172 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
172 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
173 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
173 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
174 \end{defn} |
174 \end{defn} |
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 |
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178 \[ |
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179 A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to A \cup B \cup C \cup D |
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180 \] |
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181 is a ball decomposition, but other sequences of gluings starting from $A \sqcup B \sqcup C \sqcup D$ |
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182 have intermediate steps which are not manifolds. |
178 |
183 |
179 We'll now slightly restrict the possible configurations of blobs. |
184 We'll now slightly restrict the possible configurations of blobs. |
180 %%%%% oops -- I missed the similar discussion after the definition |
185 %%%%% oops -- I missed the similar discussion after the definition |
181 %The basic idea is that each blob in a configuration |
186 %The basic idea is that each blob in a configuration |
182 %is the image a ball, with embedded interior and possibly glued-up boundary; |
187 %is the image a ball, with embedded interior and possibly glued-up boundary; |