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178 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
178 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
179 $M_0 \to M_1 \to \cdots \to M_m = X$ such that each $M_k$ is obtained from $M_{k-1}$ |
179 $M_0 \to M_1 \to \cdots \to M_m = X$ such that each $M_k$ is obtained from $M_{k-1}$ |
180 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
180 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
181 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
181 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
182 \end{defn} |
182 \end{defn} |
183 Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is |
183 |
184 splittable along it if it is the image of a field on $M_0$. |
184 Let $M_0 \to M_1 \to \cdots \to M_m = X$ be a gluing decomposition of $X$, |
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185 and let $M_0^0,\ldots,M_0^k$ be the connected components of $M_0$. |
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186 We say that a field |
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187 $a\in \cF(X)$ is splittable along the decomposition if $a$ is the image |
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188 under gluing and disjoint union of a fields $a_i \in \cF(M_0^i)$, $0\le i\le k$. |
185 |
189 |
186 In the example above, note that |
190 In the example above, note that |
187 \[ |
191 \[ |
188 A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to A \cup B \cup C \cup D |
192 A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to A \cup B \cup C \cup D |
189 \] |
193 \] |
198 %and the entire configuration should be compatible with some gluing decomposition of $X$. |
202 %and the entire configuration should be compatible with some gluing decomposition of $X$. |
199 \begin{defn} |
203 \begin{defn} |
200 \label{defn:configuration} |
204 \label{defn:configuration} |
201 A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ |
205 A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ |
202 of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and |
206 of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and |
203 for each subset $B_i$ there is some $0 \leq r \leq m$ and some connected component $M_r'$ of |
207 for each subset $B_i$ there is some $0 \leq l \leq m$ and some connected component $M_l'$ of |
204 $M_r$ which is a ball, so $B_i$ is the image of $M_r'$ in $X$. |
208 $M_l$ which is a ball, so $B_i$ is the image of $M_l'$ in $X$. |
205 We say that such a gluing decomposition |
209 We say that such a gluing decomposition |
206 is \emph{compatible} with the configuration. |
210 is \emph{compatible} with the configuration. |
207 A blob $B_i$ is a twig blob if no other blob $B_j$ is a strict subset of it. |
211 A blob $B_i$ is a twig blob if no other blob $B_j$ is a strict subset of it. |
208 \end{defn} |
212 \end{defn} |
209 In particular, this implies what we said about blobs above: |
213 In particular, this implies what we said about blobs above: |
211 they either have disjoint interiors, or one blob is contained in the other. |
215 they either have disjoint interiors, or one blob is contained in the other. |
212 We describe these as disjoint blobs and nested blobs. |
216 We describe these as disjoint blobs and nested blobs. |
213 Note that nested blobs may have boundaries that overlap, or indeed coincide. |
217 Note that nested blobs may have boundaries that overlap, or indeed coincide. |
214 Blobs may meet the boundary of $X$. |
218 Blobs may meet the boundary of $X$. |
215 Further, note that blobs need not actually be embedded balls in $X$, since parts of the |
219 Further, note that blobs need not actually be embedded balls in $X$, since parts of the |
216 boundary of the ball $M_r'$ may have been glued together. |
220 boundary of the ball $M_l'$ may have been glued together. |
217 |
221 |
218 Note that often the gluing decomposition for a configuration of blobs may just be the trivial one: |
222 Note that often the gluing decomposition for a configuration of blobs may just be the trivial one: |
219 if the boundaries of all the blobs cut $X$ into pieces which are all manifolds, |
223 if the boundaries of all the blobs cut $X$ into pieces which are all manifolds, |
220 we can just take $M_0$ to be these pieces, and $M_1 = X$. |
224 we can just take $M_0$ to be these pieces, and $M_1 = X$. |
221 |
225 |
233 \end{itemize} |
237 \end{itemize} |
234 such that |
238 such that |
235 the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace |
239 the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace |
236 $U(B_i) \subset \cF(B_i)$. |
240 $U(B_i) \subset \cF(B_i)$. |
237 (See Figure \ref{blobkdiagram}.) |
241 (See Figure \ref{blobkdiagram}.) |
238 More precisely, each twig blob $B_i$ is the image of some ball $M_r'$ as above, |
242 More precisely, each twig blob $B_i$ is the image of some ball $M_l'$ as above, |
239 and it is really the restriction to $M_r'$ that must lie in the subspace $U(M_r')$. |
243 and it is really the restriction to $M_l'$ that must lie in the subspace $U(M_l')$. |
240 \end{defn} |
244 \end{defn} |
241 \begin{figure}[t]\begin{equation*} |
245 \begin{figure}[t]\begin{equation*} |
242 \mathfig{.7}{definition/k-blobs} |
246 \mathfig{.7}{definition/k-blobs} |
243 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
247 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
244 and |
248 and |