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460 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
460 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
461 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
461 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
462 Boundary restrictions and gluing are again straightforward to define. |
462 Boundary restrictions and gluing are again straightforward to define. |
463 Define product morphisms via product cell decompositions. |
463 Define product morphisms via product cell decompositions. |
464 |
464 |
465 |
465 \subsection{Example (bordism)} |
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466 When $X$ is a $k$-ball with $k<n$, $\Bord^n(X)$ is the set of all $k$-dimensional |
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467 submanifolds $W$ in $X\times \bbR^\infty$ which project to $X$ transversely |
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468 to $\bd X$. |
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469 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes rel boundary of such $n$-dimensional submanifolds. |
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470 |
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471 There is an $A_\infty$ analogue enriched in topological spaces, where at the top level we take all such submanifolds, rather than homeomorphism classes. For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we can topologize the set of submanifolds by ambient isotopy rel boundary. |
466 |
472 |
467 \subsection{The blob complex} |
473 \subsection{The blob complex} |
468 \subsubsection{Decompositions of manifolds} |
474 \subsubsection{Decompositions of manifolds} |
469 |
475 |
470 A \emph{ball decomposition} of $W$ is a |
476 A \emph{ball decomposition} of $W$ is a |