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522 For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we |
522 For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we |
523 topologize the set of submanifolds by ambient isotopy rel boundary. |
523 topologize the set of submanifolds by ambient isotopy rel boundary. |
524 |
524 |
525 \subsection{The blob complex} |
525 \subsection{The blob complex} |
526 \subsubsection{Decompositions of manifolds} |
526 \subsubsection{Decompositions of manifolds} |
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527 Our description of an $n$-category associates data to each $k$-ball for $k\leq n$. In order to define invariants of $n$-manifolds, we will need a class of decompositions of manifolds into balls. We present one choice here, but alternatives of varying degrees of generality exist, for example handle decompositions or piecewise-linear CW-complexes \cite{1009.4227}. |
527 |
528 |
528 A \emph{ball decomposition} of a $k$-manifold $W$ is a |
529 A \emph{ball decomposition} of a $k$-manifold $W$ is a |
529 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
530 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
530 $\du_a X_a$ and each $M_i$ is a manifold. |
531 $\du_a X_a$ and each $M_i$ is a manifold. |
531 If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
532 If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
533 \[ |
534 \[ |
534 \coprod_a X_a \to W, |
535 \coprod_a X_a \to W, |
535 \] |
536 \] |
536 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
537 which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
537 A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls |
538 A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls |
538 are glued up to yield $W$, and just require that there is some non-pathological way to do this. |
539 are glued up to yield $W$, and just require that there is some non-pathological way to do this. |
539 |
540 |
540 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
541 Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
541 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
542 of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
542 with $\du_b Y_b = M_i$ for some $i$. |
543 with $\du_b Y_b = M_i$ for some $i$. |
543 |
544 |