1022 \coprod_a X_a \to W, |
1022 \coprod_a X_a \to W, |
1023 \] |
1023 \] |
1024 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
1024 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
1025 Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
1025 Roughly, a permissible decomposition is like a ball decomposition where we don't care in which order the balls |
1026 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
1026 are glued up to yield $W$, so long as there is some (non-pathological) way to glue them. |
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1027 |
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1028 (Every smooth or PL manifold has a ball decomposition, but certain topological manifolds (e.g.\ non-smoothable |
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1029 topological 4-manifolds) do nat have ball decompositions. |
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1030 For such manifolds we have only the empty colimit.) |
1027 |
1031 |
1028 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
1032 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
1029 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
1033 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
1030 with $\du_b Y_b = M_i$ for some $i$, |
1034 with $\du_b Y_b = M_i$ for some $i$, |
1031 and with $M_0,\ldots, M_i$ each being a disjoint union of balls. |
1035 and with $M_0,\ldots, M_i$ each being a disjoint union of balls. |