of one of the $M_k$. Such a gluing decomposition is \emph{compatible} with the configuration. |
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145 \end{defn} |
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146 In particular, this means that for any two blobs in a configuration of blobs in $X$, they either have disjoint interiors, or one blob is strictly contained in the other. We describe these as disjoint blobs and nested blobs. Note that nested blobs may have boundaries that overlap, or indeed coincide. Blobs may meet the boundary of $X$. |
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147 |
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148 Note that the boundaries of a configuration of $k$-blobs may cut up in manifold $X$ into components which are not themselves manifolds. \todo{example: the components between the boundaries of the balls may be pathological} |
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149 |
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150 \todo{this notion of configuration of blobs is the minimal one that allows gluing and engulfing} |
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151 |
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152 Now for the general case. |
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153 A $k$-blob diagram consists of |
126 A $k$-blob diagram consists of |
154 \begin{itemize} |
127 \begin{itemize} |
155 \item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
128 \item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
156 For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
129 For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
157 $B_i \sub B_j$ or $B_j \sub B_i$. |
130 $B_i \sub B_j$ or $B_j \sub B_i$. |
158 (The case $B_i = B_j$ is allowed. |
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159 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
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160 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
131 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
161 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
132 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
162 (These are implied by the data in the next bullets, so we usually |
133 (These are implied by the data in the next bullets, so we usually |
163 suppress them from the notation.) |
134 suppress them from the notation.) |
164 The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
135 The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
212 Thus we have a chain complex. |
183 Thus we have a chain complex. |
213 |
184 |
214 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. |
185 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. |
215 A homeomorphism acts in an obvious way on blobs and on fields. |
186 A homeomorphism acts in an obvious way on blobs and on fields. |
216 |
187 |
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188 At this point, it is time to pay back our debt and define certain notions more carefully. |
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189 |
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190 In order to precisely state the general definition, we'll need a suitable notion of cutting up a manifold into balls. |
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191 \begin{defn} |
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192 \label{defn:gluing-decomposition} |
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193 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds $M_0 \to M_1 \to \cdots \to M_m = X$ such that $M_0$ is a disjoint union of balls, and each $M_k$ is obtained from $M_{k-1}$ by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
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194 \end{defn} |
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195 Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is splittable along it if it is the image of a field on $M_0$. |
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196 |
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197 By `a ball in $X$' we don't literally mean a submanifold homeomorphic to a ball, but rather the image of a map from the pair $(B^n, S^{n-1})$ into $X$, which is an embedding on the interior. The boundary of a ball in $X$ is the image of a locally embedded $n{-}1$-sphere. \todo{examples, e.g. balls which actually look like an annulus, but we remember the boundary} |
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198 |
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199 \begin{defn} |
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200 \label{defn:ball-decomposition} |
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201 A \emph{ball decomposition} of an $n$-manifold $X$ is a collection of balls in $X$, such that there exists some gluing decomposition $M_0 \to \cdots \to M_m = X$ so that the balls are the images of the components of $M_0$ in $X$. |
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202 \end{defn} |
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203 In particular, the union of all the balls in a ball decomposition comprises all of $X$. \todo{example} |
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204 |
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205 We'll now slightly restrict the possible configurations of blobs. |
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206 \begin{defn} |
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207 \label{defn:configuration} |
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208 A configuration of $k$ blobs in $X$ is an ordered collection of $k$ balls in $X$ such that there is some gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and each of the balls is the image of some connected component of one of the $M_k$. Such a gluing decomposition is \emph{compatible} with the configuration. |
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209 \end{defn} |
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210 In particular, this implies what we said about blobs above: that for any two blobs in a configuration of blobs in $X$, they either have disjoint interiors, or one blob is strictly contained in the other. We describe these as disjoint blobs and nested blobs. Note that nested blobs may have boundaries that overlap, or indeed coincide. Blobs may meet the boundary of $X$. |
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211 |
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212 Note that the boundaries of a configuration of $k$ blobs may cut up the manifold $X$ into components which are not themselves manifolds. \todo{example: the components between the boundaries of the balls may be pathological} |
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213 |
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214 In the informal description above, in the definition of a $k$-blob diagram we asked for any collection of $k$ balls which were pairwise disjoint or nested. We now further insist that the balls are a configuration in the sense of Definition \ref{defn:configuration}. Also, we asked for a local relation on each twig blob, and a field on the complement of the twig blobs; this is unsatisfactory because that complement need not be a manifold. Thus, the official definition is |
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215 \begin{defn} |
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216 A $k$-blob diagram on $X$ consists of |
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217 \begin{itemize} |
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218 \item a configuration of $k$ blobs in $X$, |
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219 \item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
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220 \end{itemize} |
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221 such that |
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222 the restriction of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$. |
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223 \end{defn} |
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224 and |
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225 \begin{defn} |
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226 The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum of all configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, modulo identifying the vector spaces for configurations that only differ by a permutation of the balls by the sign of that permutation. The differential $bc_k(X) \to bc_{k-1}(X)$ is, as above, the signed sum of ways of forgetting one ball from the configuration, preserving the field $r$. |
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227 \end{defn} |
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228 We readily see that if a gluing decomposition is compatible with some configuration of blobs, then it is also compatible with any configuration obtained by forgetting some blobs, ensuring that the differential in fact lands in the space of $k{-}1$-blob diagrams. |
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229 A slight compensation to the complication of the official definition arising from attention to splitting is that the differential now just preserves the entire field $r$ without having to say anything about gluing together fields on smaller components. |
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230 |
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231 \todo{this notion of configuration of blobs is the minimal one that allows gluing and engulfing} |
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232 |
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233 |
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234 |
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235 |
217 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
236 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
218 to be the union of the blobs of $b$. |
237 to be the union of the blobs of $b$. |
219 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
238 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
220 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
239 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
221 |
240 |