175 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 $M_0$. |
175 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 $M_0$. |
176 |
176 |
177 In the example above, note that $$A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to A \cup B \cup C \cup D$$ is a ball decomposition, but other sequences of gluings starting from $A \sqcup B \sqcup C \sqcup D$have intermediate steps which are not manifolds. |
177 In the example above, note that $$A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to A \cup B \cup C \cup D$$ is a ball decomposition, but other sequences of gluings starting from $A \sqcup B \sqcup C \sqcup D$have intermediate steps which are not manifolds. |
178 |
178 |
179 We'll now slightly restrict the possible configurations of blobs. |
179 We'll now slightly restrict the possible configurations of blobs. |
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180 %%%%% oops -- I missed the similar discussion after the definition |
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181 %The basic idea is that each blob in a configuration |
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182 %is the image a ball, with embedded interior and possibly glued-up boundary; |
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183 %distinct blobs should either have disjoint interiors or be nested; |
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184 %and the entire configuration should be compatible with some gluing decomposition of $X$. |
180 \begin{defn} |
185 \begin{defn} |
181 \label{defn:configuration} |
186 \label{defn:configuration} |
182 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} |
187 A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ |
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188 of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and |
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189 for each subset $B_i$ there is some $0 \leq r \leq m$ and some connected component $M_r'$ of |
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190 $M_r$ which is a ball, so $B_i$ is the image of $M_r'$ in $X$. |
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191 We say that such a gluing decomposition |
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192 is \emph{compatible} with the configuration. |
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193 A blob $B_i$ is a twig blob if no other blob $B_j$ is a strict subset of it. |
183 \end{defn} |
194 \end{defn} |
184 In particular, this implies what we said about blobs above: |
195 In particular, this implies what we said about blobs above: |
185 that for any two blobs in a configuration of blobs in $X$, |
196 that for any two blobs in a configuration of blobs in $X$, |
186 they either have disjoint interiors, or one blob is contained in the other. |
197 they either have disjoint interiors, or one blob is contained in the other. |
187 We describe these as disjoint blobs and nested blobs. |
198 We describe these as disjoint blobs and nested blobs. |