163 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
163 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
164 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
164 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
165 \end{defn} |
165 \end{defn} |
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 $M_0$. |
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 $M_0$. |
167 |
167 |
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. |
168 We'll now slightly restrict the possible configurations of blobs. |
179 \begin{defn} |
169 \begin{defn} |
180 \label{defn:configuration} |
170 \label{defn:configuration} |
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. |
171 A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and for each subset $B_i$ there is some $0 \leq r \leq m$ and some connected component $M_r'$ of $M_r$ which is a ball, so $B_i$ is the image of $M_r'$ in $X$. Such a gluing decomposition is \emph{compatible} with the configuration. A blob $B_i$ is a twig blob if no other blob $B_j$ maps into the appropriate $M_r'$. \nn{that's a really clumsy way to say it, but I struggled to say it nicely and still allow boundaries to intersect -S} |
182 \end{defn} |
172 \end{defn} |
183 In particular, this implies what we said about blobs above: |
173 In particular, this implies what we said about blobs above: |
184 that for any two blobs in a configuration of blobs in $X$, |
174 that for any two blobs in a configuration of blobs in $X$, |
185 they either have disjoint interiors, or one blob is contained in the other. |
175 they either have disjoint interiors, or one blob is contained in the other. |
186 We describe these as disjoint blobs and nested blobs. |
176 We describe these as disjoint blobs and nested blobs. |
187 Note that nested blobs may have boundaries that overlap, or indeed coincide. |
177 Note that nested blobs may have boundaries that overlap, or indeed coincide. |
188 Blobs may meet the boundary of $X$. |
178 Blobs may meet the boundary of $X$. |
189 |
179 Further, note that blobs need not actually be embedded balls in $X$, since parts of the boundary of the ball $M_r'$ may have been glued together. |
190 % (already said above) |
180 |
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} |
181 \todo{Say something reassuring: that 'most of the time' all the regions are manifolds anyway, and you can take the `trivial' gluing decomposition} |
192 |
182 |
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 |
183 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 |
194 \begin{defn} |
184 \begin{defn} |
195 \label{defn:blob-diagram} |
185 \label{defn:blob-diagram} |
196 A $k$-blob diagram on $X$ consists of |
186 A $k$-blob diagram on $X$ consists of |
197 \begin{itemize} |
187 \begin{itemize} |
198 \item a configuration of $k$ blobs in $X$, |
188 \item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$, |
199 \item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
189 \item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
200 \end{itemize} |
190 \end{itemize} |
201 such that |
191 such that |
202 the restriction of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$. |
192 the restriction of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$. |
203 \end{defn} |
193 \end{defn} |
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194 \todo{Careful here: twig blobs aren't necessarily balls?} |
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195 (See Figure \ref{blobkdiagram}. \todo{update diagram}) |
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196 \begin{figure}[t]\begin{equation*} |
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197 \mathfig{.7}{definition/k-blobs} |
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198 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
204 and |
199 and |
205 \begin{defn} |
200 \begin{defn} |
206 \label{defn:blobs} |
201 \label{defn:blobs} |
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 $r$. |
202 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 blob from the configuration, preserving the field $r$: |
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203 \begin{equation*} |
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204 \bdy(\{B_1, \ldots B_k\}, r) = \sum_{i=1}^{k} (-1)^{i+1} (\{B_1, \ldots, \widehat{B_i}, \ldots, B_k\}, r) |
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205 \end{equation*} |
208 \end{defn} |
206 \end{defn} |
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. |
207 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. |
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. |
208 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. |
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. |
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223 A $k$-blob diagram consists of |
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224 \begin{itemize} |
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225 \item A collection of blobs $B_i \sub X$, $i = 1, \ldots, k$. |
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226 For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
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227 $B_i \sub B_j$ or $B_j \sub B_i$. |
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228 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
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229 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
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230 (These are implied by the data in the next bullets, so we usually |
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231 suppress them from the notation.) |
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232 The fields $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
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233 if the latter space is not empty. |
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234 \item A field $r \in \cC(X \setmin B^t; c^t)$, |
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235 where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
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236 is determined by the $c_i$'s. |
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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 $X \setmin B^t$.) |
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239 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$. |
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240 If $B_i = B_j$ then $u_i = u_j$. |
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241 \end{itemize} |
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242 (See Figure \ref{blobkdiagram}.) |
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243 \begin{figure}[t]\begin{equation*} |
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244 \mathfig{.7}{definition/k-blobs} |
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245 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
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246 |
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247 If two blob diagrams $D_1$ and $D_2$ |
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248 differ only by a reordering of the blobs, then we identify |
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249 $D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
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250 |
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251 Roughly, then, $\bc_k(X)$ is all finite linear combinations of $k$-blob diagrams. |
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252 As before, the official definition is in terms of direct sums |
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253 of tensor products: |
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254 \[ |
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255 \bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
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256 \left( \bigotimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
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257 \] |
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258 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
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259 The index $\overline{c}$ runs over all boundary conditions, |
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260 again as described above and $j$ runs over all indices of twig blobs. |
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261 The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are |
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262 splittable along all of the blobs in $\overline{B}$. |
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263 |
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264 The boundary map |
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265 \[ |
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266 \bd : \bc_k(X) \to \bc_{k-1}(X) |
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267 \] |
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268 is defined as follows. |
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269 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
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270 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
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271 If $B_j$ is not a twig blob, this involves only decrementing |
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272 the indices of blobs $B_{j+1},\ldots,B_{k}$. |
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273 If $B_j$ is a twig blob, we have to assign new local relation labels |
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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?} |
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277 If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
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278 where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
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279 Finally, define |
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280 \eq{ |
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281 \bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). |
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282 } |
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283 The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. |
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284 Thus we have a chain complex. |
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285 |
209 |
286 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, |
210 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, |
287 is immediately obvious from the definition. |
211 is immediately obvious from the definition. |
288 A homeomorphism acts in an obvious way on blobs and on fields. |
212 A homeomorphism acts in an obvious way on blobs and on fields. |
289 |
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290 |
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291 \nn{end relocated informal def} |
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292 |
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293 |
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294 |
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295 |
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296 |
213 |
297 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
214 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
298 to be the union of the blobs of $b$. |
215 to be the union of the blobs of $b$. |
299 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
216 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
300 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
217 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |