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. |
1027 |
1027 |
1028 (Every smooth or PL manifold has a ball decomposition, but certain topological manifolds (e.g.\ non-smoothable |
1028 (Every smooth or PL manifold has a ball decomposition, but certain topological manifolds (e.g.\ non-smoothable |
1029 topological 4-manifolds) do nat have ball decompositions. |
1029 topological 4-manifolds) do not have ball decompositions. |
1030 For such manifolds we have only the empty colimit.) |
1030 For such manifolds we have only the empty colimit.) |
|
1031 |
|
1032 We want the category (poset) of decompositions of $W$ to be small, so when we say decomposition we really |
|
1033 mean isomorphism class of decomposition. |
|
1034 Isomorphisms are defined in the obvious way: a collection of homeomorphisms $M_i\to M_i'$ which commute |
|
1035 with the gluing maps $M_i\to M_{i+1}$ and $M'_i\to M'_{i+1}$. |
1031 |
1036 |
1032 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
1037 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
1033 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
1038 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
1034 with $\du_b Y_b = M_i$ for some $i$, |
1039 with $\du_b Y_b = M_i$ for some $i$, |
1035 and with $M_0,\ldots, M_i$ each being a disjoint union of balls. |
1040 and with $M_0,\ldots, M_i$ each being a disjoint union of balls. |
1187 \begin{proof} |
1192 \begin{proof} |
1188 $\cl{\cC}(W)$ is a colimit of a diagram of sets, and each of the arrows in the diagram is |
1193 $\cl{\cC}(W)$ is a colimit of a diagram of sets, and each of the arrows in the diagram is |
1189 injective. |
1194 injective. |
1190 Concretely, the colimit is the disjoint union of the sets (one for each decomposition of $W$), |
1195 Concretely, the colimit is the disjoint union of the sets (one for each decomposition of $W$), |
1191 modulo the relation which identifies the domain of each of the injective maps |
1196 modulo the relation which identifies the domain of each of the injective maps |
1192 with it's image. |
1197 with its image. |
1193 |
1198 |
1194 To save ink and electrons we will simplify notation and write $\psi(x)$ for $\psi_{\cC;W}(x)$. |
1199 To save ink and electrons we will simplify notation and write $\psi(x)$ for $\psi_{\cC;W}(x)$. |
1195 |
1200 |
1196 Suppose $a, \hat{a}\in \psi(x)$ have the same image in $\cl{\cC}(W)$ but $a\ne \hat{a}$. |
1201 Suppose $a, \hat{a}\in \psi(x)$ have the same image in $\cl{\cC}(W)$ but $a\ne \hat{a}$. |
1197 Then there exist |
1202 Then there exist |