equal
deleted
inserted
replaced
741 A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls |
741 A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls |
742 are glued up to yield $W$, and just require that there is some non-pathological way to do this. |
742 are glued up to yield $W$, and just require that there is some non-pathological way to do this. |
743 |
743 |
744 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
744 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
745 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
745 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
746 with $\du_b Y_b = M_i$ for some $i$. |
746 with $\du_b Y_b = M_i$ for some $i$, and each $M_j$ with $j<i$ is also a disjoint union of balls. |
747 |
747 |
748 \begin{defn} |
748 \begin{defn} |
749 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
749 The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
750 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
750 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
751 See Figure \ref{partofJfig} for an example. |
751 See Figure \ref{partofJfig} for an example. |