133 \end{eqnarray*} |
133 \end{eqnarray*} |
134 % __ (already said this above) |
134 % __ (already said this above) |
135 %For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign |
135 %For the disjoint blobs, reversing the ordering of $B_1$ and $B_2$ introduces a minus sign |
136 %(rather than a new, linearly independent, 2-blob diagram). |
136 %(rather than a new, linearly independent, 2-blob diagram). |
137 |
137 |
138 |
138 \medskip |
139 |
139 |
140 |
140 Roughly, $\bc_k(X)$ is generated by configurations of $k$ blobs, pairwise disjoint or nested. |
141 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. |
141 The boundary is the alternating sum of erasing one of the blobs. |
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142 In order to describe this general case in full detail, we must give a more precise description of |
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143 which configurations of balls inside $X$ we permit. |
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144 These configurations are generated by two operations: |
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145 \begin{itemize} |
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146 \item For any (possibly empty) configuration of blobs on an $n$-ball $D$, we can add |
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147 $D$ itself as an outermost blob. |
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148 (This is used in the proof of Proposition \ref{bcontract}.) |
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149 \item If $X'$ is obtained from $X$ by gluing, then any permissible configuration of blobs |
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150 on $X$ gives rise to a permissible configuration on $X'$. |
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151 (This is necessary for Proposition \ref{blob-gluing}.) |
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152 \end{itemize} |
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153 Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not |
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154 a manifold. |
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155 Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
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156 |
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157 |
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158 %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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159 \begin{defn} |
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160 \label{defn:gluing-decomposition} |
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161 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
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162 $M_0 \to M_1 \to \cdots \to M_m = X$ such that each $M_k$ is obtained from $M_{k-1}$ |
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163 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
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164 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
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165 \end{defn} |
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166 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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167 |
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168 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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169 \nn{not all balls in $X$ can arise via gluing, but I suppose that's OK.} |
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170 |
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171 \nn{do we need this next def?} |
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172 \begin{defn} |
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173 \label{defn:ball-decomposition} |
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174 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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175 \end{defn} |
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176 In particular, the union of all the balls in a ball decomposition comprises all of $X$. \todo{example} |
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177 |
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178 We'll now slightly restrict the possible configurations of blobs. |
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179 \begin{defn} |
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180 \label{defn:configuration} |
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181 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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182 \end{defn} |
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183 In particular, this implies what we said about blobs above: |
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184 that for any two blobs in a configuration of blobs in $X$, |
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185 they either have disjoint interiors, or one blob is contained in the other. |
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186 We describe these as disjoint blobs and nested blobs. |
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187 Note that nested blobs may have boundaries that overlap, or indeed coincide. |
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188 Blobs may meet the boundary of $X$. |
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189 |
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190 % (already said above) |
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191 %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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192 |
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193 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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194 \begin{defn} |
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195 \label{defn:blob-diagram} |
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196 A $k$-blob diagram on $X$ consists of |
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197 \begin{itemize} |
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198 \item a configuration of $k$ blobs in $X$, |
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199 \item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
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200 \end{itemize} |
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201 such that |
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202 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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203 \end{defn} |
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204 and |
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205 \begin{defn} |
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206 \label{defn:blobs} |
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207 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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208 \end{defn} |
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209 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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210 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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211 |
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212 |
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213 |
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214 |
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215 |
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216 |
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217 |
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218 \nn{should merge this informal def with official one above} |
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219 |
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220 Before describing the general case, note that when we say blobs are disjoint, |
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221 we will only mean that their interiors are disjoint. |
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222 Nested blobs may have boundaries that overlap, or indeed may coincide. |
142 A $k$-blob diagram consists of |
223 A $k$-blob diagram consists of |
143 \begin{itemize} |
224 \begin{itemize} |
144 \item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
225 \item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
145 For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
226 For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
146 $B_i \sub B_j$ or $B_j \sub B_i$. |
227 $B_i \sub B_j$ or $B_j \sub B_i$. |
151 The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
232 The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
152 if the latter space is not empty. |
233 if the latter space is not empty. |
153 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
234 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
154 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
235 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
155 is determined by the $c_i$'s. |
236 is determined by the $c_i$'s. |
156 The field $r$ is required to be splittable along the boundaries of all blobs, twigs or not. (This is equivalent to asking for a field on of the components of $X \setmin B^t$.) |
237 The field $r$ is required to be splittable along the boundaries of all blobs, twigs or not. |
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238 (This is equivalent to asking for a field on of the components of $X \setmin B^t$.) |
157 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$. |
239 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$. |
158 If $B_i = B_j$ then $u_i = u_j$. |
240 If $B_i = B_j$ then $u_i = u_j$. |
159 \end{itemize} |
241 \end{itemize} |
160 (See Figure \ref{blobkdiagram}.) |
242 (See Figure \ref{blobkdiagram}.) |
161 \begin{figure}[t]\begin{equation*} |
243 \begin{figure}[t]\begin{equation*} |
185 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
269 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
186 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
270 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
187 If $B_j$ is not a twig blob, this involves only decrementing |
271 If $B_j$ is not a twig blob, this involves only decrementing |
188 the indices of blobs $B_{j+1},\ldots,B_{k}$. |
272 the indices of blobs $B_{j+1},\ldots,B_{k}$. |
189 If $B_j$ is a twig blob, we have to assign new local relation labels |
273 If $B_j$ is a twig blob, we have to assign new local relation labels |
190 if removing $B_j$ creates new twig blobs. \todo{Have to say what happens when no new twig blobs are created} |
274 if removing $B_j$ creates new twig blobs. |
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275 \todo{Have to say what happens when no new twig blobs are created} |
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276 \nn{KW: I'm confused --- why isn't it OK as written?} |
191 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
277 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
192 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
278 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
193 Finally, define |
279 Finally, define |
194 \eq{ |
280 \eq{ |
195 \bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). |
281 \bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). |
196 } |
282 } |
197 The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. |
283 The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. |
198 Thus we have a chain complex. |
284 Thus we have a chain complex. |
199 |
285 |
200 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. |
286 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, |
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287 is immediately obvious from the definition. |
201 A homeomorphism acts in an obvious way on blobs and on fields. |
288 A homeomorphism acts in an obvious way on blobs and on fields. |
202 |
289 |
203 At this point, it is time to pay back our debt and define certain notions more carefully. |
290 |
204 |
291 \nn{end relocated informal def} |
205 In order to precisely state the general definition, we'll need a suitable notion of cutting up a manifold into balls. |
292 |
206 \begin{defn} |
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207 \label{defn:gluing-decomposition} |
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208 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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209 \end{defn} |
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210 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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211 |
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212 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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213 |
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214 \begin{defn} |
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215 \label{defn:ball-decomposition} |
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216 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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217 \end{defn} |
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218 In particular, the union of all the balls in a ball decomposition comprises all of $X$. \todo{example} |
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219 |
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220 We'll now slightly restrict the possible configurations of blobs. |
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221 \begin{defn} |
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222 \label{defn:configuration} |
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223 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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224 \end{defn} |
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225 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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226 |
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227 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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228 |
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229 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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230 \begin{defn} |
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231 \label{defn:blob-diagram} |
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232 A $k$-blob diagram on $X$ consists of |
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233 \begin{itemize} |
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234 \item a configuration of $k$ blobs in $X$, |
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235 \item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
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236 \end{itemize} |
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237 such that |
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238 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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239 \end{defn} |
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240 and |
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241 \begin{defn} |
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242 \label{defn:blobs} |
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243 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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244 \end{defn} |
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245 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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246 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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247 |
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248 \todo{this notion of configuration of blobs is the minimal one that allows gluing and engulfing} |
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249 |
293 |
250 |
294 |
251 |
295 |
252 |
296 |
253 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
297 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |